Answer:
yes
Step-by-step explanation:
let f : (0,1) → r be a bounded continuous function. show that the function g(x) := x(1−x)f(x) is uniformly continuous.
We have shown that |g(x) - g(y)| < 12ε whenever |x - y| < δ. Since ε was arbitrary, this shows that g(x) is uniformly continuous on (0, 1).
What is uniform continuity?A stronger version of continuity known as uniform continuity ensures that functions defined on metric spaces, such as the real numbers, only vary by a small amount when their inputs change by a small amount. Contrary to uniform continuity, continuity merely demands that the function act "locally" around each point. To clarify, this means that for any given point x, there exists a tiny neighbourhood around x such that the function behaves properly inside that neighbourhood.
For the function g(x) to be continuous we need to have any ε > 0, and δ > 0 such that if |x - y| < δ, then |g(x) - g(y)| < ε for all x, y in (0, 1).
Now, g(x) is bounded as the parent function f(x) is bounded.
Suppose, (0, 1) such that |x - y| < δ.
Thus, without generality we have:
|g(x) - g(y)| = |x(1-x)f(x) - y(1-y)f(y)|
= |x(1-x)(f(x) - f(y)) + y(f(y) - f(x)) + xy(f(x) - f(y))|
≤ x(1-x)|f(x) - f(y)| + y|f(y) - f(x)| + xy|f(x) - f(y)|
< x(1-x)4ε + y4ε + xy4ε (by the choice of δ)
= 4ε(x(1-x) + y + xy)
< 4ε(x + y + xy)
≤ 4ε(1 + 1 + 1) = 12ε
Hence, we have shown that |g(x) - g(y)| < 12ε whenever |x - y| < δ. Since ε was arbitrary, this shows that g(x) is uniformly continuous on (0, 1).
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the length of a rectangle is increasing at a rate of 5 cm/s and its width is increasing at a rate of 4 cm/s. when the length is 13 cm and the width is 4 cm, how fast is the area of the rectangle increasing (in cm2/s)?
The rate of increasing the area of the rectangle is 72cm²/s
We have, The length of a rectangle is increasing at a rate of 5 cm/s and its width is increasing at a rate of 4 cm/s. When the length is 13 cm and the width is 4 cm, we have to find how fast is the area of the rectangle increasing.
The area of a rectangle is given by, A = l × w
Where l is the length and w is the width.
Now we will differentiate the equation with respect to time t.
dA/dt = d/dt (l × w)
dA/dt = l(dw/dt) + w(dl/dt)
We can use this formula to calculate how fast the area of the rectangle increases when the length is 13 cm and the width is 4 cm.
Substituting the given values, l = 13 cm and dl/dt = 5 cm/s w = 4 cm and dw/dt = 4 cm/s
dA/dt = 13(4) + 4(5)
dA/dt = 72 cm²/s
Therefore, the area of the rectangle increasing at a rate of 72 cm²/s.
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Assume you are running gradient descent, what will happen when the learning rate α is too small or too large? If you run gradient descent for 30 iterations with a=0.5 and compute J(θ) after each iteration. You find that the value of J(θ) increases over time. Based on this, how do you adjust the value of α to solve the problem?
The learning rate in gradient descent determines the step size and should be not too small or too large, as it can cause the algorithm to converge slowly or overshoot the minimum; adjusting the value of the learning rate can fix the problem, but the optimal value depends on the problem and data set.
According to the given information:
When running gradient descent,
The learning rate α determines the step size taken in each iteration toward the optimal solution.
If α is too small, the algorithm will take small steps and will converge slowly, or may even get stuck in a local minimum.
If α is too large, the algorithm may overshoot the minimum and diverge, or bounce back and forth without converging.
In the scenario described, the learning rate α of 0.5 appears too large, causing J(θ) to increase over time.
This suggests that the algorithm is not converging and is overshooting the minimum.
To fix this,
The value of α can be adjusted by reducing it to a smaller value,
Such as 0.1 or 0.01.
This should allow the algorithm to take smaller steps towards the minimum and eventually converge to a lower value of J(θ).
