A right circular cylinder has a height of $$ 201 / 2 \mathrm{ft} $$. and a diameter of $$ 13 / 4 $$ times its height. What is the volume of the cylinder? Enter your answer as a decimal in the box. Do not enter units. Use 3.14 for pi and round only your final answer to the nearest hundredth.

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UpStudy AI · Accepted Answer

To find the volume of a right circular cylinder, we can use the formula:

$$
V = \pi r^2 h
$$

where $$ V $$ is the volume, $$ r $$ is the radius, and $$ h $$ is the height of the cylinder.

### Step 1: Extract known conditions
- Height $$ h = \frac{201}{2} $$ ft
- Diameter $$ d = \frac{13}{4} \times h $$

### Step 2: Calculate the diameter
First, we need to calculate the diameter using the height:

$$
d = \frac{13}{4} \times \frac{201}{2}
$$

### Step 3: Calculate the radius
The radius $$ r $$ is half of the diameter:

$$
r = \frac{d}{2}
$$

### Step 4: Substitute and calculate the volume
Now we can substitute $$ r $$ and $$ h $$ into the volume formula.

Let's perform these calculations step by step.

1. Calculate the diameter $$ d $$.
2. Calculate the radius $$ r $$.
3. Calculate the volume $$ V $$.

Let's start with the calculations.

1. Calculate $$ d $$:
$$
d = \frac{13}{4} \times \frac{201}{2}
$$

2. Calculate $$ r $$:
$$
r = \frac{d}{2}
$$

3. Calculate $$ V $$:
$$
V = \pi r^2 h
$$

Now, I will perform these calculations.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{13}{4}\times \frac{201}{2}$$
- step1: Multiply the fractions:
  $$\frac{13\times 201}{4\times 2}$$
- step2: Multiply:
  $$\frac{2613}{8}$$
Calculate or simplify the expression $$ (13/4)*(201/2)/2 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\frac{13}{4}\times \frac{201}{2}}{2}$$
- step1: Multiply the numbers:
  $$\frac{\frac{2613}{8}}{2}$$
- step2: Multiply by the reciprocal:
  $$\frac{2613}{8}\times \frac{1}{2}$$
- step3: Multiply the fractions:
  $$\frac{2613}{8\times 2}$$
- step4: Multiply:
  $$\frac{2613}{16}$$
Calculate or simplify the expression $$ 3.14*(163.3125^2)*(201/2) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$3.14\times 163.3125^{2}\times \frac{201}{2}$$
- step1: Convert the expressions:
  $$3.14\left(\frac{2613}{16}\right)^{2}\times \frac{201}{2}$$
- step2: Multiply the terms:
  $$\frac{31557}{100}\left(\frac{2613}{16}\right)^{2}$$
- step3: Evaluate the power:
  $$\frac{31557}{100}\times \frac{2613^{2}}{16^{2}}$$
- step4: Multiply the fractions:
  $$\frac{31557\times 2613^{2}}{100\times 16^{2}}$$
- step5: Multiply:
  $$\frac{31557\times 2613^{2}}{25600}$$
### Step 1: Calculate the diameter
The diameter $$ d $$ is calculated as follows:

$$
d = \frac{13}{4} \times \frac{201}{2} = 326.625 \text{ ft}
$$

### Step 2: Calculate the radius
The radius $$ r $$ is half of the diameter:

$$
r = \frac{d}{2} = \frac{326.625}{2} = 163.3125 \text{ ft}
$$

### Step 3: Calculate the volume
Now we can substitute $$ r $$ and $$ h $$ into the volume formula:

$$
V = \pi r^2 h
$$

Substituting the values:

$$
V = 3.14 \times (163.3125)^2 \times \left(\frac{201}{2}\right)
$$

Calculating this gives:

$$
V \approx 8416558.84
$$

### Final Step: Round to the nearest hundredth
Rounding $$ 8416558.84 $$ to the nearest hundredth gives:

$$
\boxed{8416558.84}
$$

This is the volume of the cylinder.
The volume of the cylinder is approximately 8416558.84.

A right circular cylinder has a height of \( 201 / 2 \mathrm{ft} \). and a diameter of \( 13 / 4 \) times its height. What is the volume of the cylinder? Enter your answer as a decimal in the box. Do not enter units. Use 3.14 for pi and round only your final answer to the nearest hundredth.

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