If (x,y) is the solution to the system of equations above, what is the value of x?

The rate of change of:

total revenue is $500 per day

total cost is $40 per day

total profit is $460 per day

To find the rate of change of total revenue, cost, and profit with respect to time, we need to take the derivative of the given functions with respect to x and then multiply by the rate of change of x with respect to time (dx/dt).

Total revenue (R) = 60x – 0.5x²
dR/dx = 60 – x
When x = 35, dR/dx = 60 – 35 = 25
Rate of change of total revenue = (dR/dx) * (dx/dt) = 25 * 20 = $500 per day

Total cost (C) = 2x+ 10
dC/dx = 2
When x = 35, dC/dx = 2
Rate of change of total cost = (dC/dx) * (dx/dt) = 2 * 20 = $40 per day

Total profit (P) = R - C
dP/dx = (dR/dx) - (dC/dx)
When x = 35, dP/dx = 25 - 2 = 23
Rate of change of total profit = (dP/dx) * (dx/dt) = 23 * 20 = $460 per day

Therefore, the rate of change of total revenue is $500 per day, the rate of change of total cost is $40 per day, and the rate of change of total profit is $460 per day.

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The surface area of revolution about the x-axis of y=4x+2 over the interval 2 is approximately 88.99 square units.

To find the surface area of revolution about the x-axis of y=4x+2 over the interval 2, we first need to express the equation in terms of x.

Rearranging the equation, we get x = (y-2)/4.

Next, we need to determine the limits of integration.

Since we are rotating about the x-axis, the limits of integration are the x-values, which in this case are 0 and 2.

Using the formula for the surface area of revolution, S = 2π∫(y√(1+(dy/dx)^2))dx, we can plug in the values we have found.

dy/dx for y=4x+2 is simply 4, so we get:

S = 2π∫(4x+2)√(1+16)dx from 0 to 2

Simplifying this, we get:

S = 2π∫(4x+2)√17 dx from 0 to 2

Evaluating this integral using calculus, we get:

S = 32π√17/3

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1 hour =60 minutes

Step-by-step explanation:

Let M be the time in minutes . T be temperature in Farhenheit. From 45th min to end of the hour there remains a steady temperature. after that gyms starts to cools down . For time 45≤M≤60, Temperature T=77oF.To find a) Is M a function of T ? we know that Temperature changes with respect to time . So M is independent variable and T is dependent variable . so M cannot be a function of T .

Answer:

25

Step-by-step explanation:

let x = the amount of money that shelly has.

let y = the amount of money that john has.

if shelly give john 5 dollars, then they both have the same amount of money.

this leads to the equation:

x-5 = y+5

if john give shelly 5 dollars, then shelly has twice as much money as john has.

this leads to the equation:

x+5 = 2(y-5)

solve for x in each equation to get:

x-5 = y+5 leads to:

x = y+10

x+5 = 2(y-5) leads to:

x+5 = 2y-10 which becomes:

x = 2y-15

you have 2 expressions that are equal to x.

they are:

x = y+10

x = 2y-15

you can set these expressions equal to each other to get:

y+10 = 2y-15

subtract y from both sides of this equation and add 15 to both sides of this equation to get:

y = 25

since x = 2y-15, this leads to:

x = 2(25)-15 which becomes:

x = 35

the equation x = y + 10 leads to the same answer of:

y =35

you have:

x = 25

y = 35

Step-by-step explanation:

d = a2 - a1

d = 11 - 5

d = 6

[tex]Sn \: = \frac{n}{2} (2a1 + (n - 1)d) \\ S7 = \frac{7}{2} (2 \times 11 + (7 - 1)6) \\ S7 = \frac{7}{2} (22 + 36) \\ S7 = \frac{7}{2} \times 58 \\ S7 = 7 \times 29 \\ S7 = 203[/tex]

The first and fourth sequences show a geometric series.

Consequently, the sequences 2, 3, 4.5, 6.75, … and 4, 8, 16, 32, … are the geometric series.

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Complete Question:-

Which of the following show a geometric series? select all that apply.