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Simplify 9(x+7)^2/3(x-1)(x+7)
Hey i know this question is linked to Pythagorean theorem but i am still confused. So, please help me if you can ❤️
Step-by-step explanation:
From Pythagorean theorem, one of the sides can be determined as x^2 + y^2 =8^2
or y = (8^2 - x^2)^(1/2)
we can write the perimeter P as
P = 2x + 2y ---> 20 = 2x + 2(8^2 - x^2)^(1/2)
Dividing by 2, we get
10 = x + (8^2 - x^2)^(1/2)
Move the x to the other side,
10 - x = (8^2 - x^2)^(1/2)
Take the square of both sides to get rid of the radical sign:
(10 - x)^2 = 8^2 - x^2
Move everything to the left and expand the quantity inside the parenthesis:
x^2 + (100 - 20x + x^2) - 64 = 0
2x^2 - 20x + 64 = 0
or
x^2 - 10x + 32 = 0
Now we can see that a = -10 and b = 32
Answer:
a = -10, b = 18
Step-by-step explanation:
The Pythagorean Theorem is, indeed, involved. Use it to find an expression (you won't get a number!) for the height of the rectangle.
Using the right triangle, one leg has length x and hypotenuse length 8. for a moment, label the height h. Then
\(x^2+h^2=8^3\\\\h^2=64-x^2\\\\h=\sqrt{64-x^2}\)
This expression tells the height of the rectangle, so it is the length of the two vertical sides. The top and bottom sides each have length x.
Perimeter = 20 says that the total length of all the sides is 20. Set that up and do a heap of algebra!
\(x+x+\sqrt{64-x^2}+\sqrt{64-x^2}=20\\\\2x+2\sqrt{64-x^2}=20\)
Divide by 2 (to simplify a bit).
\(x +\sqrt{64-x^2}=10\)
Subtract x to get the square root by itself (you'll see why in the next step).
\(\sqrt{64-x^2}=10-x\)
Square both sides of the equation.
\((\sqrt{64-x^2})^2=(10-x)^2\\\\\\64-x^2=100-20x+x^2\\\\64=100-20x+2x^2\\\\0=36-20x+2x^2\)
Divide by 2 again (because you can)
\(0=18-10x+x^2\)
Rearrange terms to match the order in the question.
\(x^2-10x+18=0\)
The coefficient of x is a = -10. The constant is b = 18.
Does any body know Jsipes44
Answer:
No..... who's that's
Step-by-step explanation:
jsjsjdkdhshnskxjxnxbhckdkdkmfjcjcjcjncjcnxmxkxjjxjxjxhchdhjxjxjcjcjjgkfkf
-2/3x + 4y = 5 in standard form
PLEASE HELP ME TO SOLVE THIS QUESTION
3.Xavier's salary increases by 2% each year.
In 2010 , his salary was £40,100
i) Calculate his salary in 2015 and give your answer to the nearest pound.
ii) In which year did Xavier's salary first greater than £47,500
i) Based on exponential growth, Xavier's salary in 2015 is £44,274.
ii) The year that Xavier's salary first became greater than £47,500 would be 2019.
What is exponential function?Exponential functions show the relationship between two variables and a variable exponent with a periodic constant rate of growth or decay in the value of something
Annual increase in Xavier's salary = 2% or 0.02
Xavier's salary in 2010 = £40,100
The number of years between 2015 and 2010 = 5 years
Exponential Function:i) y = £40,100(1.02)^5
= £44,273.64
= £44,274
ii) £47,500 = £40,100(1.02)^t
t = 9 years
= 2019 (2010 + 9)
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You have a 35-mile commute into work. Since you leave very early, the trip going to work is easier than the trip home. You can travel to work in the same time that it takes for you to make it 28 miles on the trip back home. Your average speed coming home is 10 miles per hour slower than your average speed going to work. What is your average speed going to work?
Average speed going to work is 50 mph.
What is average speed to work?Let's assume that the average speed going to work is "x" miles per hour.
