Question

an open top box is to be made from a rectangular piece of cardboard, measuring 15 in. by 9 in,, by cutting equal squares from the corners and turning up the sides. what is the value of \\( x \\) that would represent the side length of each square cut that would produce the box with the greatest volume? round to the nearest hundredth.

Explanation:

Step1: Define Volume Function

Let \( x \) be the side length of the square cut from each corner. The length of the box will be \( 15 - 2x \), the width will be \( 9 - 2x \), and the height will be \( x \). The volume \( V \) of a box is given by \( V=\text{length}\times\text{width}\times\text{height} \), so \( V(x)=(15 - 2x)(9 - 2x)x \). Expand this: \( V(x)=(135-30x - 18x + 4x^{2})x=4x^{3}-48x^{2}+135x \), where \( 0\lt x\lt4.5 \) (since \( 9 - 2x\gt0\Rightarrow x\lt4.5 \)).

Step2: Find Derivative of Volume Function

To find the maximum volume, we take the derivative of \( V(x) \) with respect to \( x \). Using the power rule, \( V^\prime(x)=\frac{d}{dx}(4x^{3}-48x^{2}+135x)=12x^{2}-96x + 135 \).

Step3: Solve for Critical Points

Set \( V^\prime(x) = 0 \) to find critical points: \( 12x^{2}-96x + 135 = 0 \). Divide through by 3: \( 4x^{2}-32x + 45 = 0 \). Use the quadratic formula \( x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a} \), where \( a = 4 \), \( b=-32 \), \( c = 45 \). Calculate the discriminant: \( \Delta=b^{2}-4ac=(-32)^{2}-4\times4\times45=1024 - 720 = 304 \). Then \( x=\frac{32\pm\sqrt{304}}{8}=\frac{32\pm4\sqrt{19}}{8}=\frac{8\pm\sqrt{19}}{2} \). Calculate the two roots: \( x_1=\frac{8+\sqrt{19}}{2}\approx\frac{8 + 4.3589}{2}\approx6.179 \) (which is greater than 4.5, so we discard it) and \( x_2=\frac{8-\sqrt{19}}{2}\approx\frac{8 - 4.3589}{2}\approx1.8205 \).

Step4: Verify Maximum (Optional)

We can use the second - derivative test. The second derivative \( V^{\prime\prime}(x)=24x - 96 \). At \( x=\frac{8 - \sqrt{19}}{2}\approx1.82 \), \( V^{\prime\prime}(1.82)=24\times1.82-96\approx43.68 - 96=-52.32\lt0 \), so the function is concave down at this point, meaning it is a local maximum. Since this is the only critical point in the interval \( (0,4.5) \), it is the global maximum.

Answer:

\( 1.82 \)