1) 2 3 4.5 6.75 …

2) 2 2.5 3 3.5 …

3) 6 7 9 12 …

4) 4 8 16 20 …

5) 8 4 2 1 …

Answer:

Step-by-step explanation:

To find the length of a'b', we first need to know the scale factor of the dilation. The scale factor is given by the ratio of the corresponding side lengths in the original and diluted figures.

In this case, we are given that the original figure Aabc has been diluted by a factor of √2. So the length of each side in the dilated figure aa'b'c is √2 times the length of the corresponding side in Aabc.

To find the length of a'b, we can use the Pythagorean theorem in the right triangle aa'b'. Since we know that ab is one of the legs of this triangle, we can find its length as follows:

ab = (a'b' / √2) * sin(28°)

We are not given the length of ab or a in the original figure, so we cannot find their exact values. However, we can find the measure of angle A using the Law of Sines in triangle Aab:

sin(A) / ab = sin(62°) / b

where b is the length of side bc in Aabc. Solving for sin(A) and substituting the expression for ab that we found earlier, we get:

sin(A) = (sin(62°) / b) * [(a'b' / √2) * sin(28°)]

Since we know the values of sin(62°) and sin(28°), we can simplify this expression and use a value for b (if it is given in the problem) to find sin(A) and then A.

Based on the inscribed angle theorem, the measure of angle JKL in the circle is: m<JKL = 121°.

Where an inscribed angle is subtended by an arc it intercepts, the measure of the inscribed angle is equal to half of the the measure of the intercepted arc in the circle, based on the inscribed angle theorem.

Angle JKL is the inscribed angle that is subtended by arc JML. Find the measure of arc JML.

Measure of arc JML = 360 - 53 - 65

Measure of arc JML = 242°

m<JKL = 1/2(measure of arc JML) [inscribed angle theorem]

Substitute:

m<JKL = 1/2(242)

m<JKL = 121°

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Answer:

The distance between these parallel lines is 2 - (-7) = 9 units.

Answer:

21.5

Step-by-step explanation:

First we can solve for c using the pythagoreom theorem. (probably didn't spell that right)

A squared + B squared =  C squared

9 squared + 3 squared = c squared

81+9= c squared

90=c squared

90 square root is (rounded to the nearest tenth) 9.5

c=9.5

Then we can add 9.5+9+3= 21.5

The domain of the function f(x) = 1/x - 8 is (-∞, 0) U (0, +∞), and f = 1/x - 8.

The function f(x) is defined as 1/x - 8. The domain of the function is the set of all possible values of x for which the function is defined. Since the function involves division by x, x cannot be equal to zero. Therefore, the domain of f(x) is (-∞, 0) U (0, +∞), which means that x can take any value except 0.

To find f(x), we simply substitute the expression for f(x) in the definition of the function. Therefore, we have:

f(x) = 1/x - 8

This is the final answer. We cannot simplify it any further. The function f(x) represents a hyperbola with a vertical asymptote at x = 0 and a horizontal asymptote at y = -8.

As x approaches 0 from the left, f(x) goes to negative infinity, and as x approaches 0 from the right, f(x) goes to positive infinity. Similarly, as x approaches positive or negative infinity, f(x) approaches 0.

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The function f(z) = 1 + 6x + 96x^-1 has one local minimum and one local maximum. This function has a local maximum at x. The function f(x) = 1 + 6x + 96x^(-1) has a local maximum at x = -4.

To find the local maximum of the function f(x) = 1 + 6x + 96x^(-1), we first need to find the critical points. We do this by finding the first derivative of the function and setting it equal to zero.

Step 1: Find the derivative of the function
f'(x) = d/dx (1 + 6x + 96x^(-1))
f'(x) = 6 - 96x^(-2)

Step 2: Set the derivative equal to zero and solve for x
6 - 96x^(-2) = 0

Step 3: Solve for x
96x^(-2) = 6
x^(-2) = 6/96
x^(-2) = 1/16
x^2 = 16
x = ±4

Step 4: Determine which of the critical points is a local maximum. To do this, we will examine the second derivative of the function.

f''(x) = d^2/dx^2 (1 + 6x + 96x^(-1))
f''(x) = 192x^(-3)

Now we will evaluate the second derivative at each critical point:

f''(4) = 192(4)^(-3) = 3 > 0, so x = 4 is a local minimum.
f''(-4) = 192(-4)^(-3) = -3 < 0, so x = -4 is a local maximum.

