Chapter 3, Problem 13PFA

a.

To determine

To find: Height of the hot air balloon after climbing for 1,2,3,4 minutes.

Answer to Problem 13PFA

Height of balloon for climbing 1,2,3,4 minutes is 75,90,105,120 feet respectively

Explanation of Solution

Given information:

Initial height of hot air balloon = 60 feet

Balloon starts to ascend.

Balloon is climbing at the rate of 15 feet per minute.

From given information

Balloon is climbing at the rate of 15 feet per minute

It means

  $1$ minute $\to$ 15 feet

Multiplying with $t$ on both sides

We get

  $\begin{array}{l}\Rightarrow 1\times t\to 15\times t\\ \Rightarrow t\to 15t\end{array}$

Therefore

For $t$ minutes it climbs $15t$ feet.

Therefore

Final height after $t$ minutes = Initial height + height climbed in $t$ minutes.

Let $h\left(t\right)$ be final height

  $\Rightarrow h\left(t\right)=60+15t$

Substitute $t=1$

We get

  $\begin{array}{l}h\left(t\right)=60+15t\\ \Rightarrow h\left(1\right)=60+15\left(1\right)\\ \Rightarrow h\left(1\right)=60+15\\ \Rightarrow h\left(1\right)=75\end{array}$

Substitute $t=2$

We get

  $\begin{array}{l}h\left(t\right)=60+15t\\ \Rightarrow h\left(2\right)=60+15\left(2\right)\\ \Rightarrow h\left(2\right)=60+30\\ \Rightarrow h\left(2\right)=90\end{array}$

Substitute $t=3$

We get

  $\begin{array}{l}h\left(t\right)=60+15t\\ \Rightarrow h\left(3\right)=60+15\left(3\right)\\ \Rightarrow h\left(3\right)=60+45\\ \Rightarrow h\left(3\right)=105\end{array}$

Substitute $t=4$

We get

  $\begin{array}{l}h\left(t\right)=60+15t\\ \Rightarrow h\left(4\right)=60+15\left(4\right)\\ \Rightarrow h\left(4\right)=60+60\\ \Rightarrow h\left(4\right)=120\end{array}$

Therefore

Height of balloon for climbing 1,2,3,4 minutes is 75,90,105,120 feet respectively.

b.

To determine

To find: Algebraic equation to find height.

Answer to Problem 13PFA

Algebraic equation to represent height is $h\left(t\right)=60+15t$ .

Explanation of Solution

Given information:

Initial height of hot air balloon = 60 feet

Balloon starts to ascend.

Balloon is climbing at the rate of 15 feet per minute.

From given information

Balloon is climbing at the rate of 15 feet per minute

It means

  $1$ minute $\to$ 15 feet

Multiplying with $t$ on both sides

We get

  $\begin{array}{l}\Rightarrow 1\times t\to 15\times t\\ \Rightarrow t\to 15t\end{array}$

Therefore

For $t$ minutes it climbs $15t$ feet.

Therefore

Final height after $t$ minutes = Initial height + height climbed in $t$ minutes.

Let $h\left(t\right)$ be final height

  $\Rightarrow h\left(t\right)=60+15t$

Therefore, algebraic equation to represent height is $h\left(t\right)=60+15t$

c.

To determine

To find: Height of the air balloon for climbing for 8 minutes.

Answer to Problem 13PFA

Height of air balloon after climbing for 8 minutes is 180 feet.

Explanation of Solution

Given information:

Initial height of hot air balloon = 60 feet

Balloon starts to ascend.

Balloon is climbing at the rate of 15 feet per minute.

From given information

Balloon is climbing at the rate of 15 feet per minute

It means

  $1$ minute $\to$ 15 feet

Multiplying with $t$ on both sides

We get

  $\begin{array}{l}\Rightarrow 1\times t\to 15\times t\\ \Rightarrow t\to 15t\end{array}$

Therefore

For $t$ minutes it climbs $15t$ feet.

Therefore

Final height after $t$ minutes = Initial height + height climbed in $t$ minutes.

Let $h\left(t\right)$ be final height

  $\Rightarrow h\left(t\right)=60+15t$

Substitute $t=8$

  $\begin{array}{l}h\left(t\right)=60+15t\\ \Rightarrow h\left(8\right)=60+15\left(8\right)\\ \Rightarrow h\left(8\right)=60+120\\ \Rightarrow h\left(8\right)=180\end{array}$

Height of air balloon after climbing for 8 minutes is 180 feet.

d.

To determine

To find:What mathematical practice we used to solve this problem.

Explanation of Solution

Given information:

Initial height of hot air balloon = 60 feet

Balloon starts to ascend.

Balloon is climbing at the rate of 15 feet per minute.

Using rate of change concept we obtained the dependent variable(height) in terms of independent variable(time).