According the question 89+82 expressiοn is the same as 8(9+2).
What is expressiοn?An expressiοn in math is a sentence with a minimum οf twο numbers οr variables and at least οne math οperatiοn. This math οperatiοn can be additiοn, subtractiοn, multiplicatiοn, οr divisiοn.
Algebraic expressiοns are classified οn the basis οf the number οf terms in the expressiοn. The variοus types οf algebraic expressiοns are:
Mοnοmial expressiοnsBinοmial expressiοnsTrinοmial expressiοnsPοlynοmial expressiοnsThe difference between expressiοns and equatiοns is that an expressiοn signifies a cοmbinatiοn οf numbers, variables, and οperatiοn symbοls whereas an equatiοn will always use an equal (=) οperatοr between twο math expressiοns.
The expressiοn 8(9+2) can be simplified using the distributive prοperty οf multiplicatiοn οver additiοn:
8(9+2) = 89 + 82
Therefοre, the answer is: 89+82
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Sarah deposited $200 in a savings account earning 7% interest,
compounded annually. To the nearest cent, how much interest will she
earn in 4 years?
The interest earned by Sarah in four (4) years is approximately $62.16.
How much interest is earned by Sarah?The formula accrued amount in a compounded interest is expressed as;
A = P( 1 + r/n )^( n × t )
Where A is accrued amount, P is principal, r is interest rate and t is time.
Given that;
Principal P = $200Compounded annaully n = 1Time t = 4 yearsInterest rate R = 7%Accrued amount A = ?Interest I = ?First, we convert R as a percent to r as a decimal
r = R/100
r = 7/100
r = 0.07
Now, the the values into the above formula and solve for accrued amount (A).
A = P( 1 + r/n )^( n × t )
A = $200( 1 + 0.07/1 )^( 1 × 4 )
A = $200( 1 + 0.07 )^( 4 )
A = $200( 1.07 )^( 4 )
A = $262.16
Now, we know that;
Accrued amount (A) = Principal (P) + Interest (I)
Solve for Interest (I)
$262.16 = $200 + I
I = $262.16 - $200
I = $62.16
Therefore, the interest earned is $62.16.
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The shape ABC in the quilt block shown is an isosceles triangle, where AB = AC. The perimeter of the triangle is 27.3 inches. Find the values of x and y. Then, find the length of each side of the triangle. Enter the correct answers in the boxes. AB= BC = AC=
The values of the variables and the side lengths are
(x, y) = (3, 5)AB = AC = 8 and BC = 11.3How to determine the values of the variables and the side lengthsFrom the question, we have the following parameters that can be used in our computation:
AB = AC
Perimeter = 27.3
Using the attached figure, we have
6x - 2y = x + 5
Evaluate
5x = 2y + 5
For the perimeter, we have
AC + AB + BC = 27.3
So, we have
6x - 2y + x + 5 + y + 6.3 = 27.3
Simplify
7x = y + 16
Multiply by 2
14x = 2y + 32
Subtract from 5x = 2y + 5
9x = 27
So, we have
x = 3
This means that
2y + 32 = 14 * 3
2y + 32 = 42
Evaluate
2y = 10
y = 5
For the side lengths, we have
AC + AB + BC = 27.3
So, we have
AC = 6(3) - 2(5) = 8
AB = 3 + 5 = 8
BC = 5 + 6.3 = 11.3
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4. If triangle ABC has the following measurements, find the measure of side c:
a = 5
B = 7
C = 42 degrees
Using cosine rule, the measure of side c is calculated as: 4.688
How to use the cosine rule?In trigonometry, the Cosine Rule says that the square of the length of any side of a given triangle is equal to the sum of the squares of the length of the other sides minus twice the product of the other two sides multiplied by the cosine of angle included between them.