Since you can travel to work in the same time it takes for you to make it 28 miles on the trip back home, we can set up an equation using the formula:
time = distance / speed
The time it takes to travel 35 miles to work is:
time to work = 35 / x
The time it takes to travel 28 miles on the return trip is:
time to return = 28 / (x - 10)
We know that both times are equal, so we can set them equal to each other and solve for "x":
\(35 / x = 28 / (x - 10)\)
Multiplying both sides by x(x - 10), we get:
\(35(x - 10) = 28x\)
Expanding the left side and simplifying, we get:
\(35x - 350 = 28x\)
\(7x = 35\)
\(x = 50\)
Therefore, your average speed going to work is 50 miles per hour.
To solve this problem, we used the formula for speed, distance, and time. We t up an equation using the fact that the time it takes to travel to work is equal to the time it takes to travel 28 miles on the return trip. We then solved for the average speed going to work by setting the two times equal to each other and solving for "x". Finally, we found that the average speed going to work is 50 miles per hour.
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At a coffee shop, the price of a cup of coffee increased from $1.20 to $1.44. What is the percent increase in the cost of the coffee?
The percent increase in the cost of the coffee is 20%
What is Percentage?percentage, a relative value indicating hundredth parts of any quantity.
Given that At a coffee shop, the price of a cup of coffee increased from $1.20 to $1.44
We need to find the percent increase in the cost of the coffee
Original price-Change price/original price
1.20-1.44/1.20
0.20
Now multiply with 100 as it is hundredth part
0.20×100
20%
Hence, the percent increase in the cost of the coffee is 20%
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Please help .!!!!!!!!!!!!!!!
This being a piecewise function, trying to solve f(2), we must find which two equations we can use.
We can use the function '3x + 1' because its domain is 'x is greater than or equal to 2
--> f(2) = 3(2) + 1 = 7
Hope that helps!
A bag contains 4 blue marbles, 3 white marbles, and 2 pink marbles. What is the probability of NOT drawing a white marble?
Answer:
2/3
Step-by-step explanation:
Total marbles:
4 + 3 + 2 = 9
Non-white marbles:
4 + 2 = 6
Probability of not drawing a white marble:
6/9
2/3
Answer:
6/9
Step-by-step explanation:
n=numerator
d=denominator
b=blue
p=pink
w=white
m=marbles
What we know:
4 bm
2pm
3 wm
what we need to know:
we need to know the denominator (all the numbers added up) and the numerator (the marbles that aren't white marbles).
work:
n= 4+2=6
d= 4+3+2=9
so it is 6/9
Can I please have Brainliest?
Match each expression with its value -5,-3,3 or undefined
G(0)
G(0.0001)
G(2.999)
G(3)
By using the given graph, we will see that:
G(0) = -5G(0.0001) = -3G(2.999) = -3G(3) = -3How to evaluate function G?Here we have a graph of function G(x), and we want to evaluate it in different values of x.
To do that, we need to look at the interval where the lines are defined. An open dot means that the endpoint does not belong to the interval, while the closed dot means that the endpoint belongs.
First, G(0).
If you look at the graph, when x = 0 we have a closed dot on the bottom line. That is the line that we need to look to do this evaluation, there we can see that when x = 0, we have G(0) = -5.
Then we have x = 2.999.
Notice that x < 3.
And the second line (the one at y = -3) is on the interval (0, 3].
Then x = 2.999 is included on that interval, it means that:
G(2.999) = -3.
Now we have x = 0.0001
Again, notice that the second line is on the interval (0, 3]
Because 3 > 0.0001 > 0, then:
0.0001 ∈ (0, 3]
From that, we conclude that:
G(0.0001) = -3
Finally:
G(3).
Again, 3 belongs to the interval (0, 3]
Then we use the second line again.
G(3) = -3
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Answer:
G(0) = -5
G(0.0001) = -3
G(2.999) = -3
G(3) = -3
Step-by-step explanation:
Find a particular solution y, of the following equation using the Method of Undetermined Coefficients. Primes denote the derivatives with respect to x. y'' + 17y=3e^8xA particular solution is y,(x) =
The general solution for the non-homogeneous equation is the sum of the homogeneous solution and the particular solution::
y(x) = c1 * cos(√-17i * x) + c2 * sin(√-17i * x) + (3/64) * e^8x
where c1 and c2 are arbitrary constants.
