Answer :
Sure, let's solve the problem step-by-step.
### Step 1: Understand the Problem
We need to find out how long it will take an air pump to fill a queen-sized air mattress if it takes 5 minutes to fill a twin-sized air mattress.
### Step 2: Dimensions and Volume
First, let's consider the dimensions of the two mattresses:
- Twin-sized mattress dimensions: 39 inches (width) x 8.75 inches (height) x 75 inches (length)
- Queen-sized mattress dimensions: 60 inches (width) x 8.75 inches (height) x 80 inches (length)
The volume of each mattress is calculated by multiplying the three dimensions together (in cubic inches).
### Step 3: Calculate the Volume of Each Mattress
- Volume of the twin-sized mattress:
[tex]\[
39 \times 8.75 \times 75 = 25,593.75 \text{ cubic inches}
\][/tex]
- Volume of the queen-sized mattress:
[tex]\[
60 \times 8.75 \times 80 = 42,000 \text{ cubic inches}
\][/tex]
### Step 4: Calculate the Time to Fill the Queen-Sized Mattress
The air pump fills the twin mattress in 5 minutes. We can find out how many times bigger the queen mattress is compared to the twin mattress by taking the ratio of their volumes.
- Ratio of volumes (queen to twin):
[tex]\[
\frac{42,000}{25,593.75} \approx 1.6401
\][/tex]
Multiply the time it takes to fill the twin mattress by this ratio to find the time for the queen mattress:
- Time to fill the queen-sized mattress:
[tex]\[
5 \times 1.6401 \approx 8.21 \text{ minutes}
\][/tex]
Therefore, it will take approximately 8.21 minutes to fill a queen-sized air mattress.
### Step 1: Understand the Problem
We need to find out how long it will take an air pump to fill a queen-sized air mattress if it takes 5 minutes to fill a twin-sized air mattress.
### Step 2: Dimensions and Volume
First, let's consider the dimensions of the two mattresses:
- Twin-sized mattress dimensions: 39 inches (width) x 8.75 inches (height) x 75 inches (length)
- Queen-sized mattress dimensions: 60 inches (width) x 8.75 inches (height) x 80 inches (length)
The volume of each mattress is calculated by multiplying the three dimensions together (in cubic inches).
### Step 3: Calculate the Volume of Each Mattress
- Volume of the twin-sized mattress:
[tex]\[
39 \times 8.75 \times 75 = 25,593.75 \text{ cubic inches}
\][/tex]
- Volume of the queen-sized mattress:
[tex]\[
60 \times 8.75 \times 80 = 42,000 \text{ cubic inches}
\][/tex]
### Step 4: Calculate the Time to Fill the Queen-Sized Mattress
The air pump fills the twin mattress in 5 minutes. We can find out how many times bigger the queen mattress is compared to the twin mattress by taking the ratio of their volumes.
- Ratio of volumes (queen to twin):
[tex]\[
\frac{42,000}{25,593.75} \approx 1.6401
\][/tex]
Multiply the time it takes to fill the twin mattress by this ratio to find the time for the queen mattress:
- Time to fill the queen-sized mattress:
[tex]\[
5 \times 1.6401 \approx 8.21 \text{ minutes}
\][/tex]
Therefore, it will take approximately 8.21 minutes to fill a queen-sized air mattress.
