Algebra I Help - Word Problem

Question

An electricity facility has three square solar panel fields. The diagram shows their arrangement and areas (in square meters). The diagram has three boxes  4x^2 + 40x + 100  and x^2 and x^2 + 40x + 400.

 

a.) How many meters longer is the side length of box 3 than the side length of box 2? Explain how you found your answer. (Hint: you need to find the side lengths of each box first)

 

b.) For what value of x is the area of box 1 equal to the combined area of box 2 and box 3?

 

c.) The length of box 1 is 10 meters longer than the side length of Grid 3. What is the total area of three grids? Explain how you found your answer.

Answer 1

(a) Let's find the side lengths of each box. Since these boxes are square, and Area = side2, we can take the square root of the area to find the length of a side. Factor each equation:

4x2 + 40x + 100
(2x + 10)(2x + 10)
(2x + 10)2

First box's side length: 2x + 10 meters

x2

Second box's side length: x meters

x2 + 40x + 400
(x + 20)(x + 20)
(x + 20)2

Third box's side length: x + 20

So, how many meters longer is the third box than the second box? Subtract:

(x + 20) - x
x + 20 - x
20 meters


(b) Let's set the area of box 1 equal to the areas of box 2 and 3 and solve for x:

4x2 + 40x + 100 = x2 + x2 + 40x + 400
4x2 + 40x + 100 = 2x2 + 40x + 400
2x2 + 40x + 100 = 40x + 400
2x2 + 100 = 400
2x2 = 300
x2 = 150
x = √150 ≈ 3.87 meters


(c) By "Grid 3," I'll assume you just mean box 3.

The length of box 1 is 10 meters longer than the length of box 3. So:

2x + 10 - 10 = x + 20
2x = x + 20
x = 20

Plug x = 20 into each area equation and sum them together:

4(20)2 + 40(20) + 100 + (20)2 + (20)2 + 40(20) + 400
4(400) + 800 + 100 + 400 + 800 + 400
1600 + 900 + 1200 + 400
2500 + 1600
4100 square meters