Therefore, the angle between V and w is approximately 75.97 degrees.
To find the angle between V and w, we can use the dot product formula:
V · w = |V| |w| cosθ
where θ is the angle between the two vectors, and |V| and |w| are the magnitudes of the vectors.
First, let's calculate the dot product:
V · w = (-5)(4) + (8)(12)
= 61
Next, let's calculate the magnitudes:
|V| = √((-5)^2 + 8^2)
= √89
|w| = √(4^2 + 12^2)
= 4√5
Now we can solve for cosθ:
cosθ = (V · w) / (|V| |w|)
= 61 / (4√5 √89)
≈ 0.2577
Finally, we can find the angle θ:
θ = cos^(-1)(0.2577)
≈ 75.97°
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A fundamental set of solutions of x' =(1 2 0, -3 -1 3, 3 2 -2)x is: (a) x1 = e^-2t(2 -3 3), X2 = e^-t(1 -1 1), X3 = e^t(1 0 1) (b) x1 = e^2t(2 -3 3), X2 = e^-t(1 1 1), X3 = e^t(1 2 1) (c) x1 = e^2t(2 3 -)3, x2 = e^-t(-1 -1 1), X3 = e^t(1 0 -1) (d) x1 = e^-2t(-2 -3 3), X2 = e^-t(1 1 -1), X3 = e^t(1 -1 1) (e) None of the above.
The fundamental set of solutions of the given system of differential equations x' =(1 2 0, -3 -1 3, 3 2 -2) is to be identified from the given options.
The correct answer is option (a) x1 = e^-2t(2 -3 3), X2 = e^-t(1 -1 1), X3 = e^t(1 0 1).
To verify this, we can calculate the Wronskian of the three solutions and show that it is non-zero, which confirms that they form a fundamental set of solutions. Another way to check is to substitute the solutions into the differential equation and verify that they satisfy it. In this case, both methods give us the same result - the solutions satisfy the differential equation and are linearly independent, hence form a fundamental set of solutions. Therefore, the correct answer is (a).
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For each of the functions below, indicate whether the function is onto, one-to-one, neither or both. If the function is not onto or not one-to-one, give an example showing why.A = {a, b, c}, h: P(A) → P(A). For X ⊆ A, h(X) = X ∪ {a}.2. Find a function whose domain is the set of all integers and whose target is the set of all positive integers that satisfies each set of properties.(a)Neither one-to-one, nor onto.(b)One-to-one, but not onto.(c)Onto, but not one-to-one.(d)One-to-one and onto.
The function is Neither one-to-one nor onto. An example of a function that is one-to-one but not onto is f(x) = x + 1, where the domain is all integers and the target is all positive integers.
The function h is neither one-to-one nor onto.
It is not one-to-one because for example, h({a}) = h({b}) since h({a}) = {a, b} and h({b}) = {a, b}.
It is not onto because {b, c} is not in the range of h since h(X) always contains a but {b, c} does not contain a.
One example of a function with the given properties is f(x) = x + 1.
It is one-to-one because for any distinct integers x and y, f(x) = x + 1 and f(y) = y + 1 are different since x and y are different.
It is not onto because the target set of f only includes positive integers, but there is no integer x such that f(x) = 1.
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A theater has 25 seats in the first row and 35 rows in all. Each successive row contains one additional seat. How many seats are in the theater?
The theater has a total of 945 seats.
To determine the number of seats in the theater, we need to calculate the sum of seats in each row. The first row has 25 seats, and each subsequent row increases by one seat. Since there are 35 rows in total, we can calculate the sum of an arithmetic series to find the total number of seats.
The formula for the sum of an arithmetic series is Sn = (n/2) * (a1 + an), where n is the number of terms, a1 is the first term, and an is the last term. In this case, n = 35 (number of rows), a1 = 25 (number of seats in the first row), and an = a1 + (n - 1) = 25 + (35 - 1) = 25 + 34 = 59 (number of seats in the last row). Plugging these values into the formula, we get Sn = (35/2) * (25 + 59) = 17.5 * 84 = 1470. Therefore, the theater has a total of 945 seats.
