Question

Tina makes an hourglass that is 3 inches across the base and 12 inches high. If she fills the upper part of the hourglass to half its volume, how many cubic inches of sand will she need? (Her hourglass is two cones together, tip to tip.) Use 3.14 for \(\pi\). A. 7.065 in \(^3\) B. \(9.420 \mathrm{in}^3\) C. \(14.130 \mathrm{in}^3\) D. \(63.585 \mathrm{in}^3\)

Short answer

7.065 cubic inches

Step-by-step solution

Understand the Shape

The hourglass consists of two cones joined tip to tip. Each cone has a base radius of 1.5 inches (since the diameter is 3 inches) and a height of 6 inches (since the total height is 12 inches, and each cone is half the height).

Calculate the Volume of One Cone

Use the volume formula for a cone: \[ V = \frac{1}{3} \pi r^2 h \] where \( r = 1.5 \) inches and \( h = 6 \) inches. Substitute the values: \[ V = \frac{1}{3} (3.14) (1.5)^2 (6) \] Calculate \( V \): \[ V = \frac{1}{3} (3.14) (2.25) (6) \] which simplifies to: \[ V = \frac{1}{3} (42.39) \] leading to: \[ V = 14.13 \mathrm{in}^3 \]

Calculate Half the Volume

Since the sand fills half the volume of the hourglass, take half of the previously calculated volume: \[ \text{Half Volume} = \frac{14.13}{2} \] which is: \[ \text{Half Volume} = 7.065 \mathrm{in}^3 \]

Select the Correct Answer

From the options given, the correct answer is: A. 7.065 in \(^3\)

Key concepts

cone volume formula

The first step in solving any problem involving the volume of a cone is to understand the formula. The volume of a cone is given by: \[ V = \frac{1}{3} \pi r^2 h \]In this formula, \(V\) is the volume, \(\pi\) is a constant (approximately 3.14), \(r\) is the radius of the cone's base, and \(h\) is the height of the cone. This formula tells us that the volume of a cone is proportional to the base area (\(\pi r^2\)) and one-third of the height.Let's apply this to the exercise. Tina's hourglass consists of two cones joined tip to tip. Each cone has a base radius of 1.5 inches and a height of 6 inches. So for one cone, we substitute these values into our formula:\[ V = \frac{1}{3} (3.14) (1.5)^2 (6) \]Calculate the terms step by step:\[ V = \frac{1}{3} (3.14) (2.25) (6) \]\[ V = \frac{1}{3} (42.39) \]Finally:\[ V = 14.13 \text{ in}^3 \]So, each cone in the hourglass has a volume of 14.13 cubic inches. Understanding this formula and how to use it is crucial when solving geometry problems involving cones.

geometry

Geometry is the branch of mathematics focusing on the properties and relations of points, lines, surfaces, solids, and higher-dimensional analogs. In this exercise, we are dealing specifically with the geometry of cones and how to calculate their volume. Understanding basic geometrical shapes and their formulas greatly helps with problems like these.A cone, as featured in this exercise, is a three-dimensional shape that tapers smoothly from a flat base to a point called the apex. Important properties of a cone include its height and the radius of its base. For example, Tina’s hourglass consists of two identical cones, making it easier to calculate the total volume since you only need to compute one and then double it if necessary.In our exercise, the key geometric operations involve:- Identifying the radius of the cone's base from the diameter.- Using the height of each cone correctly when applying the volume formula.By mastering these basic concepts, students can solve any similar geometric problems involving cones easily.

GED math problems

General Educational Development (GED) math problems often include practical applications of mathematical concepts and are designed to assess a student’s understanding of fundamental principles. Problems like calculating the volume of a cone not only test knowledge of the cone volume formula, but also comprehension in geometry and logical problem-solving.In the given exercise, Tina’s hourglass problem is a typical GED math problem because:- It combines understanding geometric shapes with practical application.- It requires using a mathematical formula accurately.- It demonstrates real-life relevance and applicability.To tackle such problems effectively:- Break down the problem into smaller, manageable steps.- Ensure you understand and correctly apply the relevant formulas.- Double-check calculations for accuracy.In the case of Tina’s hourglass, these steps are straightforward given you follow the geometry and volume rules accurately. Students preparing for GED exams will benefit greatly from practicing similar problems to strengthen their understanding and problem-solving abilities.