Desiree is making apple juice from concentrate. The cylindrical container of concentrate has a diameter of 7 centimmeters ans a height of 12 centimeters. To make the juice, the concentrate must be diluted with water so that the mix is 75 percent water and 25 percent concentrate.
What is the total volume of juice, rounded to the nearest cubic centimeter, that will be produced if Desiree follows the direction exactly ?

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Calculate the volume of the concentrate
The volume \( V \) of a cylinder can be calculated using the formula:
\[ V = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height.

Given:
- Diameter of the container = 7 cm, so the radius \( r = \frac{7}{2} = 3.5 \) cm
- Height \( h = 12 \) cm

Substituting the values into the formula:
\[ V = \pi (3.5)^2 (12) \]
Calculating \( (3.5)^2 = 12.25 \):
\[ V = \pi (12.25)(12) = 147 \pi \text{ cubic centimeters} \]

### Step 2: Determine the ratio of concentrate to juice
The mixture is 25% concentrate and 75% water. Therefore, the ratio of concentrate to the total mixture (juice) is:
\[ \text{Concentrate : Juice} = 25 : 75 = 1 : 3 \]
This means for every 1 part of concentrate, there are 3 parts of juice.

### Step 3: Calculate the total volume of juice
From the ratio, we can express the total volume of juice \( V_{juice} \) in terms of the volume of concentrate \( V_{concentrate} \):
\[ \frac{V_{concentrate}}{V_{juice}} = \frac{1}{4} \]
Thus, we can write:
\[ V_{juice} = 4 \times V_{concentrate} \]

Substituting \( V_{concentrate} = 147 \pi \):
\[ V_{juice} = 4 \times 147 \pi = 588 \pi \]

### Step 4: Calculate the numerical value of the total volume of juice
Using \( \pi \approx 3.14 \):
\[ V_{juice} \approx 588 \times 3.14 \approx 1847.52 \text{ cubic centimeters} \]

### Step 5: Round to the nearest cubic centimeter
Rounding \( 1847.52 \) gives us:
\[ V_{juice} \approx 1848 \text{ cubic centimeters} \]

### Final Answer
The total volume of juice produced is approximately **1848 cubic centimeters**.

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