A rectangular piece of cardboard of size 16 inches by 20 inches is to be used in a factory to create boxes to transport apricots to market. The cardboard is formed into a box by cutting squares of dimensions x by x from each of the four corners and then folding up and taping the sides. The goal is to find a volume function, graph this function, and estimate the value of x that will maximize the volume of the box. a) Here is a plan view of the cardboard sheet and a three dimensional sketch of the taped box. Label the dimensions of the cutout square “x”. Label the dimensions (length, width, height) of the box in terms of x. Cardboard to be cut and folded: Resulting open-top Box: 20 in 16 in b) Write a Polynomial Function, V(x), that expresses the volume of the box in terms of x only. Factor this function fully. c) Find the x and Vintercepts of this function.

A rectangular piece of cardboard of size 16 inches by 20 inches is to be used in a factory to create boxes to transport apricots to market. The cardboard is formed into a box by cutting squares of dimensions x by x from each of the four corners and then folding up and taping the sides. The goal is to find a volume function, graph this function, and estimate the value of x that will maximize the volume of the box. a) Here is a plan view of the cardboard sheet and a three dimensional sketch of the taped box. Label the dimensions of the cutout square “x”. Label the dimensions (length, width, height) of the box in terms of x. Cardboard to be cut and folded: Resulting open-top Box: 20 in 16 in b) Write a Polynomial Function, V(x), that expresses the volume of the box in terms of x only. Factor this function fully. c) Find the x and Vintercepts of this function.

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

A rectangular piece of cardboard of size 16 inches by 20 inches is to be used in a factory to create boxes to
transport apricots to market. The cardboard is formed into a box by cutting squares of dimensions x by x
from each of the four corners and then folding up and taping the sides. The goal is to find a volume
function, graph this function, and estimate the value of x that will maximize the volume of the box.
a) Here is a plan view of the cardboard sheet and a three dimensional sketch of the taped box. Label the
dimensions of the cutout square "x". Label the dimensions (length, width, height) of the box in terms of x.
Cardboard to be cut and folded:
Resulting open-top Box:
20 in
16 in
b) Write a Polynomial Function, V(x), that expresses the volume of the box in terms of x only. Factor
this function fully.
c) Find the x and Vintercepts of this function.

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Continuation of this math problem

d) Plot the x and V intercepts on the Cartesian Plane. Label the axes and title the graph. Clearly show
your scale.
e) Construct a table of values and complete the graph. You may use a calculator!
f) Estimate the value of x that will maximize the volume of the box. Identify this on the graph.

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