Therefore, the function f(x) = 1 + 6x + 96x^(-1) has a local maximum at x = -4.

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The total inventory costs consist of two parts: the cost of storing the shoes and the cost of reordering the shoes. The cost of storing the shoes is given by the formula:

Cost of storing = $6 per shoe pair per year x number of shoe pairs

Since the shoes are stored for a year, this cost is incurred annually. The cost of reordering the shoes is given by the formula:

Cost of reordering = $14 per order + $7 per shoe pair x number of shoe pairs

This cost is incurred each time the store places an order with the manufacturer.

Let x be the number of shoe pairs in each order from the manufacturer. The number of orders needed to sell 1750 shoe pairs each year is given by:

Number of orders = 1750 shoe pairs / x shoe pairs per order

The total inventory costs can be expressed as:

Total cost = Cost of storing + Cost of reordering

Substituting the formulas for the two costs and the expression for the number of orders, we get:

Total cost = $6 per shoe pair per year x 1750 shoe pairs + ($14 per order + $7 per shoe pair x x shoe pairs) x (1750 shoe pairs / x shoe pairs per order)

Simplifying this expression, we get:

Total cost = $10,500 + $14(1750/x) + $7(1750)

Total cost = $10,500 + $24,500/x

Therefore, the function that models the total inventory costs as a function of x is:

Total cost(x) = $10,500 + $24,500/x

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The surface area of the given regular pyramid is 84 cm^2.

To find the surface area of a regular pyramid, we need to calculate the area of each face and add them together. A regular pyramid has a base that is a regular polygon, and its lateral faces are triangles that meet at a common vertex. We can use the Pythagorean theorem to find the slant height of the pyramid, which is the height of each lateral face.

Let's assume that the base of the regular pyramid is a square with side length 6 cm, and the slant height is 4 cm.

First, we need to find the area of the base of the pyramid:

Area of the base = (side length)^2

                = 6 cm x 6 cm

                = 36 cm^2

Next, we need to find the area of each triangular lateral face. Since the pyramid is a regular pyramid, all the triangular faces are congruent.

We can find the area of each triangular face using the formula:

Area of a triangle = (1/2) x base x height

The base of each triangular face is equal to the side length of the square base, which is 6 cm. The height of each triangular face is equal to the slant height, which is 4 cm.

Area of each triangular face = (1/2) x 6 cm x 4 cm

                            = 12 cm^2

Since the pyramid has 4 triangular faces, we need to multiply the area of one triangular face by 4 to get the total area of all the triangular faces:

Total area of the triangular faces = 4 x 12 cm^2

                                 = 48 cm^2

Finally, we can find the total surface area of the pyramid by adding the area of the base and the area of the triangular faces:

Total surface area = Area of the base + Total area of the triangular faces

                  = 36 cm^2 + 48 cm^2

                  = 84 cm^2

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Answer:

Problem 8:
Option C)       DE = √21

Problem 9:
Option C)     Perimeter = 30 cm

Step-by-step explanation:

1. Problem 8

Draw a line segment connecting the center of the circle, X, to A. This is the radius of the circle

The points ABX form a right triangle with AX as the hypotenuse and AB, BX as the legs of the right triangle

By the Pythagorean theorem
hypotenuse² = sum of the squares of the two legs

Plugging in line segment references

AX² = AB² + BX²

We are given AC = 8, BX = 3

Since the segment BX intersects AC at right angles, AB = BC = AC/2

So AB = 8/2= 4

Plugging these values into the Pythagorean formula
AX² = AB² + BX²

AX² = 4² +3²

AX² = 16 + 9

AX² = 25

Draw a line connecting X and D
The points DEX form a right triangle with DX as the hypotenuse and EX and DE as the legs

Again, by the Pythagorean theorem

DX² = DE² + EX²

But DX is the radius of the circle so it must be the same as AX

Substitute for DX in terms of AX
DX² = DE² + EX²

=> AX² = DE² + EX²

But AX² = 25 and EX = 2 giving EX² = 4

Therefore

AX² = DE² + EX² becomes
25 = DE² + 4

DE² = 25 - 4

DE² = 21

DE = √21

This is choice C

Problem 9

To find the perimeter, just add up the individual side lengths:

Perimeter = 10 + 8 + 7 + 5 = 30 cm

Option C

Answer: 8

Step-by-step explanation:

The solution to the equation is x = 15.