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If the cost of carpeting a floor is $2.50 per square foot, how much will it cost to carpet a rectangular floor that is 10 feet by 12 feet?
Answer:
$300
Step-by-step explanation:
The area of the area would be 10 x 12 = 120 square feet.
120 sq ft x 2.50 per sq ft = $300.
It would cost $300 to carpet this area.
A perfectly competitive painted necktie industry has a large number of potential entrants. Each firm has an identical cost structure such that long-run average cost is minimized at an output of 20 units (qi = 20). The minimum average cost is $10 per unit. Total market demand is given by Q = 1,500 - 50P a. What is the industry's long-run supply schedule? b. What is the long-run equilibrium price (P*)? The total industry output (Q*)? The output of each firm (q*i) ? The number of firms? The profits of each firm? c. The short-run total cost curve associated with each firm's long-run equilibrium output is given by STC = .5q2 - 10q + 200 where SMC = q- 10. Calculate the short-run average and marginal cost curves. At what necktie output level does short-run average cost reach a minimum?d. Calculate the short-run supply curve for each firm and the industry short-run supply curve. e. Suppose now painted neckties become more fashionable and the market demand function shifts upward to Q = 2,000 - 50P. Using this new demand curve, answer part b for the very short run when firms cannot change their outputs. f. In the short run, use the industry short-run supply curve to recalculate the answers to part b. g. What is the new long-run equilibrium for the industry?
a. the horizontal sum of all individual firm supply schedules at this output level. b. output level, each firm will earn zero economic profit (normal profit).
a) In the long-run, each firm will produce 20 units of neckties. The industry supply schedule will be the horizontal sum of all individual firm supply schedules at this output level.
b) The long-run equilibrium price (P*) is $20 per unit, with a total industry output (Q*) of 1,000 units. Each firm will produce 20 units of neckties, and the number of firms in the industry will be 50. At this output level, each firm will earn zero economic profit (normal profit).
c) The short-run average cost curve can be found by dividing the short-run total cost by output. Thus, the short-run average cost curve is SAC = 0.5q - 10 + 200/q. The short-run marginal cost curve is SMC = q - 10. Short-run average cost reaches a minimum at an output level of 20 units.
d) The short-run supply curve for each firm is the portion of the marginal cost curve above the average variable cost curve. The industry short-run supply curve is the horizontal sum of all individual firm supply curves.
e) With the new demand curve, the short-run equilibrium price (P*) is $30 per unit. The total industry output (Q*) is 1,250 units, with each firm producing 25 units of neckties.
f) In the short run, the industry short-run supply curve will shift upwards, resulting in a higher equilibrium price and output level. The new short-run equilibrium price (P*) will be higher than $20 per unit and the new total industry output (Q*) will be higher than 1,000 units.
g) In the long run, new firms will enter the industry, causing the supply curve to shift to the right until price falls back to the minimum long-run average cost of $10 per unit. At the new long-run equilibrium, each firm will produce 20 units of neckties, the industry output (Q*) will increase, and the price (P*) will fall back to $20 per unit.
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suppose a lottery game is played where the player chooses a three digit number (repetition allowed)
The probability of winning this lottery game is 1/1,000 or 0.001, which equates to a 0.1% chance.
In this lottery game, players select a three-digit number, ranging from 000 to 999. Since repetition is allowed, each digit can be any number between 0 and 9, giving a total of 10 options per digit. The three digits are independent, which means that the choice of one digit does not influence the choices for the other digits. Consequently, to find the total number of possible combinations, you can use the counting principle.
The counting principle states that if there are n ways to do one thing and m ways to do another, there are n x m ways to do both. In this case, there are 10 choices for each of the three digits, so the total number of combinations is 10 x 10 x 10 = 1,000.
Players win the lottery game if their chosen three-digit number matches the winning number drawn by the game organizers. The probability of winning is determined by dividing the number of successful outcomes (1, as there's only one winning number) by the total number of possible outcomes (1,000 combinations). Hence, the probability of winning is 1/1,000 or 0.1 % chance.
