Question
A hose dispenses 1,413 cm of water every minute into a cone with a radius of 60 cm and a height of 150 cm.
How long will it take to fill the cone?
Use 3.14 to approximate pi.
Enter your answer in the box.
How long will it take to fill the cone?
Use 3.14 to approximate pi.
Enter your answer in the box.
Answers
GPT-4o mini
To determine how long it will take to fill the cone with the water, we first need to find the volume of the cone. The formula for the volume \( V \) of a cone is:
\[
V = \frac{1}{3} \pi r^2 h
\]
where:
- \( r \) is the radius,
- \( h \) is the height, and
- \( \pi \) is approximately 3.14.
Given:
- \( r = 60 \) cm,
- \( h = 150 \) cm.
Now, we can substitute these values into the volume formula:
\[
V = \frac{1}{3} \times 3.14 \times (60)^2 \times 150
\]
Calculating \( (60)^2 \):
\[
(60)^2 = 3600
\]
Now substituting that back into the volume calculation:
\[
V = \frac{1}{3} \times 3.14 \times 3600 \times 150
\]
Calculating \( 3600 \times 150 \):
\[
3600 \times 150 = 540000
\]
Now substitute that in:
\[
V = \frac{1}{3} \times 3.14 \times 540000
\]
Calculating \( \frac{1}{3} \times 540000 \):
\[
\frac{540000}{3} = 180000
\]
Continuing with the volume calculation:
\[
V = 3.14 \times 180000
\]
Now calculating \( 3.14 \times 180000 \):
\[
3.14 \times 180000 = 565200
\]
Therefore, the volume of the cone is:
\[
V = 565200 \text{ cm}^3
\]
Next, we need to find out how long it takes to fill the cone with water dispensed by the hose. The hose dispenses \( 1413 \) cm³ of water per minute.
To find how many minutes it will take to fill the cone, we divide the volume of the cone by the rate at which water is dispensed:
\[
\text{Time} = \frac{\text{Volume of cone}}{\text{Rate}} = \frac{565200 \text{ cm}^3}{1413 \text{ cm}^3/\text{min}}
\]
Calculating that:
\[
\text{Time} \approx 400 \text{ minutes}
\]
Therefore, it will take approximately **400 minutes** to fill the cone.
\[
V = \frac{1}{3} \pi r^2 h
\]
where:
- \( r \) is the radius,
- \( h \) is the height, and
- \( \pi \) is approximately 3.14.
Given:
- \( r = 60 \) cm,
- \( h = 150 \) cm.
Now, we can substitute these values into the volume formula:
\[
V = \frac{1}{3} \times 3.14 \times (60)^2 \times 150
\]
Calculating \( (60)^2 \):
\[
(60)^2 = 3600
\]
Now substituting that back into the volume calculation:
\[
V = \frac{1}{3} \times 3.14 \times 3600 \times 150
\]
Calculating \( 3600 \times 150 \):
\[
3600 \times 150 = 540000
\]
Now substitute that in:
\[
V = \frac{1}{3} \times 3.14 \times 540000
\]
Calculating \( \frac{1}{3} \times 540000 \):
\[
\frac{540000}{3} = 180000
\]
Continuing with the volume calculation:
\[
V = 3.14 \times 180000
\]
Now calculating \( 3.14 \times 180000 \):
\[
3.14 \times 180000 = 565200
\]
Therefore, the volume of the cone is:
\[
V = 565200 \text{ cm}^3
\]
Next, we need to find out how long it takes to fill the cone with water dispensed by the hose. The hose dispenses \( 1413 \) cm³ of water per minute.
To find how many minutes it will take to fill the cone, we divide the volume of the cone by the rate at which water is dispensed:
\[
\text{Time} = \frac{\text{Volume of cone}}{\text{Rate}} = \frac{565200 \text{ cm}^3}{1413 \text{ cm}^3/\text{min}}
\]
Calculating that:
\[
\text{Time} \approx 400 \text{ minutes}
\]
Therefore, it will take approximately **400 minutes** to fill the cone.