Question

James wants to build a wooden barrel to hold rain water to use for irrigation. He wants the height of the barrel in feet to be 4 less
than the area of the base in square feet. He also wants the area of the base in square feet to be equal to its perimeter in feet. He
also needs to place a pump inside the barrel to move the collected water. After the pump is put inside the barrel, he needs it to
still hold at least 90 cubic feet of water
The cost for the materials to build the barrel will be 58 per square foot. Since the barrel is meant to catch rain water, he will not
need a top. The cost of the pump is proportional to its volume. For each cubic foot of volume that the pump takes up the cost
will be 550 James can only afford to spend up to $1,100 on this project
If x represents the area of the base of the barrel in square feet and y represents the volume of the pump in cubic feet, then which
of the following systems of inequalities can be used to determine the dimensions of the barrel and the volume of the pump?

Answer 1

Answer:

Step-by-step explanation:

To create a system of inequalities, write an inequality to model each condition that must be satisfied in the given situation.

It is given that x represents the area of the base of the barrel in square feet and y represents the volume of the pump in cubic feet.

Write an inequality to represent the amount of water that the barrel needs to hold after the pump is inserted. To find the volume of the barrel, multiply the area of the base by the height of the barrel. Then subtract the volume that will be taken up by the pump from that amount.

x(x-4) - y ≤ 90

To find the cost of the supplies for the barrel, find the surface area by multiplying the perimeter of the base by the height of the barrel and then adding the area of the base. Since there is no top, the area of the base only needs to be added once. Then, multiply the cost per square foot by the surface area.

To find the cost of the pump, multiply the cost per square foot of the pump by its volume.

Write an inequality representing the total cost. The cost must be less than or equal to $1,100.

8(x(x-4)+x) + 50y ≤ 1,100

Combining both of the inequalities gives the following system of inequalities.

x(x-4) - y ≤ 90

8(x(x-4)+x) + 50y ≤ 1,100