(i) (CUD-E) specified using set builder notation:
(CUD-E) = {x ∈ R | (x > 0 ∧ x ≤ 10) ∨ (x > 9 ∧ x ≤ 15) ∧ x ∉ {1, 2, 3}}
(ii) (CE) specified using interval notation and set operations concisely:
(CE) = (0, 10] ∩ {1, 2, 3} = {1, 2, 3}
(iii) (CDF) specified using the most concise notation:
(CDF) = (C ∩ D) ∩ F
(b) Proof using the element argument method:
Given: A and B are sets such that P(A) ≤ P(B).
To prove: A ≤ B.
Proof:
1. Let x be an arbitrary element in A.
2. Since x is in A, by definition, x is a subset of A. Hence, x ⊆ A.
3. Since x ⊆ A and A ≤ B, by the definition of ≤, x ⊆ B.
4. Therefore, x is a subset of B. Hence, x ∈ P(B), where P(B) is the power set of B.
5. Since x ∈ P(B), by definition, x is a subset of B. Hence, x ⊆ B.
6. Since x is an arbitrary element in A and x ⊆ B, by definition, A ≤ B.
7. Therefore, if P(A) ≤ P(B), then A ≤ B.
In this proof, we used the fact that if x is an element of A, then x is a subset of A. Also, if x is a subset of A and A ≤ B, then x is a subset of B. These properties are based on the definitions of subsets and the order relation between sets.
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Let A and B be two disjoint events such that P(A) = 0.24 and P(B) = 0.46. What is P(A or B)?
The probability of A or B occurring is 0.7.
How we find the probability of A or B?If A and B are disjoint events, it means they cannot occur at the same time. Therefore, the probability of A or B occurring can be found by adding the probabilities of A and B:
P(A or B) = P(A) + P(B)
However, we need to be careful when adding probabilities of events. If events are not disjoint, we may need to subtract the probability of their intersection to avoid double-counting. But in this case, since A and B are disjoint, their intersection is empty, so we don't need to subtract anything.
Substituting the given values, we have:
P(A or B) = P(A) + P(B) = 0.24 + 0.46 = 0.7
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9. A 45 rpm record has a 7-inch diameter and spins at 45 revolutions per minute. A 33 rpm record has a 12-inch diameter and spins at 33 revolutions per minute. Find the difference in speeds of a point on the edge of a 33 rpm record to that of a point on the edge of a 45 rom record, in ft/sec.
A point on the edge of a 45 rpm record is moving 0.18 ft/sec faster than a point on the edge of a 33 rpm record.
To compare the speeds of the two records, we need to find the linear velocity of a point on the edge of each record. The linear velocity is the distance traveled by a point on the edge of the record in a given time.
For the 45 rpm record, the diameter is 7 inches, which means the radius is 3.5 inches (7/2). The circumference of the record is then 2πr = 2π(3.5) = 22 inches. To convert this to feet, we divide by 12 to get 1.83 feet.
The linear velocity of a point on the edge of the 45 rpm record is then:
V45 = 1.83 ft/circumference x 45 rev/min x 1 min/60 sec = 1.91 ft/sec
For the 33 rpm record, the diameter is 12 inches, which means the radius is 6 inches (12/2). The circumference of the record is then 2πr = 2π(6) = 37.7 inches. To convert this to feet, we divide by 12 to get 3.14 feet.
The linear velocity of a point on the edge of the 33 rpm record is then:
V33 = 3.14 ft/circumference x 33 rev/min x 1 min/60 sec = 1.73 ft/sec
The difference in speeds between the two records is then:
V45 - V33 = 1.91 ft/sec - 1.73 ft/sec = 0.18 ft/sec
Therefore, a point on the edge of a 45 rpm record is moving 0.18 ft/sec faster than a point on the edge of a 33 rpm record.
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Jeff and Jason spend X dollars to purchase an annuity. Jeff buys a perpetuity- immediate, which makes annual payments of 30. Jason buys a 10-year annuity immediate, also with annual payments. The first payment is 53, with each subsequent payment K% larger than the previous year's payment. Both annuities use an annual effective interest rate of K%. Calculate K
The value of K is: K = 6%.
Here, we have,
To calculate the value of K,
we can set up an equation based on the information given.
For Jeff's perpetuity-immediate, the annual payment is $30.
For Jason's 10-year annuity immediate, the first payment is $53, and each subsequent payment is K% larger than the previous year's payment.
so, we get,
PV = 30/K
now, we get,
Let's denote the common ratio for Jason's annuity as r.
Therefore, the second payment would be (1 + r) times the first payment, the third payment would be (1 + r) times the second payment, and so on.
We can set up the equation:
53 + 53(1 + r) + 53(1 + r)(1 + r) + ... + 53(1 + r)⁹ = 30 + 30 + 30 + ...
To simplify the equation, we can use the formula for the sum of a geometric series:
Sum = a(1 - rⁿ) / (1 - r)
Here, a is the first term (53), r is the common ratio (1 + K/100), and n is the number of terms (10).
