Q16. - Algebra 1

Question

A hot air balloon was at a height of 60 feet above the ground when it began to ascend. The balloon climbed at a rate of 15 feet per minute.

 

a. Make a table that shows the height of the hot air balloon after climbing for 1, 2, 3, and 4 minutes.

 

b. Let t represent the time in minutes since the balloon began climbing. Write an algebraic equation for a sequence that can be used to find the height h, of the balloon after t minutes.

 

c. Use your equation from part b to find the height, in feet, of the hot air balloon after climbing for 8 minutes.

Step-by-Step Solution

Answer

a. 

 

Time in minutes	Height in feet
1	75
2	90
3	105
4	120

 

b. $h=15t+60$ is the required algebraic equation.

 

c. The hot air balloon would be at the height of 180 feet after climbing for 8 minutes.

Part a. Step 1. Derive the equation for the height h of the hot air balloon at time t minutes.

The balloon is climbing at a rate of 15 feet per minute.

Therefore, $\text{height }\left(h\right)=15\times \text{\hspace{0.17em}time\hspace{0.17em} }\left(t\right)$

The balloon was already at the height of 60 feet above the ground. 

Therefore, $h=15t+60$ is the required equation.

Part a. Step 2. Find the values of height h of the hot air balloon at time t = 1 , 2 , 3    &    4 minutes.

Substitute $t=1,2,3\&4$ in $h=15t+60$ to get the required heights.

At $t=1,$

$$\begin{gathered} h=15\left(1\right)+60 \\ =15+60 \\ h=75 \end{gathered}$$

At $t=2$,

$$\begin{gathered} h=15\left(2\right)+60 \\ =30+60 \\ h=90 \end{gathered}$$ 

At $t=3$, 

$$\begin{gathered} h=15\left(3\right)+60 \\ =45+60 \\ h=105 \end{gathered}$$

 

At $t=4$, 

$$\begin{gathered} h=15\left(4\right)+60 \\ =60+60 \\ h=120 \end{gathered}$$

Part a. Step 3. Write the values in tabular form.

The tabular form is given as follows.

 

Time in minutes	Height in feet
1	75
2	90
3	105
4	120

 

Part b. Step 1. State the concept of linear equation of the form h = m t + c

In the linear equation $h=mt+c$, m is the rate of change and c is the constant value.

Part b. Step 2. Derive the equation for the height h of the hot air balloon at time t minutes.

The balloon is climbing at a rate of 15 feet per minute.

Therefore, $m=15$

Therefore, height $\left(h\right)=15\times$ time (t)

The balloon was already at the height of 60 feet above the ground. 

Therefore, $c=60$

Therefore, $h=15t+60$ is the required equation.

Part b. Step 3. Write the required algebraic equation

$h=15t+60$ is the required algebraic equation.

Part c. Step 1. State the concept of linear equation of the form h = m t + c

In the linear equation $h=mt+c$

 

m is the rate of change and c is the constant value.

Part c. Step 2. Derive the equation for the height h of the hot air balloon at time t minutes.

The balloon is climbing at a rate of 15 feet per minute.

Therefore, $m=15$.

Therefore, height $\left(h\right)=15\times$ time (t)

The balloon was already at the height of 60 feet above the ground. 

Therefore, $c=60$.

Therefore, $h=15t+60$ is the required equation.

Part c. Step 3. Find the value of height h of the air balloon when time t = 8 minutes.

Substitute $t=8$ in the equation $h=15t+60$ to get the height of the air balloon.

$$\begin{gathered} h=15\left(8\right)+60 \\ =120+60 \\ h=180 \end{gathered}$$

Therefore, the hot air balloon would be at the height of 180 feet after climbing for 8 minutes.