To rewrite the exponential equation using the same base, we need to express both 8 and 32 as powers of the same base. Since both 8 and 32 are powers of 2, we can rewrite the equation as:

[tex](2^3)^(2x) = (2^5)^(x+3)[/tex]

Here, we used the fact that[tex](a^b)^c = a^(b*c)[/tex]to simplify the exponents. We also used the property that 8 is equal to 2 raised to the power of 3, and 32 is equal to 2 raised to the power of 5.

Now that we have rewritten the equation with the same base, we can equate the exponents on both sides of the equation to solve for x:

[tex]2^(6x) = 2^(5x + 15)[/tex]

Since the bases on both sides of the equation are equal, we can equate the exponents and solve for x:

6x = 5x + 15

Subtracting 5x from both sides, we get:

x = 15

Therefore, the solution to the equation is x = 15.

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A. Melissa's monthly payment is approximately $146.89.

B. The total amount of the monthly payments over the four-year period is approximately $7,052.72.

C. The finance charge on the loan is approximately $1,052.72.

The interest on a loan or principal amount can be easily calculated using simple interest. Simple interest is a notion that is employed across a wide range of industries, including banking, finance, automobiles, and more.

To calculate the monthly payment, we can use the formula:

Monthly payment = [P x r x (1 + r)ⁿ] / [(1 + r)ⁿ⁻¹]

where P is the principal (the amount of the loan), r is the monthly interest rate, and n is the number of monthly payments.

A. To find the monthly payment, we first need to calculate the values of r and n.

Since the APR is 10%, the monthly interest rate is 10% / 12 = 0.00833 (rounded to 5 decimal places).

The number of monthly payments over four years is 4 x 12 = 48.

Using these values, we can calculate the monthly payment:

Monthly payment = [6000 x 0.00833 x (1 + 0.00833)⁴⁸] / [(1 + 0.00833)⁴⁸⁻¹]

Monthly payment ≈ $146.89

So Melissa's monthly payment is approximately $146.89.

B. To find the total amount of the monthly payments, we can multiply the monthly payment by the number of payments:

Total amount of monthly payments = Monthly payment x Number of payments

Total amount of monthly payments = $146.89 x 48

Total amount of monthly payments ≈ $7,052.72

So the total amount of the monthly payments over the four-year period is approximately $7,052.72.

C. To find the finance charge, we can subtract the original principal from the total amount of the monthly payments:

Finance charge = Total amount of monthly payments - Principal

Finance charge = $7,052.72 - $6,000

Finance charge ≈ $1,052.72

So the finance charge on the loan is approximately $1,052.72.

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Declan swims faster than Madison, with a speed of 7/2 miles per hour compared to Madison's speed of 2/5 miles per hour.

Declan swims faster than Madison. While Madison swims 2/5 mile in an hour, which means her speed is 2/5 miles per hour, Declan swims 7/10 mile in 1/5 hour, which means his speed is 7/2 miles per hour. This indicates that Declan's speed is greater than Madison's speed.

In fact, we can see that Declan's speed is almost 4 times faster than Madison's speed. This means that Declan can cover a greater distance in the same amount of time compared to Madison.

Therefore, based on the given information, we can confidently say that Declan swims faster than Madison.

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Answer: 8

Step-by-step explanation:

CF looks like the radius of the circle.

ED looks like the diameter.

.: ED = 2 x CF = 2 x 4 = 8

.: ED = 8

Answer:

The length of ED, or the diameter, is 8

Step-by-step explanation:

As the other person explained, CF is the radius, as the radius is from the centermost point to the edge.  I also see that ED is the diameter, as the diameter is from edge to edge, going through the centermost point.  Therefore, since the diameter is double the radius, we can solve this with the following equation:

2 * r = d, where r is radius and d is diameter.