Using this formula, the equation becomes:
53(1 - (1 + r)¹⁰) / (1 - (1 + r)) = 30(1 - r) / (1 - 1)
again, we have,
final payment = 53(1+K%)⁹
so, total payment = 53 [(1+K%)⁹-1}/(1+K%)
so, PV = 53*10*1/(1+K)
so we have,
53*10*1/(1+K) = 30/K
=> 53 = 3 + 3K
=> K = 3/50
=> K = 6%
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Given the function w(x) = 9x + 8, evaluate w(5).
a.53
b.28
c.96
d.12
what is the conditional probability that the second card is a king given that the firstcard is a diamond?
The conditional probability that the second card is a king given that the first card is a diamond is 4/51.
What is probability?
Probability is a measure of the likelihood or chance of an event occurring. It is a number between 0 and 1, with 0 representing an impossible event and 1 representing a certain event. The probability of an event is calculated by dividing the number of ways the event can occur by the total number of possible outcomes.
To find the conditional probability that the second card is a king given that the first card is a diamond, we need to use Bayes' theorem.
Let A be the event that the first card is a diamond, and let B be the event that the second card is a king. We want to find P(B|A), the probability that B occurs given that A has occurred.
Bayes' theorem states:
P(B|A) = P(A|B) * P(B) / P(A)
We know that the probability of drawing a king from a standard deck of cards is 4/52, or 1/13 (since there are 4 kings in a deck of 52 cards). So, P(B) = 1/13.
To find P(A), the probability that the first card is a diamond, we note that there are 13 diamonds in a deck of 52 cards, so P(A) = 13/52 = 1/4.
To find P(A|B), the probability that the first card is a diamond given that the second card is a king, we note that if the second card is a king, then the first card could be any of the remaining 51 cards, of which 13 are diamonds. So, P(A|B) = 13/51.
Putting it all together, we have:
P(B|A) = P(A|B) * P(B) / P(A)
= (13/51) * (1/13) / (1/4)
= 4/51
Therefore, the conditional probability that the second card is a king given that the first card is a diamond is 4/51.
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Need help quickly this problem. Very important.
Answer:
x=-1
y=-10
x=-2
y=0
Step-by-step explanation:
we pit why into the first formula, wimplifying it and using the quadratic formula to get the two values for x, we then put the two values into x into the value for y
Brady has $20,000 in student loans with 3.3% interest that he plans to pay off in 5 years. Find the total cost of repayment.
The total cost of repayment over 5 years is $23,300.
What is the total cost of repayment?A loan repayment refers to the act of paying back money previously borrowed from a lender.
To get total cost of repayment, we must principal amount, the interest rate and the duration of the loan.
The formula to get total cost of repayment is given by [tex]Total Cost of Repayment = Principal + Interest[/tex]
Interest = Principal * Interest Rate * Time
Given:
Principal amount is $20,000
Interest rate is 3.3%
Duration is 5 years.
Interest = $20,000 * 0.033 * 5
Interest = $3,300
Total Cost of Repayment = Principal + Interest
= $20,000 + $3,300
= $23,300.
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Which of the following gap penalty functions represent affine gap penalties (k represents the number of gaps in a row) a. Cost (k) = a k2
b. Cost (k) = a k + b c. Cost(k) = log(k) + b
The correct answer is b. Cost(k) = a k + b.
Affine gap penalties in sequence alignment involve a linear function that considers the number of gaps in a row. The function typically includes two components: a linear term to represent the initial gap and an additional linear term to account for each consecutive gap.
In option a, the cost function includes a quadratic term (k^2), which does not represent a linear affine penalty.
In option c, the cost function includes a logarithmic term (log(k)), which also does not represent a linear affine penalty.
Option b, with the cost function of a k + b, correctly represents an affine gap penalty, as it includes a linear term (a k) to account for the number of gaps in a row.
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Calculate the area of each circle.
