Certainly! Let's solve the problem step by step:
### a. What is the volume of the paper cup?
The volume [tex]\( V \)[/tex] of a cone can be calculated using the formula:
[tex]\[ V = \frac{1}{3} \pi r^2 h \][/tex]
where:
- [tex]\( r \)[/tex] is the radius of the base of the cone
- [tex]\( h \)[/tex] is the height of the cone
- [tex]\( \pi \)[/tex] is a constant (approximately 3.14159)
We are given:
- The diameter of the cone is 2 centimeters.
- The height of the cone is 5 centimeters.
Firstly, we need to determine the radius [tex]\( r \)[/tex] of the cone. The radius is half of the diameter:
[tex]\[ r = \frac{\text{diameter}}{2} = \frac{2}{2} = 1 \text{ centimeter} \][/tex]
Now we can substitute the radius [tex]\( r \)[/tex] and the height [tex]\( h \)[/tex] into the volume formula:
[tex]\[ V = \frac{1}{3} \pi (1)^2 (5) \][/tex]
[tex]\[ V = \frac{1}{3} \pi (1 \times 1) \times 5 \][/tex]
[tex]\[ V = \frac{1}{3} \pi \times 5 \][/tex]
[tex]\[ V \approx \frac{1}{3} \times 3.14159 \times 5 \][/tex]
[tex]\[ V \approx 5.235987755982988 \text{ cubic centimeters} \][/tex]
So, the volume of the paper cup is approximately [tex]\( 5.24 \)[/tex] cubic centimeters.
### b. How long does it take for the cup to fill with water? Round your answer to the nearest tenth.
Given:
- The rate of water flow is 1.5 cubic centimeters per second.
To find the time [tex]\( t \)[/tex] it takes to fill the cup, we use the formula:
[tex]\[ t = \frac{\text{Volume of the cup}}{\text{Rate of water flow}} \][/tex]
We already calculated the volume of the cup, which is [tex]\( 5.235987755982988 \)[/tex] cubic centimeters. Substituting this and the rate of water flow into the formula:
[tex]\[ t = \frac{5.235987755982988}{1.5} \][/tex]
[tex]\[
t \approx 3.490658503988659 \text{ seconds}
\][/tex]
Rounding this to the nearest tenth:
[tex]\[ t \approx 3.5 \text{ seconds} \][/tex]
Therefore, it takes approximately 3.5 seconds for the cup to fill with water.