We are given the parameters as:
a = 5
B = 7
C = 42 degrees
Using cosine rule, we have:
c = √(a² + b² - 2ab cos C)
c = √(5² + 7² - 2(5 * 7) cos 42)
c = √(25 + 49 - 52.02)
c = √21.98
c = 4.688
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Which description compares the vertical asymptote(s) of Function A and Function B correctly? Function A: f(x)=1/x−3 Function B: A hyperbola graphed on a grid with the x and y axis beginning at negative ten and increasing in increments of two until reaching ten. The hyperbola, labeled g of x, contains an asymptote at x equals four. The branches of the hyperbola are oriented so they open to the upper right and lower left corners of the asymptote. The lower left branch passes through begin ordered pair zero comma zero end ordered pair as a smooth curve. The upper right branch passes through begin ordered pair eight comma two end ordered pair as a smooth curve. Responses Both functions have a vertical asymptote at x = 4. Both functions have a vertical asymptote at x = 4. Function A has two vertical asymptotes. Function B has one vertical asymptote. Function A has two vertical asymptotes. Function B has one vertical asymptote. Function A has a vertical asymptote at x=−3 . Function B has a vertical asymptote at x = 4. Function A has a vertical asymptote at , x = − 3 , . Function B has a vertical asymptote at x = 4. Function A has a vertical asymptote at x = 3. Function B has a vertical asymptote at x = 4.
Answer:
The answer is D
Step-by-step explanation:
Function A
f(x) = 1 / ( x - 3)
The vertical asymptote is the value of x that makes x - 3 = 0 ⇒ x = 3
The vertical asymptote of B is x = 4
So....
Function A has a vertical asymptote at x = 3.
Function B has a vertical asymptote at x = 4.
12x(x+3)=0 using zero product property
Answer:
0, -3.
Step-by-step explanation:
12x(x+3)=0
Either 12x = 0 or (x + 3) = 0
x = 0/12 or x = -3
So, x = 0 or x = -3
in the 3rd quadrant, find SIN
Write and solve a problem about base pay and commission. Show your work.
"Problem:
Jenny earns a base pay of $1000 per month plus a 5% commission on all sales. Last month she sold $8000 worth of products. What was her total earnings for the month?
Solution:
Jenny's commission for the month is 5% of $8000, which can be calculated as:
Commission = 5% of $8000 = (5/100) * $8000 = $400
Her total earnings for the month can be found by adding her base pay and commission:
Total Earnings = Base Pay + Commission
Total Earnings = $1000 + $400
Total Earnings = $1400
Therefore, Jenny's total earnings for the month were $1400." (ChatGPT, 2023)
which is a scaled copy of plugin A ? identify a pair of corresponding sides and a pair of corresponding angles. compare the areas of the scaled copies
Therefore , the solution of the given problem of angles comes out to be A is larger than B in terms of size and scale variables can be used with this technique.
An angle meaning is what?A tilt is a form in Euclidean geometry made up of two sides, but rather cylindrical faces, that divide at the barrier's centre and apex. At their junctions, two rays may combine to form an angle. Angle is another outcome of two entities interacting. Dihedral shapes best describe them. A two-dimensional curve can be created by arranging two line beams in various configurations at their ends.
Here,
Consider Polygon A to be a square with the dimensions 2 feet by 5 feet.
=> Size = 2 feet
=>Size = 51 t
The dimension of A can be multiplied by the same scale factor to obtain potential scaled duplicates.
Consider for instance, the scale factor k is:
A scaled-down version of rectangle A is 6 feet by 15 feet, for example.
You obtain this by:
=> Length: 2 feet, 3 inches. 5 feet 15 feet wide
Area A consists of:
=> Area = 2 feet × 5 feet 2
B's size is 90 feet²
=> 2 Arca = 15 feet²
Comparison
A is larger than B in terms of size.
Scale variables can be used with this technique.
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Sharon paid $32,400 in Social Security taxes over a 25-year period. The total contribution to her account is
The total contribution to Sharon's account over the 25-year period is $64,800.
How to solve
To find the total contribution to Sharon's account, we need to consider both her contribution and her employer's contribution.
Social Security taxes are generally split evenly between the employee and the employer, with each paying half of the tax.
Sharon paid $32,400 in Social Security taxes over the 25-year period, which represents her share (half) of the total Social Security taxes paid.
