Question

Milk is poured from a full rectangular container with dimensions 4 inches by 9 inches by 10 inches into a cylindrical container with a diameter of 6 inches. If the milk does not overflow, how many inches high will the milk reach? $$\begin{array}{l} \frac{60}{\pi} \\ 24 \\ \frac{40}{\pi} \\ 10 \\ 3 \pi \end{array}$$

Short answer

The height of the milk in the cylindrical container will be \( \frac{40}{\pi} \) inches.

Step-by-step solution

Volume of the Rectangular Container

First, calculate the volume of the rectangular container. Use the formula for the volume of a rectangular prism: \[ V = l \times w \times h \]Plug in the values:\[ V = 4 \text{ inches} \times 9 \text{ inches} \times 10 \text{ inches} \]This simplifies to:\[ V = 360 \text{ cubic inches} \]

Volume of the Cylindrical Container

Next, we need to find the height of the milk in the cylindrical container. Start by using the volume formula for a cylinder:\[ V = \frac{\text{volume of rectangular container}}{\text{height of milk}} = \text{base area} \times \text{height} \]where the base area of the cylindrical container is given by \[ \text{base area} = \frac{\text{volume of rectangular container}}{\text{height}} = \frac{360}{h}\]

Height of the Milk

Simplify this further by expressing the height of the milk as:\[ h = \frac{360}{\text{base area}} \]The diameter of the cylindrical container is 6 inches, so the radius is 3 inches. The area of the base is:\[ \text{base area} = \pi \times \text{radius}^2 = \pi \times (3)^2 = 9\pi \]Thus,\[ h = \frac{360}{9\pi} \]

Simplify the Expression

Finally, simplify the expression to get the height:\[ h = \frac{360}{9\pi} = \frac{40}{\pi} \]

Key concepts

Volume Calculation

Understanding volume calculation is a fundamental part of solving many geometry problems. Volume measures the amount of space an object occupies. In general, it’s computed by multiplying the area of the base by the height of the object.

For different shapes, different formulas are used. It's essential to know the correct formula for each shape to compute the volume accurately.

In this exercise, you'll see how to deal with volumes of both a rectangular prism and a cylinder.

Rectangular Prism Volume

A rectangular prism is a three-dimensional shape with six faces, all of which are rectangles. To calculate its volume, use the formula: \[ V = l \times w \times h \] where \( l \) is the length, \( w \) is the width, and \( h \) is the height.

In our example, the dimensions given are 4 inches (length), 9 inches (width), and 10 inches (height). Plugging in these values, we get: \[ V = 4 \times 9 \times 10 = 360 \text{ cubic inches} \] This volume tells us how much space the milk occupies before being poured into the cylindrical container.

Cylindrical Volume

A cylinder is a three-dimensional shape with two parallel circular bases connected by a curved surface. To find its volume, use the formula: \[ V = \text{base area} \times \text{height} \] The area of the base can be found using the formula for the area of a circle, \( \text{base area} = \pi \times r^2 \).

The problem we're solving involves a cylindrical container with a diameter of 6 inches, so the radius is 3 inches. The base area is: \[\text{base area} = \pi \times 3^2 = 9\text{pi} \]

Then, to find the height, we rearrange the volume formula: \[ h = \frac{V}{\text{base area}} \] Given the volume is the same as the rectangular container's, 360 cubic inches, we get: \[ h = \frac{360}{9\text{pi}} = \frac{40}{\pi} \] This tells us the height of the milk in the cylindrical container.

Mathematics Formulas

Mathematics formulas are essential tools that help solve various problems efficiently. Knowing how to apply these formulas correctly can make a big difference.

In this problem, we used these key formulas:

• Volume of a rectangular prism: \[ V = l \times w \times h \]
• Volume of a cylinder: \[ V = \pi \times r^2 \times h \]
• Area of a circle: \[\text{Area} = \pi \times r^2 \]

By correctly applying these formulas, we were able to find the volume of the milk initially in the rectangular container and subsequently find out how high the milk would reach in the cylindrical container after being poured.

Remember, understanding how and when to use these formulas is the key to solving similar problems on your GMAT!