Question

Suppose that a family wants to increase the size of the garden which is currently 16 feet long and 8 feet wide without changing the basic shape of the garden. That is, the family wants to ensure that the length of the garden remains twice the width (see figure).

1. Write an equation for the area of the new garden.
2. Write an equation that calculates how much more area the new garden will have as compared to the current garden. Simplify the equation into a quadratic equation in standard form.
3. Using the equation from Exercise 2, determine what effect increasing the width of the garden by two feet will have on the length of the garden and the total area of the garden.
4. Using the equation from Exercise 2 and factoring, determine how much the width of the garden will need to be increased in order for the garden area to be 160 square feet larger than its original area.
5. Using the equation from Exercise 2 and factoring, determine how much the width of the garden will need to be increased in order for the garden area to be four times its original area.

Answer 1

See below

Explanation:

Sides of the garden

• 16 feet and 8 feet

New dimensions of the garden

• 16 + 2x and 8 + x

1. The area of the new garden

• (16 + 2x)(8 + x) =
• 2x^2 + 32x + 128

2. The difference in area

• 2x^2 + 32x + 128 - 16*8 =
• 2x^2 + 32x

3. x= 2 effect on the area

• 2*2^2 + 32*2 =
• 8 + 64 =
• 72 ft²

4. Area difference = 160

• 2x^2 + 32x = 160
• x^2 + 16x - 80 = 0
• Solving we get positive root of 4 ft

5. Area difference = 4 times

• 2x^2 + 32x = 128*3
• 2x^2 + 32x - 384 = 0
• x^2 + 16x - 192= 0
• Solving we get positive root of 8 ft