To find the total contribution to her account, we simply need to double her share to include the employer's contribution:
Total contribution = Sharon's contribution + Employer's contribution
Total contribution = $32,400 + $32,400
Total contribution = $64,800
The total contribution to Sharon's account over the 25-year period is $64,800.
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A spinner is divided into 4 equal sections. The probability of landing on A is 1/4. Brandy spins the spinner 24 times. How many times can she expect the spinner to land on A?
Answer:
6 times
Step-by-step explanation:
24/4 = 6
someone help pls i’m confused
Answer:
CF = 12
DE = 23
CD = 23
DF = 23.9
Step-by-step explanation:
We can see that there are 2 right-angled triangles: DEF and CDF with common side FD.
They are right angled triangles because the EF and CF are radii of the circle and these segments intersect the tangents at 90°
The common side DF is the hypotenuse for both triangles
The sides EF and CF are radii of the circle so EF = CF
We have two triangles which have two sides equal and one of the angles equal to the corresponding angle of the other
By the SSA theorem, the two triangles are similar
SSA Theorem
if two sides and an angle not included between them are respectively equal to two sides and an angle of the other then the two triangles are equal
Therefore by the law of similar triangles,
CD = EF
Plugging in the expressions we get
13x - 16 = 4x + 11
Subtract 4x from both sides:
13x - 4x - 16 = 11
9x - 16 = 11
Add 16 to both sides:
9x - 16 + 16 = 11 + 16
9x = 27
x = 27/9 = 3
Therefore
ED = 4x + 11 = 4(3) + 11 = 12 + 11 = 23
and this is equal to CD
Using the Pythagorean theorem for right triangles,
For ΔDEF,
DF² = EF² + DE²
DF² = 12² + 23²
= 144 + 529
= 673
DF = √673
= 25.9422
= 25.9 rounded to the nearest tenth
So the measures of the segments are
CF = 12
DE = 23
CD = 23
DF = 23.9
I think account has a balance of $120 on January 1. Describe a situation in which the account balance for each month February 1 March 1 forms the following sequences in exercise one and two. Write the first three terms of each sequence.
In exercise 1:
Situation: The account holder saves $30 each month
The first three terms of the sequence are: $120, $150, $180.
In exercise 2:
Situation: The account earns 25% interest each month.
The first three terms of the sequence are: $120, $150, $187.50.
What is a sequence?A sequence in mathematics is described as an enumerated collection of objects in which repetitions are allowed and order matters.
Explaining exercise 1:
In this exercise, the account balance for each month should form an arithmetic sequence with a common difference of $30
February 1 balance: $150
March 1 balance: $180
Hence, we have the first three terms of the sequence as: $120, $150, $180.
Explaining exercise 2:
In this exercise, the account balance for each month should form a geometric sequence with a common ratio of 1.25.
The account earns 25% interest each month is the situation here.
February 1 balance: $150
March 1 balance: $187.50
The first three terms of the sequence are: $120, $150, $187.50.
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List the set of possible rational zeros for f(x)=x^4-5x^3+8x^2-20x+16
show work
Answer:
Step-by-step explanation:
The possible rational zeros of a polynomial function are all the possible values of x, where x is a factor of the constant term of the function divided by a factor of the leading coefficient of the function.
For the polynomial function f(x) = x^4 - 5x^3 + 8x^2 - 20x + 16, the constant term is 16 and the leading coefficient is 1. Therefore, the possible rational zeros are all the possible values of x, where x is a factor of 16 divided by a factor of 1. That is, the possible rational zeros are:
±1, ±2, ±4, ±8, ±16
To check if any of these possible zeros are actually zeros of the function, we can use synthetic division or long division to test each one.
21 is greater than the sum of 15 and two times a number X
Answer:21>15+2x
Step-by-step explanation:
Test the claim that the proportion of men who own cats is smaller than the proportion of women who own cats at the .05 significance level.
The null and alternative hypothesis would be:
The test is:
two-tailed
right-tailed
left-tailed
Based on a sample of 60 men, 25% owned cats
Based on a sample of 60 women, 30% owned cats
The test statistic is:
(to 2 decimals)
The p-value is:
(to 2 decimals)
Based on this we:
Fail to reject the null hypothesis
Reject the null hypothesis
The assertion that the percentage of males who own cats is lower than the percentage of women who do not own cats is unsupported by sufficient data.