The method of undetermined coefficients is used to find a particular solution for a non-homogeneous linear ordinary differential equation of the form:
dy/dx + p(x)y = g(x)
where p(x) and g(x) are known functions. To use this method, we first find the general solution of the corresponding homogeneous equation:
dy/dx + p(x)y = 0
The general solution of this homogeneous equation can be written as:
y = c1 * y1(x) + c2 * y2(x)
where c1 and c2 are arbitrary constants and y1(x) and y2(x) are the linearly independent solutions of the homogeneous equation.
Next, we guess a particular solution yp of the non-homogeneous equation of the form:
yp = A * f(x)
where A is an unknown constant and f(x) is a function of x that matches the form of the forcing function g(x).
For the given equation: y'' + 17y = 3e^8x
The corresponding homogeneous equation is: y'' + 17y = 0
The characteristic equation is: r^2 + 17 = 0
The roots are: r = ±√-17i
So the general solution of the homogeneous equation is:
y = c1 * cos(√-17i * x) + c2 * sin(√-17i * x)
where c1 and c2 are arbitrary constants.
For the non-homogeneous equation, we guess a particular solution of the form:
yp = A * e^8x
where A is an unknown constant. Substituting this into the non-homogeneous equation, we have:
yp'' + 17yp = A * 8^2 * e^8x = A * 64e^8x
Comparing the right-hand sides of the two equations, we see that yp'' + 17yp = A * 64e^8x = 3e^8x, so A = 3/64.
Therefore, a particular solution for the non-homogeneous equation is:
y = yp = (3/64) * e^8x
The general solution for the non-homogeneous equation is the sum of the homogeneous solution and the particular solution:
y = c1 * cos(√-17i * x) + c2 * sin(√-17i * x) + (3/64) * e^8x
where c1 and c2 are arbitrary constants.
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Write x/5 = y/3 in standard form
Answer:
3x-5y=0
The standard form for linear equations in two variables is Ax+By=C, so just plug it in.
Answer:
Step-by-step explanation:
Standard form is ax+by=c
\(\frac{x}{5} =\frac{y}{3} \\\) cross multiply to get rid of fractions
3x=5y now bring the 5y to the other side, by subtracting both sides by 5y
3x-5y=0
find all values of c so that v = 1, 6, c and w = 1, −6, c are orthogonal. (enter your answers as a comma-separated list.
The only possible values of c that would make the vectors v and w orthogonal are the square roots of 35 and their negatives.
To find the values of c that make v and w orthogonal, we need to use the dot product formula:
v · w = (1)(1) + (6)(-6) + (c)(c) = 1 - 36 + \(c^2\)
We know that v and w are orthogonal when their dot product is equal to 0. So, we can set the equation we just formed equal to 0 and solve for c:
1 - 36 + \(c^2\) = 0
\(c^2\) = 35
c = ± √35
Therefore, the values of c that make v and w orthogonal are √35 and -√35. We can write the answer as a comma-separated list:
c = ± √35
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Marry me if you can answer this-
Please provide an explanation for this problem, thank you!Car A (traveling north at 70 mph) and car B (traveling west at 50 mph) are heading toward the same intersection. Car A is 5 miles from the intersection while car B is 6 miles from the intersection. Find parametric equations that describe the motion of cars A and B.A) Car A: x = 0, y = 70t-5; Car B: x = 6 - 50 t, y = 0B) Car A: x = 0, y = 50t - 6; Car B: x = 70t - 5, y = 0C) Car A: x = -70t + 5, y = 0; Car B: x = 6 - 50t, y = 0D) Car A: x = 50t - 6, y = 0; Car B: x = 0, y = 50 - 70t
The correct parametric equations for the motion of Car A and Car B are: Car A: x = 70t, y = 5 + 70t; Car B: x = 6 - 50t, y = 50t. Option C is correct.
Car A is traveling north at a speed of 70 mph. Therefore, its velocity vector can be represented as <0, 70>.
Car B is traveling west at a speed of 50 mph. Therefore, its velocity vector can be represented as <-50, 0>.
Let's assume that at time t = 0, Car A is at position (0, -5) and Car B is at position (6, 0).
