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.
Desiree is going to serve the apple juice in a cylindrical pitcheer with a diameter of 10 centimeters. What is the minimum height of the pitcher, rounded to the nearest centimeter, required for it to hold the apple juice that Desiree made ?

Answer

To find the minimum height of the pitcher required to hold the apple juice made by Desiree, we will follow these steps:

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

The diameter of the concentrate container is 7 cm, so the radius \( r \) is:
\[ r = \frac{7}{2} = 3.5 \text{ cm} \]
The height \( h \) of the concentrate container is 12 cm.

Now, substituting these values into the volume formula:
\[ V = \pi (3.5)^2 (12) = \pi (12.25)(12) = 147\pi \text{ cm}^3 \]

### Step 2: Determine the ratio of concentrate to juice
According to the problem, the juice is made up of 25% concentrate and 75% water. Therefore, the ratio of the volume of concentrate to the total volume of juice is:
\[ \text{Volume of concentrate} : \text{Volume of juice} = 25 : 100 = 1 : 4 \]

### Step 3: Calculate the total volume of juice
Let \( V_j \) be the volume of juice. From the ratio, we have:
\[ \frac{V_c}{V_j} = \frac{1}{4} \]
where \( V_c \) is the volume of concentrate. We know \( V_c = 147\pi \), so:
\[ \frac{147\pi}{V_j} = \frac{1}{4} \]
Cross-multiplying gives:
\[ V_j = 4 \times 147\pi = 588\pi \text{ cm}^3 \]

### Step 4: Calculate the volume of the pitcher
The pitcher has a diameter of 10 cm, so the radius \( r \) is:
\[ r = \frac{10}{2} = 5 \text{ cm} \]
The volume \( V_p \) of the pitcher is given by:
\[ V_p = \pi r^2 h = \pi (5)^2 h = 25\pi h \]

### Step 5: Set the volume of the pitcher equal to the volume of juice
To find the height \( h \) of the pitcher, we set the volume of the pitcher equal to the volume of juice:
\[ 25\pi h = 588\pi \]
Dividing both sides by \( \pi \):
\[ 25h = 588 \]
Now, solving for \( h \):
\[ h = \frac{588}{25} = 23.52 \text{ cm} \]

### Step 6: Round to the nearest centimeter
Rounding \( 23.52 \) to the nearest centimeter gives:
\[ h \approx 24 \text{ cm} \]

### Final Answer
The minimum height of the pitcher required to hold the apple juice is **24 cm**.

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