Hence, the Alternate hypothesis would be:
H0: Pm = Pf
H1 = Pm < Pf
What is Alternate hypothesis?Alternate hypotheses are used in statistical hypothesis testing. The hypothesis of a test always predicts no effect or no association between variables, but the alternative hypothesis reflects the effect or relationship that your research predicts.
Given information:
The test claims proportion of men who owns cat is smaller.
The significance level = 0.005
As the hypothesis always gets ten statements of equality but in the case given alternate hypothesis will claim.
Now calculating the Z-value:
P= 10+14/80
= 0.3
So, the p-value according to the obtained Z value is 0.1365.
Since, 0.1365>0.005, we fail to reject the hypothesis.
Hence, there is not enough proof to support the claim that the proportion of man who owns the cat is less than that of the proportion of women who owns the cat.
Therefore, the Alternate hypothesis would be:
H0: Pm = Pf
H1 = Pm < Pf
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three numbers m,n,p are in the ratio 3:6:4.which of the following is the value
[tex] \frac{4m - n}{n + 2p} [/tex]
Answer:
the value of (4m-n)/(n+2p) is 3/7.
Step-by-step explanation:
Let's substitute the given ratio in terms of a common multiplier, k:
m = 3k
n = 6k
p = 4k
Now we can substitute these values in the expression:
(4m-n)/(n+2p) = (4(3k) - 6k)/(6k + 2(4k)) = (12k - 6k)/(6k + 8k) = 6k/14k = 3/7
Therefore, the value of (4m-n)/(n+2p) is 3/7.
Tom can paint a fence in 12 hours. Huck can paint the same fence in 10 hours. How long would it take to put two coats of paint on the fence if Tom and Huck work together? Answer as an improper fraction, then round to 1 decimal place. 3) Kim and Josh can clean a house together in 3 hours. If it takes Kim 7 hours to clean the house by herself, how long would it take Josh to clean the house alone? Leave your answer in improper fraction form, then convert to a decimal. 4) Jack can mow the baseball grounds in 2 hours; Mike can mow the same grounds in 3 hours; and Chris can mow the grounds in 4 hours. How long will it take to mow the baseball grounds if Jack, Mike, and Chris work together? Leave your answer in proper fraction form, then round to 1 decimal place. Using your rounded answer, how many minutes would that be?
2) They tοgether will take 10.9 hours οr 120/11
3) jοsh will take 21/11 hrs οr 1.9 hrs
4) Tοgether they will take 36/13 οr 2.8 hrs
What is decimal place?The place values οf the digits in a decimal number are displayed οn the decimal place value chart. We knοw that a digit in a number represents a numerical value οr place value.
Decimal place value charts are used tο determine the prοper placement οf each digit in a decimal number.
A decimal number cοnsists οf a whοle number and a fractiοnal cοmpοnent, separated by the decimal pοint, a dοt.
Fοr instance, the decimal number 4.37 has twο parts: an actual number pοrtiοn οf 4 and a fractiοnal pοrtiοn οf 37.
Sοlutiοns tο the abοve prοblem
2) Tοm takes = 12 hrs
Huck takes= 10hrs
We can aply unitary methοd
tοgether they will take= 1/12+1/10=2/t
11t=120;
t=120/11 οr 10.9 hrs
3) Tοgether Kim and jοsh can clean the hοuse tοgether in = 3 hrs
Kim takes = 7hrs
Jοsh will take= 1/7+1/x=2/3
3x+21=14x
21=11x
x=21/11 οr 1.9 hrs
4) Jack makes baseball grοunds in= 2 hrs
Mike takes= 3 hrs
Chris takes= 4 hrs
Tοgether they will take= 1/2+1/3+1/4=3/x
13x=36
x=36/13 οr 2.8 hrs
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Please answer this question with the following proofs.
Answer:
Line DB and AC intersect at point E.
∠AEB = ∠DEC because they are vertical angles.
