Question

Jordan is tracking a recent online purchase. The shipping costs state that the item will be shipped in a 24-inch long box with a volume of 2,880 cubic inches. The width of the box is seven inches less than the height.

The volume of a rectangular prism is found using the formula V = lwh, where l is the length, w is the width, and h is the height.

Complete the equation that models the volume of the box in terms of its height, x, in inches.

Answer 1

Answer:

The expression for volume in terms of height 'x' is $V=24(x^2-7x)$.

The height of the box is 15 inches and width is 8 inches.

Step-by-step explanation:

Let the height of the box be 'x'.

Given:

Length of the box is, $l=24\ in$

Width is 7 less than its height. Therefore, width is, $w=x-7$

Volume of the box is, $V=2880\ in^3$

The volume of the box is given as:

$V=lwh\\2880=24\times (x-7)\times x\\x(x-7)=\frac{2880}{24}\\x^2-7x=120\\x^2-7x-120=0$

The above equation is a standard quadratic equation of the form $ax^2+bx+c=0$

Here, $a=1,b=-7,c=-120$

Using the quadratic formula to find the values of 'x', we get:

$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\\x=\frac{-(-7)\pm \sqrt{(-7)^2-4(1)(-120)}}{2(1)}\\x=\frac{7\pm \sqrt{49+480}}{2}\\x=\frac{7\pm \sqrt{529}}{2}\\x=\frac{7\pm 23}{2}\\x=\frac{7-23}{2}\ or\ x=\frac{7+23}{2}\\x=\frac{-16}{2}\ or\ x=\frac{30}{2}\\x=-8\ or\ x=15$

$x = -8$ is rejected as height can't be negative.

Therefore, the height of the box is 15 in.

Width of the box is, $w=15-7=8\ in$