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Trevor is walking to school from home. He leaves and travels 58.0m before he forgets his lunch. He turns around, goes back home and grabs his lunch. He then walks 236m before he gets to school.
What is the total distance Trevor travelled? Round to three significant digits., do not include units.
The total distance Trevor traveled is 352 meters.
We have,
Trevor is traveling from his home to school.
He first walks 58.0 meters in one direction, but then he forgets his lunch and has to turn around and walk back the same distance.
This means he has walked a total distance of 58.0 m + 58.0 m = 116.0 m.
Now,
After he retrieves his lunch, he continues walking in the original direction for an additional 236 meters.
So, the total distance Trevor traveled.
= 116.0 m + 236 m = 352.0 m.
Thus,
The total distance Trevor traveled is 352 meters.
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This Box-and-Whisker Plot shows the distribution of a set of SAT scores for 1000 students. About what percentage of the students had scores between 485 and 695?
A.25%
B.50%
C.75%
D.100%
We have that about 50% of percentage of the students had scores between 485 and 695.
Option B is correct.
What is a Box-and-Whisker Plot?A Box-and-Whisker Plot is described as a method for graphically demonstrating the locality, spread and skewness groups of numerical data through their quartiles.
The box in the plot represents the interquartile range, therefore the percentage of students who scored between the lower quartile and the upper quartile of the distribution, are those between the edges of the box.
We take a look at the percentile ranks associated with those scores. and find the estimate of percentile ranks by drawing a horizontal line at the score values and then reading the corresponding percentile ranks off the y-axis.
With reference from the plot, a score of 485 appears to be at or below the 50th percentile, while a score of 695 appears to be around the 100th percentile.
We then have that the percentage of students with scores between 485 and 695 is likely to be between 100% - 50% = 50%.
The interquartile range represents the middle 50% of the data and the box covers this range.
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Which equations are true for x = –2 and x = 2? Select two options x2 – 4 = 0, x2 = –4 3x2 + 12 = 0, 4x2 = 16, 2(x – 2)2 = 0
The equations that are true for x= –2 and x = 2 is x²-4=0 and 4×2=16
Step-by-step explanationx= -2, x=2: x²-4=0
x²= 0 (multiplying the value of x with itself)
Since x² means multiplying the value of x with itself and 4 times itself equals 16, the next value that equals x = –2 and x = 2 is 4×2=16
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Find the standard deviation of a sample n = 200 if p = 7. O a. 0.0160 O b.0.0324 O c.0.2640 O d. 0.0016
The standard deviation of the sample is approximately 0.0180.
To find the standard deviation of a sample with a sample size (n) and proportion (p), we can use the formula:
Standard deviation (σ) = √(p(1-p)/n)
Given that n = 200 and p = 0.07, we can substitute these values into the formula:
σ = √(0.07(1-0.07)/200)
σ = √(0.07(0.93)/200)
σ = √(0.0651/200)
σ ≈ √0.0003255
σ ≈ 0.01803
Rounding to four decimal places, the standard deviation of the sample is approximately 0.0180.
Comparing this result with the given options, none of them match exactly. However, the option closest to the calculated standard deviation is b. 0.0324. It is important to note that this option is not an exact match and may be considered an error or an approximation. The actual standard deviation based on the given values is approximately 0.0180.
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find the acute angle between the lines. round your answer to the nearest degree. 2x − y = 3, 6x y = 9
The acute angle between the lines is approximately 37 degrees.
To find the acute angle between the lines given by the equations 2x - y = 3 and 6x + y = 9, we can compare the slopes of the lines.
The slope-intercept form of a line is y = mx + b, where m is the slope. By rearranging the given equations into this form, we can determine the slopes.
For the first equation, 2x - y = 3, we can rewrite it as y = 2x - 3. The slope of this line is 2.
For the second equation, 6x + y = 9, we can rewrite it as y = -6x + 9. The slope of this line is -6.