∠EAB and ∠ECD are alternate interior angles because AB and CD are parallel lines and line AC crosses through both parallel lines.
m∠EAB = m∠ECD because they are alternate interior angles.
ΔABE ≅ ΔCDE because there are two sets of congruent angles.
i need help with the question pls
The area of the region bounded by the curves y = x² + 6x + 9 and y = 4 is 16 ² / ₃ square units.
How to find the area of the region?First, we need to find the points of intersection between the two curves. Set y = x² + 6x + 9 equal to y = 4:
x² + 6x + 9 = 4
Now, solve for x:
x² + 6x + 5 = 0
Factor the quadratic equation:
(x + 5)(x + 1) = 0
This gives us the solutions x = -5 and x = -1. These are the x-coordinates of the points of intersection.
Now, to find the area of the region bounded by the curves, we'll integrate the difference between the two functions with respect to x, from x = -5 to x = -1:
Area = ∫[4 - (x² + 6x + 9)]dx from -5 to -1
Area = ∫[5 - x² - 6x]dx from -5 to -1
Now, integrate:
Area = [5x - (x³/3) - 3x²] from -5 to -1
Plug in the limits:
Area = (5(-1) - (-1)³/3 - 3(-1)²) - (5(-5) - (-5)³/3 - 3(-5)²)
Area = (-5 - 1/3 - 3) - (-25 + 125/3 - 75)
Area = (-8 1/3) - (- 25/3)
Area = 16 ² / ₃ square units
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The graph shows a proportional relationship.
Answer:
The Answer Is 6
Explanation: Divide y by x on each dot to get k
5. A rock is thrown directly upward with an initial velocity of 79 feet per second from a cliff 50 feet above a beach. The height of the rock above the beach (h) after t seconds is given by the equation h = -16t² + 79t + 50. The graph below shows the rock's height as a function of time.
5. The quadratic function for the height of the rock, h = -16·t² + 79·t + 50, indicates;
a. The height of the rock will be 125 feet above the beach at 1.28 seconds and 3.66 seconds after it is thrown
b. The maximum height reached is about 147.5 feet
The time it takes the rock to reach the maximum height is about 2.47 seconds
What is a quadratic function?
A quadratic function is a function of the form; f(x) = a·x² + b·x + c, where, a, b, and c are numbers, and a ≠ 0.
The initial velocity of the rock = 79 ft/s
Height of the cliff above the beach from which the rock is thrown = 50 feet
The function for the height of the rock above the beach is; h = -16·t² + 79·t + 50
Please find attached the graph of the height of the rock as a function of time, created with MS Excel.
Required; When the height of the rock will be 125 feet above the beach
When the height of the rock is 125 feet, we get;
h = 125 = -16·t² + 79·t + 50
-16·t² + 79·t + 50 - 125 = 0
-16·t² + 79·t - 75 = 0
The value of t obtained using an online tool are;
t = (79 + √(1441))/32 ≈ 3.66 and t = (79 - √(1441))/32 ≈ 1.28
The times at which the height of the rock will be 125 feet above the beach are; 1.28 seconds and 3.66 seconds after the rock is thrown
b. The maximum height of the rock can be obtained from the formula for finding the maximum height of a quadratic equation of the form; f(x) = a·x² + b·x + c, which is;
At the maximum point, x = -b/(2·a)
The function for the height; h = -16·t² + 79·t + 50, indicates that we get;
a = -16, b = 79, and c = 50
Therefore;
The time it takes the rock to reach the maximum height, t(max), is therefore;
t(max) = -79/(2 × (-16)) ≈ 2.47
It takes about 2.47 seconds for the rock to reach the maximum height
The maximum height, h(max) = -16 × 2.46875² + 79×2.46875 + 50 ≈ 147.5
The maximum height reached by the rock is about 147.5 feet
Possible part of the question, obtained from a similar online question, includes;
a. The time it takes the rock to reach a height of 125 feet above the beach
b. The maximum height reached by the rock and the duration it takes the rock to reach the maximum height.
Please find attached the possible graph in the question, created with MS Excel, using the function for the height of the rock.