grade V lvl
R-radius and D-diameter
1.R-8yd
2.D-18in
3.R-10yd
4.R-6in
5.R-4ft
6.D-10ft
7.R-1yd
8.D-14in
9.R-2yd
[tex]{ \pmb{ \hookrightarrow}} \: \underline{\boxed{\pmb{\sf{Area_{(Circle)} \: = \: \pi \: {r}^{2} }}}} \: \pmb{\red{\bigstar}} \\ [/tex]
_________________________________________________
1) Radius = 8 yd[tex]\longrightarrow \sf \: Area_{(Circle)} \: = \: \pi \: {r}^{2} [/tex]
[tex]\longrightarrow \sf \: Area_{(Circle)} \: = \: \frac{22} {7} \: \times {8}^{2} \\ [/tex]
[tex]\longrightarrow \sf \: Area_{(Circle)} \: = \: \frac{22} {7} \: \times 8 \times 8 \\ [/tex]
[tex]\longrightarrow \sf \: Area_{(Circle)} \: = \: \frac{22} {7} \: \times 64 \\ [/tex]
[tex]\longrightarrow \sf \: Area_{(Circle)} \: = \: \frac{1408} {7} \: \: {yard}^{2} \: \\ [/tex]
_________________________________________________
2) Diameter = 18 inch→ Radius = 18/2 = 9 inch
[tex]\longrightarrow \sf \: Area_{(Circle)} \: = \: \pi \: {r}^{2} [/tex]
[tex]\longrightarrow \sf \: Area_{(Circle)} \: = \: \frac{22} {7} \: \times {9}^{2} \\ [/tex]
[tex]\longrightarrow \sf \: Area_{(Circle)} \: = \: \frac{22} {7} \: \times 9 \times 9 \\ [/tex]
[tex]\longrightarrow \sf \: Area_{(Circle)} \: = \: \frac{22} {7} \: \times 81 \\ [/tex]
[tex]\longrightarrow \sf \: Area_{(Circle)} \: = \: \frac{1782} {7} \: \: {inch}^{2} \: \\ [/tex]
_________________________________________________
3) Radius = 10 yd[tex]\longrightarrow \sf \: Area_{(Circle)} \: = \: \pi \: {r}^{2} [/tex]
[tex]\longrightarrow \sf \: Area_{(Circle)} \: = \: \frac{22} {7} \: \times {10}^{2} \\ [/tex]
[tex]\longrightarrow \sf \: Area_{(Circle)} \: = \: \frac{22} {7} \: \times 10 \times 10 \\ [/tex]
[tex]\longrightarrow \sf \: Area_{(Circle)} \: = \: \frac{22} {7} \: \times 100 \\ [/tex]
[tex]\longrightarrow \sf \: Area_{(Circle)} \: = \: \frac{2200} {7} \: \: {yard}^{2} \: \\ [/tex]
_________________________________________________
4) Radius = 6 inch[tex]\longrightarrow \sf \: Area_{(Circle)} \: = \: \pi \: {r}^{2} [/tex]
[tex]\longrightarrow \sf \: Area_{(Circle)} \: = \: \frac{22} {7} \: \times {6}^{2} \\ [/tex]
[tex]\longrightarrow \sf \: Area_{(Circle)} \: = \: \frac{22} {7} \: \times 6 \times 6 \\ [/tex]
[tex]\longrightarrow \sf \: Area_{(Circle)} \: = \: \frac{22} {7} \: \times 36 \\ [/tex]
[tex]\longrightarrow \sf \: Area_{(Circle)} \: = \: \frac{792} {7} \: \: {inch}^{2} \: \\ [/tex]
_________________________________________________
5) Radius = 4 ft[tex]\longrightarrow \sf \: Area_{(Circle)} \: = \: \pi \: {r}^{2} [/tex]
[tex]\longrightarrow \sf \: Area_{(Circle)} \: = \: \frac{22} {7} \: \times {4}^{2} \\ [/tex]
[tex]\longrightarrow \sf \: Area_{(Circle)} \: = \: \frac{22} {7} \: \times 4 \times 4 \\ [/tex]
[tex]\longrightarrow \sf \: Area_{(Circle)} \: = \: \frac{22} {7} \: \times 16 \\ [/tex]
[tex]\longrightarrow \sf \: Area_{(Circle)} \: = \: \frac{352} {7} \: \: {ft}^{2} \: \\ [/tex]
_________________________________________________
6) Diameter= 10 ft→ Radius = 10/2 = 5 ft
[tex]\longrightarrow \sf \: Area_{(Circle)} \: = \: \pi \: {r}^{2} [/tex]
[tex]\longrightarrow \sf \: Area_{(Circle)} \: = \: \frac{22} {7} \: \times {5}^{2} \\ [/tex]
[tex]\longrightarrow \sf \: Area_{(Circle)} \: = \: \frac{22} {7} \: \times 5 \times 5 \\ [/tex]
[tex]\longrightarrow \sf \: Area_{(Circle)} \: = \: \frac{22} {7} \: \times 25 \\ [/tex]
[tex]\longrightarrow \sf \: Area_{(Circle)} \: = \: \frac{550} {7} \: \: {ft}^{2} \: \\ [/tex]
_________________________________________________
7) Radius = 1 yd[tex]\longrightarrow \sf \: Area_{(Circle)} \: = \: \pi \: {r}^{2} [/tex]
[tex]\longrightarrow \sf \: Area_{(Circle)} \: = \: \frac{22} {7} \: \times {1}^{2} \\ [/tex]
[tex]\longrightarrow \sf \: Area_{(Circle)} \: = \: \frac{22} {7} \: \times 1 \times 1 \\ [/tex]
[tex]\longrightarrow \sf \: Area_{(Circle)} \: = \: \frac{22} {7} \: \times 1 \\ [/tex]
[tex]\longrightarrow \sf \: Area_{(Circle)} \: = \: \frac{22} {7} \: \: {yard}^{2} \: \\ [/tex]