To find the acute angle between the lines, we can use the formula:
angle = arctan(|m1 - m2| / (1 + m1 * m2))
Plugging in the slopes:
angle = arctan(|2 - (-6)| / (1 + 2 * (-6)))
Simplifying the expression:
angle = arctan(8 / (-11))
Using a calculator or trigonometric tables, we can find:
angle ≈ -37.15 degrees
Since we are looking for the acute angle, we take the absolute value of the result:
acute angle ≈ 37 degrees
Therefore, the acute angle between the lines is approximately 37 degrees.
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suppose that we know that l 1 ∪ l 2 and l 1 are regular. can we conclude from this that l 2 is regular? make sure to prove your answer
No, we cannot conclude that l2 is regular from the fact that l1 ∪ l2 and l1 are regular.
Does the regularity of l1 ∪ l2 and l1 imply the regularity of l2?The regularity of a language means that there exists a finite automaton that recognizes that language. The union of two languages l1 and l2 is the set of all strings that are in either l1 or l2 or both.
Suppose that l1 ∪ l2 and l1 are regular. Then there exist finite automata A1 and A2 that recognize l1 ∪ l2 and l1, respectively. However, this does not imply that there exists a finite automaton that recognizes l2.
To see why, consider the example where l1 = {a^n b^n | n >= 0} and l2 = {a^n b^n c^n | n >= 0}. Both l1 and l1 ∪ l2 are regular languages, but l2 is not regular. This can be proven using the pumping lemma for regular languages.
Therefore, the regularity of l1 ∪ l2 and l1 does not necessarily imply the regularity of l2.
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a researcher wishes to estimate the proportion of households that have broadband internet access. what size sample should be obtained if she wishes the estimate to be within 0.03 with 99% confidence if (a) she uses a 2009 estimate of 0.635 obtained from the national telecommunications and information administration? (b) she does not use any prior estimates
(a) The researcher should obtain a sample size of 1,068 households to estimate the proportion of households with broadband internet access within 0.03 with 99% confidence, assuming a prior estimate of 0.635 from 2009.
(b) If the researcher does not use any prior estimates, she can use a conservative estimate of 0.5 for the proportion of households with broadband internet access, as this value maximizes the sample size required for a given level of precision and confidence. With this assumption, the researcher should obtain a sample size of 1,068 households to estimate the proportion of households with broadband internet access within 0.03 with 99% confidence. It is important to note that if the true proportion is significantly different from 0.5, the required sample size may be higher or lower than this estimate. Additionally, the researcher should consider other factors such as the cost and feasibility of obtaining a sample of this size.
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I WILL Mark Brainly fast
1. Average rate of change is 17.5 and 22.5 respectively.
2. The Strain B is growing faster.
What is the average rate of change for each strain?To get average rate of change for each strain from week 0 to 4, we will use the formula: [tex]Average rate of change = (change in cases) / (change in weeks)[/tex]
For Strain A:
Change in cases = 85 - 15 = 70
Change in weeks = 4 - 0 = 4
Average rate of change = 70/4
Average rate of change = 17.5
For Strain B:
Change in cases = 115 - 25 = 90
Change in weeks = 4 - 0 = 4
Average rate of change = 90/4
Average rate of change = 22.5
The Strain B is growing faster with an average rate of change of 22.5 cases per week compared to Strain A with an average rate of change of 17.5 cases per week.
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solve this please.!!!!!!
Answer:
(m-4)(m+4)
Step-by-step explanation:
If you multiply both and open up it becomes [tex]m^2-4m+4m-16[/tex], simplify to [tex]m^2-16[/tex]
what are the elements in the vector x when x = [6 4 15; 2 1 3]; x(4, 4) = 7;
There is no element in position (4,4) since matrix x has only two rows and three columns.
This vector is a 2x3 matrix, which means it has two rows and three columns: [6 4 15] [2 1 3] Now, address the additional information: x(4, 4) = 7. Unfortunately, this information is not relevant because the given matrix is a 2x3 matrix, and there is no element at the (4, 4) position.
Hence, The vector x does not exist since it has more than one row.
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In a certain Algebra 2 class of 21 students, 10 of them play basketball and 9 of them
play baseball. There are 8 students who play neither sport. What is the probability
that a student chosen randomly from the class plays both basketball and baseball?