2 * 4 = d

8 = d

With regard to the similar triangles,

In ΔACD

BE ∥ CD

In ΔACD and ΔABE

BE ∥ CD

∠ACD =∠ABE           (corresponding angles)

∠ADC = ∠AEB          (corresponding angles)

∠A = ∠A                    (common angle)

ACD ∼ ΔABE

So,  The corresponding sides are in proportion.

Now, find ED

AB/BC = AE/ED
ED = AE (BC/AB)
ED = 26(5/20)
ED = 6.5cm


For BE

AB/AC = BE/CD
BE = CD (AB/AC)
BE = 18 (20/25)
BE  = 14.4cm

Now, find BE

Therefore, the length of ED is 6.5cm and the length of BE is 14.4 cm.

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A. Approximately 9.85% of New York City commutes are less than 30 minutes

B. Approximately 16.91% of New York City commutes are between 30 and 35 minutes.

A. To find the percent of New York City commutes that are less than 30 minutes, we need to calculate the z-score using the formula:

z = (x - μ) / σ

where x is the value we are interested in (30 minutes), μ is the mean commute time (39.7 minutes), and σ is the standard deviation (7.5 minutes).

z = (30 - 39.7) / 7.5 = -1.29

We can use a standard normal distribution table or calculator to find the area to the left of z = -1.29, which gives us:

P(z < -1.29) = 0.0985

Therefore, approximately 9.85% of New York City commutes are less than 30 minutes.

B. To find the percent of New York City commutes that are between 30 and 35 minutes, we need to calculate the z-scores for both values using the same formula:

z1 = (30 - 39.7) / 7.5 = -1.29

z2 = (35 - 39.7) / 7.5 = -0.62

We can then find the area between these two z-scores using a standard normal distribution table or calculator, which gives us:

P(-1.29 < z < -0.62) = P(z < -0.62) - P(z < -1.29) = 0.2676 - 0.0985 = 0.1691

Therefore, approximately 16.91% of New York City commutes are between 30 and 35 minutes.

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The factors that help to determine sample size are the desired confidence level and the margin of error. Therefore, the correct option is D) both b and c.

The desired confidence level and margin of error are two important factors that help to determine the sample size. The confidence level represents the level of certainty that the sample mean is close to the true population mean, while the margin of error is the range of error that is acceptable in the estimation of the population mean.

Both of these factors are interdependent, and an increase in either of them would require a larger sample size to achieve a certain level of accuracy. Therefore, carefully considering these factors and determining an appropriate sample size is essential for obtaining valid and reliable results.

The population size can also have an impact on the sample size calculation, but it is not a direct factor. So, the correct answer is D).

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The maximum value of f on the ellipse is approximately 1.779 and the minimum value is approximately -1.779.

To find the maximum and minimum values of f(x, y) = 3x + y on the ellipse x^2 + 4y^2 = 1, we can use the method of Lagrange multipliers.

First, we define the Lagrangian function as L(x, y, λ) = 3x + y - λ(x^2 + 4y^2 - 1). We then find the partial derivatives of L with respect to x, y, and λ and set them equal to zero:

∂L/∂x = 3 - 2λx = 0

∂L/∂y = 1 - 8λy = 0

∂L/∂λ = x^2 + 4y^2 - 1 = 0

Solving these equations simultaneously, we obtain the critical points (±1/3√5, ±1/√20). We can then evaluate f at these critical points to find the maximum and minimum values:

f(1/3√5, 1/√20) ≈ 0.593

f(1/3√5, -1/√20) ≈ -0.593

f(-1/3√5, 1/√20) ≈ 1.779

f(-1/3√5, -1/√20) ≈ -1.779

Intuitively, the Lagrange multiplier method allows us to optimize a function subject to a constraint, which in this case is the ellipse x^2 + 4y^2 = 1.

The critical points of the Lagrangian function are the points where the gradient of the function is parallel to the gradient of the constraint, which correspond to the maximum and minimum values of the function on the ellipse.

