A graphing calculator is recommended. In this problem you are asked to find a fu

Question

A graphing calculator is recommended. In this problem you are asked to find a function that models a real-life situation and then use the model to answer questions about the situation. Use the guidelines on page 237 to help you. A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 12 in, by 20 in. by cutting out equal squares of side x at eech corner and then folding up the sides (see the figure). 20 in. 12 in. (a) Find a function that models the volume V of the box. (b) Find the values of x for which the volume is greater than 230 in2. (Rgund your answers to three decimal places. Enter your answer using interval notation) (c) Find the largest volume that such a box can have. (Round your answer to three decimal places) in2

Explanation / Answer

solution:

the volume of a box is V = width (w) * height(h) * length (l)

If you draw your box you will notice that the height will be x and that the height and length depend on x. In fact l = 20-2x and w = 12 - 2x

So the volume is:

V(x) = whl = (12-2x)(x)(20-2x)

expand to get

V(x) = 4x^3 - 64x^2 + 240x

[We can know find what value of x gives the largest volume by differentiating and finding where the gradient of the curve is zero.

V' = 12x^2 - 128x + 240

0 = 12x^2 - 128x + 240

x = 8.4 or 2.4

Sub these back into our equation for V(x) and see which gives the bigger volume...V(8.4) = -129.024 which is obviously incorrect as it's negative so the answer must be x = 2.4 which gives a volume of 262.656.

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.