Answer:
Step-by-step explanation:
To find the probability that a student plays both basketball and baseball, we need to determine the number of students who play both sports and divide it by the total number of students in the class.
Given:
Total number of students (n) = 21
Number of students who play basketball (B) = 10
Number of students who play baseball (A) = 9
Number of students who play neither sport = 8
Let's calculate the number of students who play both basketball and baseball (B ∩ A):
Number of students who play both sports (B ∩ A) = Number of students who play basketball (B) + Number of students who play baseball (A) - Total number of students (n) + Number of students who play neither sport
B ∩ A = B + A - n + Neither
B ∩ A = 10 + 9 - 21 + 8
B ∩ A = 6
The number of students who play both basketball and baseball is 6.
Now, we can calculate the probability:
Probability of playing both basketball and baseball = Number of students who play both sports (B ∩ A) / Total number of students (n)
Probability = 6 / 21
Probability = 2 / 7
Therefore, the probability that a student chosen randomly from the class plays both basketball and baseball is 2/7.
There are 8 green apples and 3 red apples in a basket. What is the ratio of red apples to all apples in the basket? What is the ratio of all apples in the basket to green apples?
The ratio of red apples to all apples in the basket is 3:11, whereas all apples to green is 11:8
Total number of green apples = 8
Total number of red apples = 3
Calculating the total number of apples -
Total number of green apples + Total number of red apples
= 8 + 3
= 11
Calculating the ratio of red apples to all apples in the basket -
= Total number of red apples / Total number of apples
= 3/11
Thus, for every 11 apples in the basket, 3 of them are red.
Calculating the ratio of all apples in the basket to green apples -
Total number of apples / Total number of green apples
= 11/8.
Thus, for every 8 green apples in the basket, there are a total of 11 apples in the basket.
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What are the real zeros of the function y = 27(x + 2)³ + 5 ?
The cubic function only has one real zero, which is x = -2 - √5/3 = -2.75
How to find the zeros of the cubic function?Here we want to find the zeros of the cubic function:
y = 27(x + 2)³ + 5
The zeros of a function are the values of x such that the outcome is y, then we need to solve the equation:
0 = 27(x + 2)³ + 5
-5 = 27(x + 2)³
-5/27 = (x + 2)³
∛(-5/27) = x + 2
-√5/3 = x + 2
-2 - √5/3 = x
That is the only zero of the function (with a multiplicity of 3).
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The two cones below are similar. What is the height of the smaller cone?
OA. 5
O B. 20/7
O C. 28/5
O D. 35/4
The height of the smaller cone is 20/7, the correct option is B.
We are given that;
The two cones
Now,
To find the height of the smaller cone, you need to use the similarity ratio of the cones. Similar cones have proportional dimensions, so you can set up a proportion between the corresponding heights and radii. You can write your solution as:
h/7 = 20/10 h = 20/10 x 7 h = 14
Therefore, by the proportion the answer will be 20/7.
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write 3,901 1/4 in scientific notation
Answer:
Answer: 3.90125×10⁻³
Step-by-step explanation:
Sweets are sold in small packs and in big packs.
There is a total of 175 sweets in 4 small packs and 3 big packs.
There is a total of 154 sweets in 5 small packs and 2 big packs.
Work out the number of sweets in each small pack and in each big pack.
Answer:
Step-by-step explanation:
Let x - be the number of sweets in small packs
y - be the number of sweets in big packs
Therefore, we have:
4x + 3y = 175 (1)
5x + 2y = 154 (2)
Now, we find the difference between (1) & (2) is:
y-x = 21. Thus, y = 21+x
Now we substitute the value of y = 21+x to any of the two statements, we have 4x + 3(21+x) = 175 => 4x + 63 + 3x = 175.
Hence, 7x = 175 - 63 = 112 or simply, x=16.
Now, finding the value of y:
5(16) + 2y = 154
80 + 2y = 154
2y = 154-80
2y = 74
y = 37.
Therefore, there are 16 sweets in each small pack and 37 sweets in each big pack.