_________________________________________________
8) Diameter= 14 inch→ Radius = 14/2 = 7 inch
[tex]\longrightarrow \sf \: Area_{(Circle)} \: = \: \pi \: {r}^{2} [/tex]
[tex]\longrightarrow \sf \: Area_{(Circle)} \: = \: \frac{22} {7} \: \times {7}^{2} \\ [/tex]
[tex]\longrightarrow \sf \: Area_{(Circle)} \: = \: \frac{22} {7} \: \times 7 \times 7 \\ [/tex]
[tex]\longrightarrow \sf \: Area_{(Circle)} \: = \: \frac{22} {7} \: \times 49 \\ [/tex]
[tex]\longrightarrow \sf \: Area_{(Circle)} \: = \: \frac{1078} {7} \: \: \: \\ [/tex]
[tex]\longrightarrow \sf \: Area_{(Circle)} \: = \: 154 \: \: {inch}^{2} \: \\ [/tex]
_________________________________________________
9) Radius = 2 yd[tex]\longrightarrow \sf \: Area_{(Circle)} \: = \: \pi \: {r}^{2} [/tex]
[tex]\longrightarrow \sf \: Area_{(Circle)} \: = \: \frac{22} {7} \: \times {2}^{2} \\ [/tex]
[tex]\longrightarrow \sf \: Area_{(Circle)} \: = \: \frac{22} {7} \: \times 2 \times 2 \\ [/tex]
[tex]\longrightarrow \sf \: Area_{(Circle)} \: = \: \frac{22} {7} \: \times 4 \\ [/tex]
[tex]\longrightarrow \sf \: Area_{(Circle)} \: = \: \frac{88} {7} \: \: {yard}^{2} \: \\ [/tex]
_________________________________________________
Find the area of the surface given parametrically by r(s, t) = (t sinh(s), tcosh(s), t), -2 < s < 2, 0 0 for all s. sinh?(s) = 1 and
The area of the surface is 8π.
Given, r(s, t) = (t sinh(s), t cosh(s), t)
Taking partial derivative with respect to s
[tex]\frac{\hat{a}r}{\hat{a}s}[/tex] = (t cosh(s), t sinh(s), 0)
Taking partial derivative with respect to t
[tex]\frac{\hat{a}r}{\hat{a}t}[/tex] = ( sinh(s), cosh(s), 1)
The cross product of these partial derivatives is given by
[tex]|\frac{\hat{a}r}{\hat{a}s} \times\frac{\hat{a}r}{\hat{a}t} |[/tex] = | (t sinh(s), t cosh(s), t)|
= t √(sinh²(s) + cosh²(s) + 1)
= t √(cosh²(s) - 1 + 1)
= t cosh(s)
So, the area of the surface is given by the integral:
A = ∫∫[tex]|\frac{\hat{a}r}{\hat{a}s} \times\frac{\hat{a}r}{\hat{a}t} |[/tex] ds dt
= 2 Ï [tex]\hat{a_0}^{\hat{a} tcosh(s)ds}[/tex]
(Integrating over s)
= 2 Ï [tex]\hat{a_0}^{\hat{a} t(\frac{e^s+e^{-s}}{2} ) ds}[/tex]
= 2 Ï [tex]\hat{a_0}^{\hat{a} \frac{t}{2} (e^s+e^{-s}) ds}[/tex]
= 2 Ï [tex][\frac{t}{2}e^s]_0^\hat{a}[/tex]
= 2π (∞ - 0)
= 8π
Therefore, the area of the surface is 8π.
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could you help me please
Answer:
Angle PRQ = 28 degrees.
Step-by-step explanation:
Angles P and Q are the same! That's bc this is an isosceles triangle.
The total of all 3 angles = 180. So to find R, subtract the other 2 angles from 180.
So 180-76-76 = 28. That's angle R
3. A student has a rectangular block. It is 2 cm wide, 3 cm tall, and 25 cm long. It has a mass of 600 g. First, calculate the volume of the block. Then, use that answer to determine the density of the block.
Answer:
4 (g cm³)
Step-by-step explanation:
Volume of a rectangular block (prism) = length X width X height
we have 2 X 3 X 25 = 150 cm³
Formula for density is
density = mass/volume
= 600/150
= 4 (g cm³)
x^2+2x-8/x^2+3x-10 • x+5/x^2 - 16 <<< help?
perform the indicated operations. Assume that no denominator has a value of 0.
To solve the expression (x^2 + 2x - 8)/(x^2 + 3x - 10) * (x + 5)/(x^2 - 16), we can begin by factoring the quadratic expressions in the numerator and denominator of the first fraction:
(x^2 + 2x - 8)/(x^2 + 3x - 10) = ((x + 4)(x - 2))/((x + 5)(x - 2))
Similarly, we can factor the quadratic expression in the denominator of the second fraction:
(x + 5)/(x^2 - 16) = (x + 5)/((x + 4)(x - 4))
Substituting these expressions back into the original expression, we get:
((x + 4)(x - 2))/((x + 5)(x - 2)) * (x + 5)/((x + 4)(x - 4))
We can then cancel out the x - 2 and x + 4 factors in the numerator and denominator:
(x + 5)/(x - 4)
Therefore, the simplified expression is (x + 5)/(x - 4).
the correlation between the two variables of interest is 0.81, which is significant at the 0.0337 level. this means ______.