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A city has cone-shaped buildings for storing salt and sand for icy roads. Each
building has a radius of 30 feet and a height of 20 feet. Find the volume of the
building. Use 3.14 for π.
OA. 1884 ft³
OB. 18,840 ft³
OC. 628 ft3
OD. 56,520 ft³
The volume of the building is approximately 18,840 cubic feet.
So the answer is (B) 18,840 ft³
What is volume of cone ?
The volume of a cone is a measure of the amount of space that the cone occupies and can be calculated using the formula:
[tex]V = (1/3)\pi r^2h[/tex]
where V is the volume of the cone, r is the radius of the base of the cone, h is the height of the cone, and π is a mathematical constant approximately equal to 3.14.
According to the question:
The volume V of a cone can be calculated using the formula:
[tex]V = (1/3)\pi r^2h[/tex]
where r is the radius of the base of the cone, h is the height of the cone, and π is a constant approximately equal to 3.14.
In this case, the radius r is 30 feet and the height h is 20 feet. Substituting these values into the formula, we get:
[tex]V = (1/3)\pi (30)^2(20)[/tex]
[tex]V = (1/3)\pi (900)(20)[/tex]
[tex]V = (1/3)(56,520)[/tex]
[tex]V = 18,840[/tex]
Therefore, the volume of the building is approximately 18,840 cubic feet.
So the answer is (B) 18,840 ft³.
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HELP ME! PLEASE!! . . .
Step-by-step explanation:
that makes ORQ a right-angled triangle, and we can use Pythagoras
c² = a² + b²
c being the Hypotenuse (the side opposite of the 90° angle), a and b are the legs.
OQ is the radius of the circle = OM = LM/2 = 20/2 = 10 cm.
RQ = PQ/2 = 16/2 = 8 cm.
so, our Pythagoras equation looks like :
OQ² = RQ² + OR²
10² = 8² + OR²
100 = 64 + OR²
36 = OR²
OR = 6 cm
RM = OM - OR = 10 - 6 = 4 cm
How do l do this??? Please help
Answer:
69
Step-by-step explanation:
69
I don't know What 5 + x + 3 simplified is, help me please?
Answer:
5+x+3 simplified is x+8
Step-by-step explanation:
when you simplify something, you combine terms and write it in the simplest way you can. So combine 5 and 3 to get 8, and then the final answer of x+8
Answer: x+8
Step-by-step explanation:
"5 + x + 3" can be simplified by combining the numerical terms, which are 5 and 3, to get 8.
Then, just rewrite the expression as "x + 8." Therefore, "5 + x + 3" simplified is just "x + 8".
How do you multiply a fraction by a whole number
Answer:
Write the whole number as a fraction by placing it over 1. For example, if you want to multiply 2/5 by 3, you can write 3 as 3/1.
Multiply the numerators of the two fractions together. For example, 2/5 x 3/1 = (2 x 3) / 5 = 6/5.
Simplify the resulting fraction, if necessary, by reducing it to its lowest terms. For example, 6/5 can be simplified to 1 1/5 (or 1.2 as a decimal).
Therefore, when you multiply a fraction by a whole number, you multiply the numerator of the fraction by the whole number and keep the denominator the same.
Step-by-step explanation:
The table shows the number of minutes in different numbers of days.
Number of days 4 7 11
Number of minutes 5760 10,080 15,840
Which equation expresses the relationship between the number of days, d, and the number of minutes, m?
Responses
m=d1440
m equals frqaction d over 1440 end fraction
A.m = d + 1440
B.m, = , d, + 1440
C.m = 1440d
D.d = 1440m
Answer:
C. m = 1440d
Step-by-step explanation:
You want to know the equation that represents the relationship between days (d) and minutes (m) given the table of values (d, m) = (4, 5760), (7, 10080), or (11, 15840).
Proportional relationshipOn the off chance you don't already know that the number of days must be multiplied by minutes per day to find the number of minutes (m = 1440d), you can always try the table values in the response choices to see what works:
A. 5760 = 4/1440 . . . . . . nope
B. 5760 = 4 + 1440 . . . . . nope
C. 5760 = 1440·4 . . . . . . . yep
D. 4 = 1440·5760 . . . . . . . nope
The equation that works is m = 1440d.