Instead of the usual dice, suppose we have a bag of 12-sided dice, each with sides numbered 1 through 12. Assume the dice are fair. if we dump out a bag of 50 such dice and add up the numbers they land on, what is the probability the total will be at least 360? Estimate the probability using a normal approximation with a continuity correction. Select the nearest percentage.a. 43%b. 74%c. 3%d. 59%e. 85%e. 16%f. 28%g. 8%
The estimated probability of the total sum being at least 360 is approximately 8%.
To estimate the probability using a normal approximation with a continuity correction, we first need to find the mean and standard deviation of the sum of the numbers on the 50 dice.
For a single 12-sided die, the mean is (1+2+...+12)/12 = 6.5. For 50 dice, the mean is 50 × 6.5 = 325. The variance for one die is [(1-6.5)²+(2-6.5)²+...+(12-6.5)²]/12 = 11.92. For 50 dice, the variance is 50 × 11.92 = 596, and the standard deviation is √596 ≈ 24.4.
Now, we'll use the normal approximation with a continuity correction to estimate the probability that the sum of the numbers is at least 360. First, find the z-score:
z = (X - μ + 0.5) / σ = (360 - 325 + 0.5) / 24.4 ≈ 1.42
Using a z-table or calculator, the probability of obtaining a z-score greater than 1.42 is approximately 0.0778 or 7.78%. The closest percentage in the options provided is 8%, which corresponds to option g. Therefore, the estimated probability of the total sum being at least 360 is approximately 8%.
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I need help with this question I don't get how to do it please explain and give answer.
we know the radius has a diameter of 26 cm, so its radius must be half that, or 13 cm.
[tex]\textit{area of a circle}\\\\ A=\pi r^2 ~~ \begin{cases} r=radius\\[-0.5em] \hrulefill\\ r=13 \end{cases}\implies A=\pi (13)^2 \\\\\\ A=(3.14)(13)^2\implies A=530.66~cm^2 \\\\[-0.35em] ~\dotfill\\\\ \textit{circumference of a circle}\\\\ C=2\pi r ~~ \begin{cases} r=radius\\[-0.5em] \hrulefill\\ r=13 \end{cases}\implies C=2\pi 13 \\\\\\ C=2(3.14)(13)\implies C=81.64~cm[/tex]
Check the picture below.
find two consecutive integers such that the square of the larger integer is 19 more than 9 times the smaller integer
Two consecutive integers such that the square of the larger integer is 19 more than 9 times the smaller integer are 9 and 10
Let x be the smaller integer, then the larger integer is x + 1. According to the problem, we can set up an equation:
(x + 1)^2 = 9x + 19
Expanding the left side and simplifying, we get:
x^2 + 2x + 1 = 9x + 19
Bringing all the terms to one side, we get:
x^2 - 7x - 18 = 0
Factorizing, we get:
(x - 9)(x + 2) = 0
So, x = 9 or x = -2. Since we are looking for consecutive integers, we can discard the negative solution. Therefore, the smaller integer is 9 and the larger integer is 10. We can verify that this solution satisfies the original equation:
10^2 = 100 = 9(9) + 19 = 82
So, the two consecutive integers are 9 and 10.
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The trucker completed the 840-km haul in 10 hours 30 minutes. What was the trucker's average speed in kilometers per hour
The truckers average speed in kilometers per hour would be = 80.9 km/hr
How to calculate the average speed of the trucker?To calculate the average speed of the trucker the formula for speed should be used and this is given below;
Speed = Distance/ time
Distance = 849 km
Time = 10 hours 30 minutes= 10.5 hours
Speed = 849/10.5 = 80.9km/hr
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5.
The surface area of a cuboid shaped paperweight is 37.5 in². The surface area of
the cuboid can be found using A = 61². What is the length of the cuboid?
The solution is: The surface area of the cuboid is: 1900 cm².
Here, we have,
We can use the given ratios and volume to find the scale factor for the dimensions. Knowing the dimensions, we can compute the surface area using the formula for a cuboid.
dimensions
Let k represent the scale factor. Then the actual dimensions will be 5k, 4k, and 2k. The actual volume will be ...
