Question

air conditioning a commercial air conditioner can cool a large room with a volume of up to 30120 cubic feet. a rectangular room in a warehouse is designed to have a length of 15 feet, a width of 30 feet, and a height of 16 feet. the builder wants to make the room larger by increasing the length and the width by the same amount. by how many feet can the length and width be increased and still not exceed the capacity of the air conditioner? round to the nearest tenth of a foot if necessary. \boxed{} feet

Explanation:

Step1: Define variable for increase

Let $x$ = number of feet added to length/width.

Step2: Set up volume inequality

The maximum volume is 36,200 cubic feet. The new length is $55+x$, new width is $35+x$, height stays 10.
$$(55+x)(35+x)(10) \leq 36200$$

Step3: Simplify the inequality

Divide both sides by 10:
$$(55+x)(35+x) \leq 3620$$
Expand the left side:
$$x^2 + 90x + 1925 \leq 3620$$
Rearrange to standard quadratic form:
$$x^2 + 90x - 1695 \leq 0$$

Step4: Solve quadratic equation

Use quadratic formula $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$ where $a=1$, $b=90$, $c=-1695$:
$$x = \frac{-90 \pm \sqrt{90^2 - 4(1)(-1695)}}{2(1)}$$
Calculate discriminant:
$$\sqrt{8100 + 6780} = \sqrt{14880} \approx 122.0$$
Take positive root (since length can't decrease):
$$x = \frac{-90 + 122.0}{2} = 16.0$$
Check if $x=16$ is valid:
$$(55+16)(35+16)(10) = 71 \times 51 \times 10 = 36210$$
This exceeds 36200, so test $x=15$:
$$(55+15)(35+15)(10) = 70 \times 50 \times 10 = 35000 \leq 36200$$

Answer:

15