The correlation between the two variables of interest is 0.81, which is significant at the 0.0337 level. This means that there is a strong positive relationship between the variables, and the likelihood of obtaining a correlation of 0.81 or higher due to chance alone is less than 3.37%.
The correlation coefficient measures the strength and direction of the linear relationship between two variables. In this case, the correlation coefficient of 0.81 indicates a strong positive relationship between the variables. The significance level of 0.0337 suggests that the observed correlation is unlikely to occur by chance alone.
It implies that there is evidence to support the conclusion that the correlation is statistically significant and not just a result of random variation. Therefore, the correlation of 0.81 is considered meaningful and reliable in representing the relationship between the variables.
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What’s 5 times 2 plus 3 minus 100 times 542 divided 1 plus 8 plus 3000 plus 30000 plus 5000 times 2 times 2 minus 15284
The answer is −63,868. To get this just put it into a calculator.
A spinner has 4 equal-sized sections labeled A, B, C, and D. It is spun and a fair coin is tossed. What is the probability of spinning "C” and flipping "heads”?
A "heads" flip with a spinning "C" has a 0.125 or 12.5% chance of happening.
The chance of spinning "C" is 1/4, or 0.25, if the spinner is fair and has four parts of equal size.
The chance of flipping "heads" is half, or 0.5, if the coin is fair. We multiply the individual probabilities in order to get the likelihood that both occurrences will occur:
P = (Probability of spinning "C") (Probability of flipping "heads")
P = 0.25 × 0.5
P = 0.125
The probability is 0.125 as a result.
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please help me it is diffiult
Answer:
Sin(C) = [tex]\frac{3}{5}[/tex].
Cos(B) = [tex]\frac{3}{5}[/tex]
Sin(C) and Cos(B) = [tex]\frac{3}{5}[/tex]
Step-by-step explanation:
To solve this we need to remember the trigonometry ratios. Sin is the side opposite the angle over the hypotenuse. Cos is the side touching the angle divided by the hypotenuse and Tan is the side opposite the angle divided by the side touching the angle.
Knowing this can help us solve our problem.
Let us first solve for sin C. To solve for this let us look for the side opposite (or directly across) of angle c. This side is side AB or 39. The next thing we need to look for is the hypotenuse. The hypotenuse is the side directly across the right angle. This side is side CB or 65.
Dividing these gives us our first fraction [tex]\frac{39}{65}[/tex]. Now we need to simplify this. 39/65 implied is [tex]\frac{3}{5}[/tex]. This is sin of C.
------------------------------------------------------------------------------------------------
To solve for cos of B we have to look for the side directly next to the angle. This is side AB or 39. The next step is to identify the hypotenuse. The hypotenuse is 65, so Cos of b is [tex]\frac{39}{65}[/tex] or [tex]\frac{3}{5}[/tex].
The volume of the right triangular prism is 19 in . If DE is equal to 16 inches and EF is equal to 24 inches, what is the length of EB
The length of EB is approximately 0.099 inches.
To solve this problem, we can use the formula for the volume of a right triangular prism, which is:
V = (1/2)bh × l
where V is the volume, b is the base of the triangle, h is the height of the triangle, and l is the length of the prism.
We are given that the volume of the prism is 19 in^3, so we can plug in that value:
19 = (1/2)(DE)(EF) × EB
Substituting the given values for DE and EF:
19 = (1/2)(16)(24) × EB
Simplifying:
19 = 192 × EB
Dividing both sides by 192:
EB = 19/192
So the length of EB is approximately 0.099 inches.
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2. An importer purchases two types of baseball helmet: standard helmets cost $80 each and deluxe helmets cost $120 each. The importer wants to spend a maximum of $4800, and because of government protection to local industry, can import no more than 50 helmets. Suppose the importer purchases x standard helmets and y deluxe helmets. List the constraints on the variables x and y.
Answer: the three constraints on the variables x and y are:
80x + 120y ≤ 4800
x + y ≤ 50
x ≥ 0, y ≥ 0
Step-by-step explanation:
Cost Constraint: The importer wants to spend a maximum of $4800, so the cost of the purchased helmets should not exceed $4800. The cost of x standard helmets and y deluxe helmets can be calculated as 80x + 120y, so the constraint can be written as:
80x + 120y ≤ 4800
Quantity Constraint: The importer cannot import more than 50 helmets in total. Therefore, the sum of standard and deluxe helmets purchased cannot exceed 50. The constraint can be written as:
x + y ≤ 50
Non-negativity Constraint: The importer cannot purchase negative helmets, so the variables x and y should be non-negative. The constraint can be written as:
x ≥ 0, y ≥ 0
Thus, the three constraints on the variables x and y are:
80x + 120y ≤ 4800
x + y ≤ 50
x ≥ 0, y ≥ 0
1. Trouve l'aire totale des cylindres suivants.
a) le rayon mesure 3 cm et la hauteur mesure 10 cm.
h
Answer:
Step-by-step explanation:
Pour trouver l'aire totale d'un cylindre, il faut ajouter l'aire de la base circulaire du cylindre à l'aire de sa surface latérale.