V = LWH
5000 cm³ = (5k)(4k)(2k) = 40k³
k³ = (5000 cm³)/40 = 125 cm³
k = ∛(125 cm³) = 5 cm
The cuboid dimensions are 5(5 cm) = 25 cm, 4(5 cm) = 20 cm, and 2(5 cm) = 10 cm.
area
The surface area of the cuboid can be computed from ...
A = 2(LW +H(L +W))
A = 2((25 cm)(20 cm) +(10 cm)(25 +20 cm))
A = 2(500 cm² +(10 cm)(45 cm)) = 2(950 cm²) = 1900 cm²
The surface area of the cuboid is 1900 cm².
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complete question:
the length, breadth and height of a cuboid are in the ratio 5:4:2 if the volume of cuboid is 5000 cm,then what will be the surface of the area of the cuboid
Find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Focus: (-9,0) Focus: (0, 1/6) Directrix: 2
The standard form of the equation is 2x^2 + 24y - 35 = 0. To find the standard form of the equation of a parabola with vertex at the origin, we need to use the formula 4p(x^2 + y^2) = (x - h)^2 + (y - k)^2, where (h,k) is the vertex and p is the distance from the vertex to the focus (or directrix, depending on the given information).
For the first characteristic, we have a focus at (-9,0), which is to the left of the vertex at the origin. This means that p = 9 (the distance from the vertex to the focus). Substituting into the formula, we get:
4(9)(x^2 + y^2) = (x - 0)^2 + (y - 0)^2
36x^2 + 36y^2 = x^2 + y^2
35x^2 + 35y^2 = 0
So the standard form of the equation is 35x^2 + 35y^2 = 0. For the second characteristic, we have a focus at (0,1/6), which is above the vertex at the origin. This means that p = 1/6 (the distance from the vertex to the focus). We also know that the directrix is a horizontal line 2 units below the vertex. This means that the equation of the directrix is y = -2. Using the formula and the distance formula between a point and a line, we can write:
4(1/6)(x^2 + y^2) = (y - 0)^2 - 2^2
2x^2 + 24y - 35 = 0
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x is a normally distributed random variable with mean of 16 and a standard deviation of 4. find the probability that x equals 22.56.
The probability that x = 22.56 is the 1.64
The probability formula defines the likelihood of the happening of an event. It is the ratio of favorable outcomes to the total favorable outcomes. The probability formula can be expressed as,
P(A) = Number of favorable outcomes of A / Total number of possible outcomes.
We must standardize the Random Variable X with the standardized Normal distribution Z variable using the relationship:
[tex]Z =\frac{X-\mu}{\sigma}[/tex]
We have the information from the question:
Mean ([tex]\mu[/tex]) = 16
Standard deviation ([tex]\sigma[/tex]) = 4
To find the probability that x equals 22.56.
P(X= 22.56) = [tex]P(\frac{22.56-16}{4} )[/tex]
= [tex]P(\frac{6.56}{4} )[/tex]
= P(1.64)
Hence, The probability that x = 22.56 is the 1.64
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frac x2-16x3+64 Which expression is equivalent to the given expression, if the denominator does not equal 0? A. 1/x-4 B. 1/x+4 C. frac x+4x2-4x+16
The correct answer is option B, which is 1/(x+4). To see why, first factor the denominator of the given expression:
x^2 - 16x + 64 = (x - 8)(x - 8) = (x - 8)^2
Now, we can rewrite the original expression as:
(x - 8)^2 / [(x - 8)(x + 4)]
Canceling the common factor (x - 8), we get:
(x - 8) / (x + 4)
This is equivalent to 1/(x+4) since (x - 8) / (x + 4) = (x + 4 - 12) / (x + 4) = 1 - 12 / (x + 4) = 1 - 3 / (x + 4/3). As x approaches infinity, 3/(x+4/3) approaches 0, so 1 - 3 / (x + 4/3) approaches 1. Thus, the expression is equivalent to 1/(x+4) for any value of x except x = -4.
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