Question

select the correct answer.
rachel has a box in the shape of a prism with a square base. the capacity of the box is 150 cubic inches. the height of the box (y) is $1\frac{1}{2}$ times the base length of the box (x). find the approximate height and the length of the box.
\\(\bigcirc\\) a. \\(x = 4\\) and \\(y \approx 6\\)
\\(\bigcirc\\) b. \\(x = 5\\) and \\(y = 7.5\\)
\\(\bigcirc\\) c. \\(x = 4.2\\) and \\(y = 6.3\\)
\\(\bigcirc\\) d. \\(x = 4.6\\) and \\(y = 6.9\\)

Explanation:

Step1: Set up height relation

$y = 1\frac{1}{2}x = \frac{3}{2}x$

Step2: Volume formula for prism

Volume $V = x^2 y = 150$

Step3: Substitute y into volume

$x^2 \times \frac{3}{2}x = 150 \implies \frac{3}{2}x^3 = 150$

Step4: Isolate $x^3$

$x^3 = 150 \times \frac{2}{3} = 100$

Step5: Solve for x

$x = \sqrt[3]{100} \approx 4.64$

Step6: Calculate y

$y = \frac{3}{2} \times 4.64 \approx 6.96$

Answer:

D. $x=4.6$ and $y=6.9$