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The particular solution of the given differential equation with the initial condition f(6) = -2.

To find the particular solution of the given differential equation dy/dx = (x-3)/y with the initial condition f(6) = -2, we first need to solve the differential equation. This is a first-order separable equation, so we can rewrite it as:

y dy = (x - 3) dx

Now, integrate both sides:

∫y dy = ∫(x - 3) dx

(1/2)y^2 = (1/2)x^2 - 3x + C

Now, apply the initial condition f(6) = -2:

(1/2)(-2)^2 = (1/2)(6)^2 - 3(6) + C

(1/2)(4) = (1/2)(36) - 18 + C

2 = 18 - 18 + C

C = 2

So the particular solution is:

(1/2)y^2 = (1/2)x^2 - 3x + 2

This is the particular solution of the given differential equation with the initial condition f(6) = -2.

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The 21st digit in the decimal expansion of 1/7 is 2 and the 5280th digit in the decimal expansion of 5/17 is 5.

a. To find the 21st digit in the decimal expansion of 1/7 we need to find the decimal expansion. The decimal expansion of 1/7 is a repeating decimal

= 1/7 = 0.142857142857142857…

The sequences 142857 repeat indefinitely. To find the 21st digit, we can divide 21 by the length of the repeating sequence,

= 21 / 6 = 3

Therefore, the third digit in the repeating sequence is 2

b.To find the 5280th digit in the decimal expansion of 5/17 we need to find the decimal expansion. The decimal expansion of 5/17 is a repeating decimal is

=  5/17 = 0.2941176470588235294117647…

The repeating sequences are 2941176470588235

The 5280th digit = 5280 / length of the repeating sequence,

5280 / 16 = 0

Therefore, the 5280th digit is the last digit in the repeating sequence, which is 5.

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Answer:

the probability of selecting a blue sock from the laundry basket is 2/7 or approximately 0.2857.

Step-by-step explanation:

The probability of selecting a blue sock can be found by dividing the number of blue socks by the total number of socks in the basket:

Probability of selecting a blue sock = Number of blue socks / Total number of socks

Probability of selecting a blue sock = 4 / 14

Simplifying the fraction by dividing both the numerator and denominator by 2 gives:

Probability of selecting a blue sock = 2 / 7

Answer:

Para comparar las propuestas de los bancos, debemos llevar las tasas de interés a una misma unidad de tiempo. Podemos convertir la tasa del Banco A de bimestral a mensual multiplicándola por 2 (ya que hay 6 bimestres en 1 año):

Tasa del Banco A: 1,5% * 2 = 3% mensual

Tasa del Banco B: 0,5% mensual

Para calcular los intereses que se obtendrán en cada banco, podemos utilizar la fórmula del interés compuesto:

I = C * ((1 + r/n)^(n*t) - 1)

Donde:

I es el interés

C es el capital inicial

r es la tasa de interés en forma decimal

n es el número de veces que se capitaliza al año

t es el tiempo en años

Para el Banco A, como la tasa está en meses, capitalizaremos mensualmente (n=12):

I = 40 000 * ((1 + 0,03/12)^(12*2) - 1) = 4 896,18 soles

Para el Banco B, como la tasa ya está en meses, capitalizaremos mensualmente (n=12):

I = 40 000 * ((1 + 0,005)^(12*2) - 1) = 4 225,48 soles

Por lo tanto, la propuesta más conveniente es la del Banco A, ya que ofrece una tasa de interés mayor y genera un interés total de 4 896,18 soles. En cambio, el Banco B genera un interés total de 4 225,48 soles.

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Answer:

  (D)  7/10

Step-by-step explanation:

You want the rate of change of y with respect to x for the relation ...

  (2/5)x -(4/7)y = 3/2

Solving for y, we have ...

  2/5x -3/2 = 4/7y . . . . . . . . . . add 4/7y -3/2

  (7/4)(2/5)x -(7/4)(3/2) = y . . . . multiply by 7/4

  7/10x -21/8 = y . . . . . . . . . . simplify

The rate of change is the coefficient of x: 7/10.