91Ó°ÊÓ

Understanding how to calculate the volume of a rectangular prism is an essential skill when tackling many GMAT math problems related to volume. A rectangular prism is a 3-dimensional shape with six faces, all of which are rectangles. Think of it like a box. To find its volume, you multiply its length, width, and height. This formula looks like this: \[ V = \text{Length} \times \text{Width} \times \text{Height} \] So, if you have a rectangular box with a length of 6 feet, a width of 5 feet, and a height of 7 feet, you simply plug these values into the formula to find the volume: \[ V = 6 \times 5 \times 7 = 210 \text{ cubic feet} \] This calculated volume is crucial because it is the starting point for determining the volume of other shapes related to the problem.

Calculating the volume of a cone is slightly different from calculating the volume of a rectangular prism. A cone is a 3-dimensional shape with a circular base and a pointed top. The volume of a cone is determined by the formula: \[ V = \frac{1}{3} \pi r^2 h \] Where:

• \( \pi \) is approximately 3.14159.
• \( r \) is the radius of the base.
• \( h \) is the height of the cone.

In this problem, we know the radius of the conical box's base is 6 feet and the volume is 630 cubic feet. Using these values and the known formula, you solve for the height \( h \) of the cone: \[ \frac{1}{3} \pi (6)^2 h = 630 \] Reorganizing the equation allows us to isolate \( h \): \[ 36\pi h = 1890 \] \[ h = \frac{1890}{36\pi} \approx 16.7 \text{ feet} \] Understanding the formula for a conical volume helps slice through what might seem like a complex problem.

Knowing the rate at which water flows into a container is key to solving "how long will it take to fill" problems. Here, we're dealing with a water flow rate of 6 cubic feet per minute. This means that every minute, 6 cubic feet of water are added to the box. To find out how long it takes to fill the entire conical box, use the formula: \[ \text{Time} = \frac{\text{Volume}}{\text{Rate}} \] In our problem, we already know that the conical box has a total volume of 630 cubic feet. We just need to divide this by the flow rate to find the time: \[ \text{Time} = \frac{630}{6} = 105 \text{ minutes} \] This calculation tells us that it would take 105 minutes to completely fill the conical box with water.

Breaking down a math problem step-by-step is a powerful technique for understanding and solving complex questions. Here's how it works in this problem: 1. **Identify Known Values:** Gather all the given information: dimensions of the rectangular box and conical box properties. 2. **Calculate Basic Volumes:** First, calculate the volume of the rectangular box. Use this to find the related conical box volume. 3. **Apply Relevant Formulas:** Use the volume formulas for both the rectangular prism and cone to work through the problem. 4. **Plug and Solve:** Substitute the known values into the formulas and solve for the unknowns, such as the height for the cone or time. 5. **Answer Verification:** Finally, review to ensure all steps lead to the correct solution by rechecking calculations. Following a step-by-step methodical approach helps break down even the most complex problems into manageable parts, making it easier to solve.