La formule pour l'aire de la base circulaire est:
Aire de la base = πr²
où r est le rayon du cylindre.
La formule pour l'aire de la surface latérale est:
Aire latérale = 2πrh
où r est le rayon du cylindre et h est la hauteur du cylindre.
Donc, pour le cylindre donné avec un rayon de 3 cm et une hauteur de 10 cm, l'aire de la base est:
Aire de la base = πr² = π(3²) = 9π cm²
L'aire de la surface latérale est:
Aire latérale = 2πrh = 2π(3)(10) = 60π cm²
Pour trouver l'aire totale, on ajoute l'aire de la base et l'aire de la surface latérale:
Aire totale = Aire de la base + Aire latérale = 9π + 60π = 69π cm²
Donc, l'aire totale du cylindre est 69π cm².
A study was designed to explore subjects’ ability to judge the distance between two objects placed in a
dimly lit room. The researcher suspected that the subjects would generally overestimate the distance
between the objects in the room and that this overestimation would increase the farther apart the objects
were.
The two objects were placed at random locations in the room before a subject estimated the distance (in
feet) between those two objects. After each subject estimated the distance, the locations of the objects
were randomized before the next subject viewed the room.
After data were collected for 40 subjects, two linear models were fit in an attempt to describe the
relationship between the subjects’ perceived distances (y) and the actual distance, in feet, between the two
objects.
Model 1:
y x 0.238 1.080 ( )
The standard errors of the estimated coefficients for Model 1 are 0.260 and 0.118, respectively.
Model 2:
y x 1.102
The standard error of the estimated coefficient for Model 2 is 0.393.
a) Provide an interpretation in context for the estimated slope in Model 1.
b) Explain why the researcher might prefer Model 2 to Model 1 in this context.
c) Using Model 2, test the researcher’s hypothesis that in dim light participants overestimate the distance,
with the overestimate increasing as the actual distance increases. (Assume appropriate conditions for
inference are met.)
The researchers also wanted to explore whether the performance on this task differed between subjects
who wear contact lenses and subjects who do not wear contact lenses. A new variable was created to
indicate whether or not a subject wears contact lenses. The data for this variable were coded numerically
(1 = contact wearer, 0 = noncontact wearer), and this new variable, named "contact" was included in the
following model.
Model 3:
y x contact x 1.05 0.12
The standard errors of the estimated coefficients for Model 3 are 0.357 and 0.032, respectively.
d) Using Model 3, sketch the estimated regression model for contact wearers and the estimated regression
model for noncontact wearers on the grid below.
a) The estimated slope in Model 1 (0.238) means that, on average, for each additional foot between the two objects, the subjects' perceived distance increased by 0.238 feet.
b) The researcher might prefer Model 2 because it has a simpler equation with fewer parameters, which makes it easier to interpret and apply. Additionally, the estimated slope in Model 2 (1.102) is closer to the researcher's hypothesis that subjects would generally overestimate the distance between the objects in the room and that this overestimation would increase the farther apart the objects were.
c) To test the researcher's hypothesis using Model 2, we can set up the null hypothesis as H0: β1=0 (there is no relationship between actual distance and perceived distance) and the alternative hypothesis as Ha: β1>0 (perceived distance increases as actual distance increases). Using a t-test with 38 degrees of freedom (since we estimated one parameter in the model), we find a t-value of 2.803 and a p-value of 0.008, which is less than the significance level of 0.05. Therefore, we reject the null hypothesis and conclude that there is evidence to support the researcher's hypothesis that in dim light, participants overestimate the distance, with the overestimate increasing as the actual distance increases.
d) Without information about the range of values for x and y, it's difficult to provide a precise sketch of the estimated regression models for contact wearers and noncontact wearers. However, we can say that the estimated regression line for contact wearers would have an intercept of 1.05 and a slope of x (0.238 for every non-contact wearer and 0.358 for every contact wearer), while the estimated regression line for noncontact wearers would have an intercept of 0.12 and a slope of x (0.238 for every non-contact wearer and 0.358 less for every contact wearer).
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find the first partial derivatives of the function. (sn = x1 2x2 ... xn; i = 1, ..., n. give your answer only in terms of sn and i.) u = sin(x 1 2x2 ⋯ nxn) ∂u ∂xi =
To find the partial derivative of the function u = sin(x1 2x2 ⋯ nxn) with respect to xi, where i is an integer between 1 and n, we need to use the chain rule. The answer can be expressed as follows: ∂u/∂xi = cos(x1 2x2 ⋯ nxn) * 2ixi * x1 2x2 ⋯ xi-1 2xi-1 xi+1 2xi+1 ⋯ xn.
