Chapter 18, Problem 17A

a. If the linear dimensions of an object are doubled, how much does the total area increase?

b. How much does the volume increase?

To determine

The increase in the total area of the object due to the change in dimensions.

Answer to Problem 17A

The total area of the object is increases four times, when the linear dimensions are doubled.

Explanation of Solution

Given:

Linear dimensions of an object are doubled.

Formula used:

The area $\left(A\right)$ of the object is,

  $A=lb$

Calculation:

When the linear dimensions of the object is increased by a factor x, then the surface area of the object will be increased by a factor of $x^2$ .

Consider the linear dimensions- length, breadth, and height of the object are denoted by $l$, $b$, and $h$ respectively.

The area $\left(A\right)$ of the object is,

  $A=lb$

Consider the linear dimensions of the object are doubled.

The new linear dimensions of the object- length, breadth, and height of the object are denoted by $2l$, $2b$, and $2h$ respectively.

The new area $\left(A^{\prime }\right)$ of the object is,

  $\begin{array}{l}{A}^{\prime }=\left(2l\right)\left(2b\right)\\ A^{\prime }=4lb\\ A^{\prime }=4A\end{array}$

The new area is four times the initial area of the object.

Conclusion:

Thus, the total area of the object is increases four times, when the linear dimensions are doubled.

To determine

The increase in the total volume of the object for the given changes.

Answer to Problem 17A

The volume area of the object increases eight times, when the linear dimensions are doubled.

Explanation of Solution

Given:

Linear dimensions of an object are doubled.

Formula used:

The volume $\left(V\right)$ of the object is,

  $V=lbh$

Calculation:

When the linear dimensions of the object is increased by a factor $x$ , then the volume of the object will be increased by a factor of $x^3$ .

Consider the linear dimensions- length, breadth, and height of the object are denoted by $l$, $b$, and $h$ respectively.

The initial volume $\left(V\right)$ of the object is,

  $V=lbh$

Consider the linear dimensions of the object are doubled.

The new linear dimensions of the object- length, breadth, and height of the object are denoted by $2l$, $2b$, and $2h$ respectively.

The new volume $\left(V^{\prime }\right)$ of the object is,

  $\begin{array}{l}{V}^{\prime }=\left(2l\right)\left(2b\right)\left(2h\right)\\ V^{\prime }=8lbh\\ V^{\prime }=8V\end{array}$

The new volume is eight times the initial volume of the object.

Conclusion:

Thus, the total volume of the object is increases eight times, when the linear dimensions are doubled.