Question

Adriana wishes to sell her special cactus apple preserves in cans holding 296 cubic centimeters each. Her canning machine works only with cans 7.5 centimeters in diameter. She needs to know how tall the cans will be so she can get labels printed. How tall will the cans be? A. 44.00 centimeters B. 39.47 centimeters C. 6.70 centimeters D. 3.75 centimeters

Short answer

C. 6.70 centimeters

Step-by-step solution

Understand the problem

The goal is to determine the height of the cans that each hold 296 cubic centimeters and have a diameter of 7.5 centimeters.

Write down the volume formula for a cylinder

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

Calculate the radius

The diameter of the can is 7.5 cm, so the radius is half of that. \[ r = \frac{7.5}{2} = 3.75 \text{ cm} \]

Substitute known values into the volume formula

We know that the volume \( V \) is 296 cubic centimeters and the radius \( r \) is 3.75 cm. Substitute these values into the formula: \[ 296 = \pi (3.75)^2 h \]

Solve for the height

First, calculate \( (3.75)^2 \): \[ (3.75)^2 = 14.0625 \]Next, substitute this back into the equation: \[ 296 = \pi \times 14.0625 \times h \]Now, divide both sides of the equation by \pi \times 14.0625 to isolate \( h \): \[ h = \frac{296}{\pi \times 14.0625} \approx \frac{296}{44.1967} \approx 6.70 \text{ cm} \]

Select the correct answer

Based on the calculations, the height of the cans is approximately 6.70 centimeters. So, the correct answer is C. 6.70 centimeters.

Key concepts

Volume of a Cylinder

When calculating the volume of a cylinder, we use the formula \( V = \pi r^2 h \). This formula tells us that the volume (V) is determined by three components: \( \pi \), the radius (r), and the height (h). The radius is squared, then multiplied by \( \pi \), and finally multiplied by the height of the cylinder. Making sure to use this formula accurately will give us the volume of any cylindrical object. Remember, knowing the volume can help in various practical situations, such as determining how much liquid a can can hold or how much space it will occupy.

Radius and Diameter

The radius and diameter are key terms in geometry, especially when dealing with cylinders. The diameter is the full length from one side of the circle to the other, passing through the center. To find the radius, which is half of the diameter, you simply divide the diameter by 2. For Adriana's can, the diameter was given as 7.5 centimeters. Thus, the radius is calculated as follows: \( r = \frac{7.5}{2} = 3.75 \) cm. Understanding this relationship helps us to find a critical part of the volume formula.

Height Calculation

After substituting the known values (volume and radius) into the volume formula, we need to solve for the unknown height. First, we calculate the radius squared: \( (3.75)^2 = 14.0625 \). Then, we place this into the equation: \( 296 = \pi \times 14.0625 \times h \. To isolate the height (h), we divide both sides by \pi \times 14.0625: \( h = \frac{296}{\left(\pi \times 14.0625\right) } \). This results in the height being \approx 6.70 \) cm. This step-by-step approach ensures accuracy and simplifies the problem.

Geometry

Understanding basic geometric concepts is crucial for solving problems involving shapes like cylinders. Geometry allows us to work with properties such as volume, radius, and height systematically. Cylinders feature prominently in various real-life applications, from containers to pillars. Recognizing how formulas and properties interlink will significantly enhance your problem-solving skills. The exercise with Adriana’s can illustrates how geometry can be applied to everyday scenarios, making math both practical and relevant.