To explain further, we start by applying the chain rule to u = sin(x1 2x2 ⋯ nxn) with respect to xi. We treat all the variables except xi as constants, so we get:
∂u/∂xi = cos(x1 2x2 ⋯ nxn) * ∂(x1 2x2 ⋯ nxn)/∂xi
Next, we use the product rule to differentiate x1 2x2 ⋯ nxn with respect to xi. We treat all the variables except xi as constants, so we get:
∂(x1 2x2 ⋯ nxn)/∂xi = 2ixi * x1 2x2 ⋯ xi-1 2xi-1 xi+1 2xi+1 ⋯ xn
Substituting this result back into our original equation, we get:
∂u/∂xi = cos(x1 2x2 ⋯ nxn) * 2ixi * x1 2x2 ⋯ xi-1 2xi-1 xi+1 2xi+1 ⋯ xn
Therefore, the partial derivative of the function u = sin(x1 2x2 ⋯ nxn) with respect to xi is cos(x1 2x2 ⋯ nxn) multiplied by 2ixi multiplied by the product of all the variables except xi in the original function.
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find the 3 × 3 matrix that produces the described transformation, using homogeneous coordinates. (x, y) → (x+7, y+4)
The transformation can be represented as:
\begin{bmatrix} x' \ y' \ w' \end{bmatrix} = \begin{bmatrix} 1 & 0 & 7 \ 0 & 1 & 4 \ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \ y \ 1 \end{bmatrix}
where (x', y') is the transformed point, and w' is the homogeneous coordinate (usually taken as 1 for 2D transformations).
In matrix form, the transformation can be written as:
\begin{bmatrix} x' \ y' \ 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 7 \ 0 & 1 & 4 \ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \ y \ 1 \end{bmatrix}
So the 3x3 matrix that produces the described transformation is:
\begin{bmatrix} 1 & 0 & 7 \ 0 & 1 & 4 \ 0 & 0 & 1 \end{bmatrix}
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find the radius of the sphere which passes through the point (−1, 4, 3) and has center (8, 1, 3).
We can use the distance formula to calculate the distance between the center and the point. This distance is equal to the radius of the sphere. the radius of the sphere that passes through the point (-1, 4, 3) and has center (8, 1, 3) is √90 units.
In this problem, the center of the sphere is given as (8, 1, 3) and the point it passes through is (-1, 4, 3). To find the radius, we need to calculate the distance between these two points.
Using the distance formula, we get:
√[(8 - (-1))^2 + (1 - 4)^2 + (3 - 3)^2] = √(81 + 9) = √90
Therefore, the radius of the sphere is √90 units.
In summary, to find the radius of a sphere that passes through a given point and has a known center, we can use the distance formula to calculate the distance between the center and the point. In this problem, the radius of the sphere that passes through the point (-1, 4, 3) and has center (8, 1, 3) is √90 units.
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Which compound equalities have x = 2 as a solution? Check all that apply.
Answer: 4 < 5x – 1 < 10 4 < 5x – 3 < 10 4 < 2x + 1 < 10 4 < 2x + 3 < 10
Use parallelogram WXYZ for questions 10 and 11. 6. If mXYZ = 68° and mWXZ = 71°, find mWZX. 7. If XZ = 8x - 18 and RZ = 2x + 5, find XR.
From the parallelogram WXYZ, the angle of mWZX will be 68° and angle of XR = XZ = 8x - 18.
Understanding ParallelogramUsing the properties of parallelograms and the given information, we can solve the parallelogram WXYZ
6. Given:
mXYZ = 68°
mWXZ = 71°
mWZX = ?
In a parallelogram, opposite angles are congruent.
Therefore, mWZX will also be 68°.
7. Given:
XZ = 8x - 18
RZ = 2x + 5,
XR = ?
In a parallelogram, opposite sides are congruent.
That is, XR = XZ
Recall that
XZ = 8x - 18
Substitute XZ with its value:
XR = XZ = 8x - 18
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Part A Given two vectors A⃗ =4.00i^+6.00j^ and B⃗ =2.00i^−7.00j^ , find the vector product A⃗ ×B (expressed in unit vectors). Part B What is the magnitude of the vector product?
a) the vector product of A and B is -12.00i^ - 22.00j^ - 24.00k^. b) the magnitude of the vector product is √(1441).
Part A:
The vector product of two vectors A and B is given by:
A × B = (A_yB_z - A_zB_y)i^ + (A_zB_x - A_xB_z)j^ + (A_xB_y - A_yB_x)k^
where A_x, A_y, A_z, B_x, B_y, and B_z are the components of vectors A and B in the x, y, and z directions.