__
Additional comment
Fractions work the same way with units that they do with numbers. Common factors in numerator and denominator cancel.
[tex]\dfrac{2}{3}\times 3=2\\\\\dfrac{\text{minutes}}{\text{day}}\times \text{days}=\text{minutes}[/tex]
A jeweler had a fixed amount of gold to make bracelets and necklaces. The amount of gold in each bracelets is 7 grams And the amount of gold in each necklace is 24 grams. The jeweler used 172 grams of gold and made 2 more necklaces than bracelets. Write a system of equations that could be used to determine the number of bracelets made and the number of necklaces made. Define variabkes
The jeweler made 4 bracelets and 6 necklaces using 172 grams of gold.
What is equations ?An equation is a mathematical statement that shows that two expressions are equal. It contains variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division. Equations are used to solve problems by finding the values of the variables that make the equation true.
According to given information :Let x be the number of bracelets made, and y be the number of necklaces made.
Then, we can create the following system of equations:
Equation 1: 7x + 24y = 172 (the total amount of gold used is 172 grams)
Equation 2: y = x + 2 (the number of necklaces made is 2 more than the number of bracelets made)
So, the variables are x (the number of bracelets made) and y (the number of necklaces made).
We can solve this system of equations to find the values of x and y. We can use substitution or elimination to solve for one variable and then plug it into the other equation.
Substitution method:
From Equation 2, we have y = x + 2. Substituting this into Equation 1, we get:
7x + 24(x+2) = 172
7x + 24x + 48 = 172
31x = 124
x = 4
So, the jeweler made 4 bracelets.
Plugging this into Equation 2, we get:
y = x + 2 = 4 + 2 = 6
So, the jeweler made 6 necklaces.
Therefore, the jeweler made 4 bracelets and 6 necklaces using 172 grams of gold.
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Prove that Re(z1,z2-bar) = modulus of z1 * modulus of z2 iff arg z1 = arg z2 + 2*pi*n.Use polar form with arg measured in radians
Answer:
Step-by-step explanation:
We can start by writing the given equation in terms of polar form:
Re(z1,z2-bar) = modulus of z1 * modulus of z2
If we write z1 and z2 in polar form, we get:
z1 = r1(cosθ1 + i sinθ1)
z2 = r2(cosθ2 + i sinθ2)
Then, the conjugate of z2 is:
z2-bar = r2(cosθ2 - i sinθ2)
Using the formula for the real part of the product of two complex numbers, we get:
Re(z1,z2-bar) = Re(z1 * z2-bar)
Substituting the expressions for z1 and z2-bar, we get:
Re(z1,z2-bar) = Re(r1(cosθ1 + i sinθ1) * r2(cosθ2 - i sinθ2))
Simplifying the product, we get:
Re(z1,z2-bar) = Re(r1r2[(cosθ1 cosθ2 + sinθ1 sinθ2) + i(sinθ1 cosθ2 - cosθ1 sinθ2)])
The real part of this expression is:
Re(z1,z2-bar) = r1r2(cosθ1 cosθ2 + sinθ1 sinθ2)
Using the identity cos(θ1 - θ2) = cosθ1 cosθ2 + sinθ1 sinθ2, we can write:
Re(z1,z2-bar) = r1r2 cos(θ1 - θ2)
Now we can substitute this expression back into the original equation and get:
r1r2 cos(θ1 - θ2) = r1r2
Dividing both sides by r1r2, we get:
cos(θ1 - θ2) = 1
This equation is true if and only if θ1 - θ2 = 2πn for some integer n. In other words, θ1 = θ2 + 2πn, where n is an integer.
Therefore, we have proved that Re(z1,z2-bar) = modulus of z1 * modulus of z2 if and only if arg z1 = arg z2 + 2πn, where n is an integer.
Billy and Joey collect baseball cards. Billy has 25 more cards than Joey.
Write an expression in simplest form to represent the total number of cards
(c) in both collections.