In this set of exercises, you will use polynomials to study...

In this set of exercises, you will use polynomials to study real-world problems. A rectangular solid has height $h$ and a square base. One side of the square base is 3 inches greater than the height. (a) Find an expression for the volume of the solid in terms of $h$ (b) Sketch a graph of the volume function. (c) For what values of $h$ does the volume function make sense?

In this set of exercises, you will use polynomials to study real-world problems.
A rectangular solid has height $h$ and a square base. One side of the square base is 3 inches greater than the height.
(a) Find an expression for the volume of the solid in terms of $h$
(b) Sketch a graph of the volume function.
(c) For what values of $h$ does the volume function make sense?

Key Concepts

Volume Modeling

This concept involves using geometric formulas to represent the volume of a solid by relating its dimensions. In many problems, one or more dimensions are expressed in terms of a variable, which leads to an algebraic expression—often a polynomial—that models the physical quantity, such as volume, in a real-world setting.

Cubic Polynomial Functions

When the dimensions of a solid are defined in relation to one variable, the resulting volume expression can become a cubic polynomial. Such functions are useful in modeling because their algebraic structure allows for analysis of features like roots, turning points, and end behavior, which are crucial for understanding how changes in the variable affect the volume.

Domain Restrictions in Real-World Applications

In practice, mathematical models must be constrained by the conditions of the physical world. Determining the valid set of input values (the domain) ensures that the dimensions and derived quantities, such as the volume, are physically meaningful. For instance, restrictions such as positive lengths are imposed to reflect realistic scenarios.

Graphical Analysis of Functions

Graphing a function provides a visual interpretation of the mathematical relationship and behavior of a model. This includes identifying trends, intercepts, and the overall shape of the function, which is particularly beneficial in understanding how the modeled volume changes with respect to a dimension. It also verifies that the function behaves as expected within its restricted domain.

Transcript

00:01 We are told that we have a box.

00:04 And so we're given that the height of the box is h, and that we have two sides that are equal, and that they are both three more than the height.

00:16 So we have h plus three.

00:18 And so since we have a square base, they both are the same.

00:21 So we have h plus three, h plus three, and h.

00:24 So remember, volume is equal to length times width, times height.

00:31 And so since we have length times width times height, so we know we would have h times h plus 3 and then times another h plus 3.

00:40 So that can just make it, we can make it h plus 3 squared.

00:44 And so let's see if we can sketch this function.

00:49 And so we know we have, we have two variables.

00:53 So we have, we have, or sorry, we have two x intercepts, or in this case, h intercepts, if our if our graph looks something like, if our graph looks something like volume with respect to height.

01:07 So we have h is equal to 0, and we have h is equal to, we have h is equal to negative 3.

01:19 And so what this means is that we will have the volume function cross or touch or cross the x -axis, so for our h is equal to 0, we have a multiplicity of 1, which means since the multiplicity is odd, we will cross the x -axis.

01:42 And since we have h is equal to negative 3, a multiplicity of 2, we know that this will touch the x -axis.

01:50 So this will touch the x -axis.

01:53 And so let's look at when h approaches infinity.

01:58 So when h approaches infinity, what does our...

02:01 Volume function do.

02:02 So volume approaches.

02:06 So this we have infinity times an infinity squared.

02:10 So volume would approach positive infinity and if and if we have that we have h approaches we have h approaches negative infinity, well then we have a negative infinity times the squared term.

02:27 So remember the square term is always positive.

02:29 So we have that the volume would approach a negative times a positive, and therefore we have negative infinity.

02:37 And so now we can look at the y intercept, or in this case, the v intercept, the volume intercept.

02:46 And so that is just when h is equal to zero.

02:49 So if we have, we have zero times zero plus three, we have zero times zero plus three square...

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