Chapter 11, Problem 30CR

a.

To determine

To find: Scale factor.

Answer to Problem 30CR

Scale factor $=k=1.5$

Explanation of Solution

Given information:

Original dimensions of photograph is

  $5$ inch by $7$ inch.

That means

  $\left(x,y\right)=\left(5,7\right)$

Dimensions after enlargement are

  $\left(x\text{'},y\text{'}\right)=\left(7.5,10.5\right)$

Let $k$ be the scale factor The dilation is $\left(x,y\right)\to \left(kx,ky\right)$ .

Multiply the coordinates of each vertex by $k$ .

  $\left(5,7\right)\to \left(5k,7k\right)$

But we know dimensions after enlargement are $\left(7.5,10.5\right)$

Therefore,

On equating we get

  $\left(5k,7k\right)=\left(7.5,10.5\right)$

On comparing

We get

  $\begin{array}{l}5k=7.5\\ \Rightarrow \frac{5k}{5}=\frac{7.5}{5}\\ \Rightarrow k=\frac{7.5}{5}\\ \Rightarrow k=1.5\end{array}$

Therefore,

Scale factor $=k=1.5$

b.

To determine

To find: Scale factor.

Answer to Problem 30CR

Scale factor $=k=2$

Explanation of Solution

Given information:

Original dimensions of photograph is

  $5$ inch by $7$ inch.

That means

  $\left(x,y\right)=\left(5,7\right)$

Dimensions after enlargement are

  $\left(x\text{'},y\text{'}\right)=\left(10,14\right)$

Let $k$ be the scale factor The dilation is $\left(x,y\right)\to \left(kx,ky\right)$ .

Multiply the coordinates of each vertex by $k$ .

  $\left(5,7\right)\to \left(5k,7k\right)$

But we know dimensions after enlargement are $\left(10,14\right)$

Therefore,

On equating we get

  $\left(5k,7k\right)=\left(10,14\right)$

On comparing

We get

  $\begin{array}{l}5k=10\\ \Rightarrow \frac{5k}{5}=\frac{10}{5}\\ \Rightarrow k=\frac{10}{5}\\ \Rightarrow k=2\end{array}$

Therefore,

Scale factor $=k=2$

d.

To determine

To find: New dimensions after enlargement.

Answer to Problem 30CR

New dimensions of photograph are

  $11$ inch by $15.4$ inch

Explanation of Solution

Given information:

Original dimensions of photograph is

  $5$ inch by $7$ inch.

That means

  $\left(x,y\right)=\left(5,7\right)$

Scale factor $=k=2.2$

Let $k$ be the scale factor The dilation is $\left(x,y\right)\to \left(2.2x,2.2y\right)$ .

Multiply the coordinates of each vertex by $2.2$ .

  $\begin{array}{l}\left(5,7\right)\to \left(5\left(2.2\right),7\left(2.2\right)\right)\\ \Rightarrow \left(5,7\right)\to \left(11,15.4\right)\end{array}$

Therefore,

New dimensions of photograph are

  $11$ inch by $15.4$ inch