Using the given values, we have:
A_x = 4.00, A_y = 6.00, A_z = 0
B_x = 2.00, B_y = -7.00, B_z = 0
Substituting these values into the formula, we get:
A × B = (0)(0) - (0)(0)i^ + (0)(4.00) - (2.00)(0)j^ + (4.00)(-7.00) - (6.00)(2.00)k^
Simplifying, we get:
A × B = -12.00i^ - 22.00j^ - 24.00k^
Therefore, the vector product of A and B is -12.00i^ - 22.00j^ - 24.00k^.
Part B:
The magnitude of the vector product is given by:
|A × B| = √[(A_yB_z - A_zB_y)^2 + (A_zB_x - A_xB_z)^2 + (A_xB_y - A_yB_x)^2]
Substituting the values from Part A, we get:
|A × B| = √[(-6.00)(0) + (0)(2.00) + (4.00)(-7.00 - (-6.00)(-7.00 - (-2.00)(6.00))]
Simplifying, we get:
|A × B| = √(1441)
Therefore, the magnitude of the vector product is √(1441).
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find the exact value of sin(u v) given that sinu= 3/5 and cosv=-24/25
The exact value of sin(uv) is -44/125. We can use the trigonometric identity sin(uv) = sinucosv - cosusinv to find the value of sin(uv), given sinu and cosv.
From the given information, we have sinu = 3/5 and cosv = -24/25. We can find cosu using the Pythagorean identity:
cos^2 u + sin^2 u = 1
cos^2 u = 1 - sin^2 u = 1 - (3/5)^2 = 16/25
Taking the square root of both sides, we get cos u = ±4/5. However, since sinu is positive, we can conclude that cosu is also positive, and so cosu = 4/5.
Now we can use the sin(uv) identity to find:
sin(uv) = sinucosv - cosusinv
= (3/5)(-24/25) - (4/5)sinv (substituting the given values)
= -72/125 - (4/5)sinv
To find sinv, we can use the Pythagorean identity:
sin^2 v + cos^2 v = 1
sin^2 v = 1 - cos^2 v = 1 - (-24/25)^2 = 49/625
Taking the square root of both sides, we get sinv = ±7/25. However, since cosv is negative, we can conclude that sinv is also negative, and so sinv = -7/25.
Substituting sinv into the expression we found for sin(uv), we get:
sin(uv) = -72/125 - (4/5)(-7/25) = -72/125 + 28/125
= -44/125
Therefore, the exact value of sin(uv) is -44/125.
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WILL GIVE BRAINLIEST PLS HURRY Triangle UVW has vertices at U(−1, 0), V(−4, 1), W(−4, 4). Determine the vertices of image U′V′W′, if the preimage is rotated 90° clockwise.
U′(0, −1), V′(−1, −4), W′(−4, −4)
U′(0, 1), V′(1, 4), W′(4, 4)
U′(1, 0), V′(4, −1), W′(4, −4)
U′(−1, 0), V′(−4, 0), W′(4, −4)
Question 2(Multiple Choice Worth 2 points)
(Volume of Cylinders MC)
A bakery is making cupcakes using a cylindrical mold. The cupcake mold has a diameter of 6.5 centimeters and is 4 centimeters tall. Which of the following shows a correct method to calculate the amount of cupcake batter needed to fill the mold all the way to the top? Use 3.14 for π.
V = (3.14)(6.5)2(4)
V = (3.14)(4)2(6.5)
V = (3.14)(4)2(3.25)
V = (3.14)(3.25)2(4)
Question 3(Multiple Choice Worth 2 points)
(Circumference MC)
The diameter of a child's bicycle wheel is 15 inches. Approximately how many revolutions of the wheel will it take to travel 3,000 meters? Use 3.14 for π and round to the nearest whole number. (1 meter ≈ 39.3701 inches)
4,702 revolutions
2,508 revolutions
200 revolutions
64 revolutions
Question 4(Multiple Choice Worth 2 points)
(Scale Factor MC)
An engineer has a 60:1 scale drawing of a bridge. The dimensions of the scaled bridge deck are 36 inches by four and four fifths inches. What is the area of the actual bridge deck in square feet?
6,912 square feet
4,320 square feet
576 square feet
72 square feet
find a formula for f(x) given that f is continuous and −2x5 x3 4x=∫x0f(t)dt.
The formula for f(x) is f(x) = -10x^4 + 3x^2 + 4.
How can we find the derivative of both sides of the equation?
We can find the derivative of both sides of the equation using the Fundamental Theorem of Calculus:
d/dx [−2x^5 + x^3 + 4x] = d/dx [∫_0^x f(t) dt]
-10x^4 + 3x^2 + 4 = f(x)
Therefore, the formula for f(x) is:
f(x) = -10x^4 + 3x^2 + 4
We can check that this formula satisfies the original equation by taking the definite integral:
∫_0^x f(t) dt = ∫_0^x (-10t^4 + 3t^2 + 4) dt = [-2t^5 + t^3 + 4t]_0^x = -2x^5 + x^3 + 4x
Thus, the formula for f(x) is f(x) = -10x^4 + 3x^2 + 4.
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