Then, the position vectors for Car A and Car B at any time t can be given as:
Position of Car A = <0, -5> + t<0, 70> = <0, 70t-5>
Position of Car B = <6, 0> + t<-50, 0> = <6-50t, 0>
Therefore, the parametric equations for the motion of Car A and Car B are:
Car A: x = 0, y = 70t-5
Car B: x = 6 - 50t, y = 0
So, option C is the correct answer.
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a rectangular solid (with a square base) has a surface area of square centimeters. find the dimensions that will result in a solid with maximum volume.
The dimensions will be x = 7.5 cm and y = 7.5 cm where x is the side of the base and y is the height of the solid.
We are given that:
The solid has a square base.
So, the area of the base will be:
A = x² ( where x = side of the square)
Also, it is a rectangular solid, so let the height be y.
Volume will be:
V = x² × y
V = x²y
The surface area is given as:
S = 337.5 cm²
x² + x² + xy + xy + xy+ xy = 337.5 cm²
2x² + 4 x y = 337.5 cm²
y = (337.5- 2 x²) / (4 x)
Plug into the equation of volume:
V = x² (337.5- 2 x²) / (4 x)
V = (337.5 x- 2 x³) / 4
Differentiate with respect to x:
V' = (337.5 - 6 x²) / 4
Put it equal to 0:
337.5 - 6 x² = 0
x² = 337.5 / 6
x = ±7.5
Dimension will be:
x = 7.5 cm ( as it cannot be negative)
y = 7.5 cm.
Therefore, the dimensions will be x = 7.5 cm and y = 7.5 cm where x is the side of the base and y is the height of the solid.
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Your question was incomplete. Please refer the content below:
A rectangular solid (with a square base) has a surface area of 337.5 square centimeters. Find the dimensions that will result in a solid with maximum volume.
Which properties can be used to solve 7y − 15 = −29? Select all that apply. A. Identity Property of Multiplication
B. Addition Property of Equality
C. Distributive Property
D. Inverse Property of Multiplication
E. Commutative Property of Addition
Answer:
A and E
Step-by-step explanation:
Other 2 wouldn't make sense
Only definite of A
Addition Property of Equality ----If two expressions are equal to each other, and you add the same value to both sides of the equation, the equation will remain equal.
Commutative property of addition- changing the order of the numbers we are adding, does not change the sum. Here's an example of how the sum does NOT change, even if the order is changed
The properties that can be used to solve 7y - 15 = -29 are
Addition property of equality and inverse property of multiplication.
Given equation is 7y - 15 = -29.
We will solve for y and see which of the given following properties are applied.
What is addition property of equality and the inverse property of multiplication?Addition property of equality
If two expressions are equal to each other, and you add the same value to both sides of the equation, the equation will remain equal.
Inverse property of multiplication
It states that every non-zero x multiplied with 1/x will equal 1.
Solve for y.
7y - 15 = -29
We will add 17 on both sides of the equation.
7y - 15 + 15 = -29 + 15
7y = -14
Now, multiply 1/7 into both sides of the equation.
7y / 7 = -14 / 7
y = -2
While solving for y we see that we have to use the addition property of equality and inverse property of multiplication.
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a real estate agent believes that the value of houses in the neighborhood she works in has increased from last year. to test this claim, she randomly selects houses in this neighborhood and compares their estimated market value in the current year to their estimated market value in the previous year. suppose that data were collected for a random sample of 8 houses, where each difference is calculated by subtracting the market value of the previous year from the market value of the current year. assume that the values are normally distributed. using a test statistic of t≈7.496, the significance level α
The hypothesis is called the alternative hypothesis test.
According to the statement
We have to explain about the alternative hypothesis.
So, For this purpose, we know that the
The alternative hypothesis is one among ll|one amongst |one in every of} two mutually exclusive hypotheses in a hypothesis test. the choice hypothesis states that a population parameter doesn't equal a specified value.
From the given information:
she randomly selects houses during this neighborhood and compares their estimated market price within the current year to their estimated value within the previous year. suppose that data were collected for a random sample of 8 houses, where each difference is calculated by subtracting the market price of the previous year from the value of the present year.
Then
the alternative hypothesis test usually suggests that there's an opportunity of variation (difference) within the data observed.
