A farmer’s harvest in bushels varies directly with the number of acres planted.

A. Express in sentence form what this means to the farmer.

B.Translate the variation statement into the corresponding variation equation.

C. If 8 acres can produce 144 bushels, find the constant of variation, k.

D. Rewrite the variation equation using the results from part c).

E. Use this equation to determine how many acres are needed to produce 1296 bushels.

64

A. This means that as the number of acres planted increases, the farmer's harvest also increases by the same ratio.

B. The variation equation can be written as: harvest = k * acres planted, where k is the constant of variation.

C. To find the constant of variation, we can use the given information: 8 acres can produce 144 bushels.
So, we have: 144 = k * 8.
Solving for k, we get k = 144 / 8 = 18.

D. Rewriting the variation equation using the constant of variation from part C, we have: harvest = 18 * acres planted.

E. To determine how many acres are needed to produce 1296 bushels using the variation equation, we can substitute the values: 1296 = 18 * acres planted.
Solving for acres planted, we get acres planted = 1296 / 18 = 72 acres.

A. This means that the farmer's harvest will increase or decrease proportionally with the number of acres planted. If the farmer plants more acres, the harvest will be larger, and if the farmer plants fewer acres, the harvest will be smaller.

B. To translate the variation statement into the corresponding variation equation, we can use the equation: harvest = k * acres, where k represents the constant of variation.

C. Given that 8 acres can produce 144 bushels, we can substitute these values into the variation equation to find the constant of variation:
144 = k * 8

To solve for k, divide both sides of the equation by 8:
k = 144 / 8
k = 18

Therefore, the constant of variation, k, is 18.

D. The variation equation can be written as:
harvest = k * acres

Using the result from part C, the rewritten variation equation becomes:
harvest = 18 * acres

E. To determine how many acres are needed to produce 1296 bushels, we can use the variation equation:
harvest = 18 * acres

Now we can substitute the desired harvest value into the equation and solve for the number of acres:
1296 = 18 * acres

To solve for acres, divide both sides of the equation by 18:
acres = 1296 / 18
acres = 72

Therefore, 72 acres are needed to produce 1296 bushels.