Question

The height of the doorway is 7 feet less than 5 times its width. if the area is 24 square feet, find the dimensions of the door

(use polynomials)

i feel like it’s 8 and 3 but i don’t know if it’s right or if im even doing it right in the first place

Answer 1

height: 8 ft

width: 3 ft

Explanation:

You can check your answer against the problem criteria:

area = (8 ft)(3 ft) = 24 ft^2

height = -7 +5(3) = -7 +15 = 8 . . . . your height

Your numbers match the requirements.

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The problem statement asks you to "use polynomials." You have not shown work, so we don't know whether you did. (I like to use a graphing calculator.)

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Let x represent the width and y represent the height, both in feet.

y = -7 +5x . . . . . height is 7 ft less than 5 times width

xy = 24 . . . . . . . area is 24 square feet

Using the first equation to substitute for y in the second equation, we have ...

x(-7+5x) = 24

5x^2 -7x -24 = 0 . . . . . put in standard form (the required polynomial)

5x^2 -15x +8x -24 = 0 . . . . . facilitate factoring by grouping*

5x(x -3) +8(x -3) = 0

(5x +8)(x -3) = 0

The positive solution is x = 3. Then y = 24/3 = 8.

The dimensions are 3 ft wide by 8 ft high.

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* To factor this, we look for factors of 5·(-24) = -120 that sum to -7. Those factors are -15 and +8. There are several ways to get to the binomial factors once you have these numbers, but the "factor by grouping" method is the most easily explained.