Question

The dimensions of a rectangle is 30cm x 20cm. When each dimension is

decreased by the same amount, the area of the new rectangle is
100cm^2. What are the new dimensions of the new rectangle (round to
one decimal place)? Hint: you will need to use the quadratic equation.

Answer 1

The new dimensions are 6.18 cm by 16.18 cm.

Explanation:

Original dimensions were 30 cm by 20 cm.

We decrease length and width by x and calculate the area:

Area = (length)(width)

= (30 - x)(20 - x) = 100

Performing the indicated multiplication, we get:

600 - 30x - 20x + x^2 = 100, or, after simplification,

x^2 - 50x + 500 = 0

Let's solve this by completing the square:

x^2 - 50x + 500 = x^2 - 50x + 625 - 625 + 500 = 0

This simplifies to (x - 25)^2 - 125 = 0, or (x - 25)^2 = 125

Taking the square root of both sides, we get:

x - 25 = ±√125, or

x = 25 ± 5√5

The two results are x = 36.18 (not possible, because we DECREASED the original dimensions) and x = 13.82 (possible)

The dimensions of the new rectangle are

(30 - 13.82) cm by (20 - 13.82) cm, or

16.2 cm by 6.18 cm

Check: With these dimensions is the area 100 cm^2, as expected?

(6.18)(16.18) = 99.9979 YES