Hence, since we are told the "real real estate agent believes that the values of homes within the neighborhood she works in have increased (the chance of variation) from last year," which "each difference is calculated by subtracting the market price of the previous year from the market price of this year," we are able to reach the conclusion that this can be an example of an alternate hypothesis test.
So, The hypothesis is called the alternative hypothesis test.
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what is the inverse of the function
f(x)=x/3-2
Answer:
Step-by-step explanation:
To find the inverse of the function f(x) = (x/3) - 2, we can follow these steps:
Step 1: Replace f(x) with y: y = (x/3) - 2.
Step 2: Interchange x and y: x = (y/3) - 2.
Step 3: Solve the equation for y.
To do this, we can start by isolating the y-term:
x + 2 = y/3.
Next, multiply both sides of the equation by 3 to eliminate the fraction:
3(x + 2) = y.
Simplifying further:
3x + 6 = y.
Finally, replace y with f^(-1)(x) to represent the inverse function:
f^(-1)(x) = 3x + 6.
Therefore, the inverse of the function f(x) = (x/3) - 2 is f^(-1)(x) = 3x + 6.
A banner is made of 8 equel parts five of the parts are green three of the parts are yellow
Your question is...?
The largest Ferry in the puget Sound fleet can carry 2,500 passengers every hour
Answer:
A: 2,500×H. B=20,000 passengers
round 13.43 to the nearest whole number.
Which of these strategies would eliminate a variable in the system of equations?
2x - 5y = 13
-3x+2y = 13
Choose all answers that apply:
Subtract the bottom equation from the top equation.
B
Multiply the top equation by 2, multiply the bottom equation by 3, then add the equations.
Multiply the top equation by 3. multiply the bottom equation by 2, then add the equations.
Allison works in a department store selling clothing. She makes a guaranteed salary of $350 per week, but is paid a commision on top of her base salary equal to 15% of her total sales for the week. How much would Allison make in a week in which she made $300 in sales? How much would Allison make in a week if she made xx dollars in sales?
how to do exponents
Answer:
I- I’m sorry I can’t help
Step-by-step explanation:
Answer:
395
If xx dollar then she will earn 350+ (0.15*xx)
~~~~~~~~~~~~~~~~~~~~
Double Check :)
If f(x,y,z)=ln(x^2y+sin^2(x+y))+125x^126y^2z^127, then ∂4f/∂x^2∂y∂z at (1,1,1) is equal to
__________
The value of ∂4f/∂x^2∂y∂z at (1,1,1) is -125. The partial derivative ∂4f/∂x^2∂y∂z is the fourth order partial derivative of f with respect to x, y, and z. It is evaluated at the point (1,1,1).
To calculate ∂4f/∂x^2∂y∂z, we can use the chain rule. The chain rule states that the partial derivative of a composite function is equal to the product of the derivative of the outer function and the derivative of the inner function.
In this case, the outer function is ln(x^2y+sin^2(x+y)) and the inner function is x^2y+sin^2(x+y). The derivative of the outer function is 1/(x^2y+sin^2(x+y)). The derivative of the inner function is 2xy + 2sin(x+y)*cos(x+y).
Using the chain rule, we get the following expression for ∂4f/∂x^2∂y∂z:
∂4f/∂x^2∂y∂z = (2xy + 2sin(x+y)*cos(x+y)) / (x^2y+sin^2(x+y))^2
Evaluating this expression at (1,1,1), we get the answer of -125.
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Calvin starts a business and has to
take out a loan of $500. He makes
a profit of $200 during the first
month and then for the next two
months records a profit of $-20
and the fourth month made a
profit of $300. What is the total
profit for the first four months of
Calvin's business?
Answer:
-40
Step-by-step explanation:
Rafael has two coupons for a phone.
Coupon A:
Coupon B:
$8 rebate on a $57 phone
20% off of a $57 phone
Choose the coupor that gives the lower price.
Then fill in the blank with the correct value.
Coupon A gives the lower price.
The price with coupon A is $] less than the price with coupon B.
Coupon B gives the lower price.
The price with coupon B is $] less than the price with coupon A.
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