Problem 104 - General Chemistry

Question

A cylinder with circular cross section has a radius of \(2.56 \mathrm{~cm}\) and a height of \(56.32 \mathrm{~cm}\). What is the volume of the cylinder? Express the answer to the correct number of significant figures.

Step-by-Step Solution

Answer

The volume is approximately 1160 cm³, considering significant figures.

Step 1: Understand the Formula for Volume of a Cylinder

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

Step 2: Substitute the Known Values into the Formula

Substitute the known values into the formula: the radius \( r = 2.56 \) cm and the height \( h = 56.32 \) cm, so the formula becomes \( V = \pi (2.56)^2 (56.32) \).

Step 3: Calculate the Square of the Radius

Calculate the square of the radius: \( (2.56)^2 = 6.5536 \).

Step 4: Calculate the Volume

Substitute \( 6.5536 \) back into the volume formula to get \( V = \pi \times 6.5536 \times 56.32 \). Multiplying these values gives \( V \approx 1161.797 \) cubic centimeters.

Step 5: Consider Significant Figures

The radius has 3 significant figures and the height has 4. Thus, the answer should have 3 significant figures. Therefore, the volume is rounded to \( 1160 \) cm³.

Key Concepts

Understanding Significant Figures

When calculating with measurements, like the volume of a cylinder, significant figures help us express how precise the measurements are. Each digit in your measurement contributes to the accuracy of your answer. It's important because:

• A measurement with more significant figures is more precise.
• Calculations should reflect the least precise measurement used.

In our example, the radius has 3 significant figures (2.56), and the height has 4 (56.32). When performing multiplication, as in calculating volume, your final answer must be rounded to match the measurement with the least number of significant figures—in this case, to 3 significant figures. This limitation ensures the precision of our result reflects the precision of our inputs. Therefore, we rounded the calculated volume of 1161.797 cubic centimeters to 1160 cm³ during the rounding step.

Mathematical Formulas in Cylinder Volume Calculation

Mathematical formulas are essential tools in solving geometry problems. For calculating the volume of a cylinder, we use the formula:

• \[ V = \pi r^2 h \]

In this context, \( r \) represents the radius of the base (a circle) and \( h \) is the cylinder's height. Both geometrical dimensions are pivotal in calculating how much space the cylinder occupies.
Here's a simple breakdown:

• The \( pi \) (π) represents the ratio of the circumference of any circle to its diameter. It's a fundamental constant in geometry, approximately equal to 3.14159.
• Multiplying the square of the radius, \( r^2 \), with π gives you the area of the circular base.
• Multiplying by the height gives you the full volume, as height stacks these circular base areas into a 3D shape known as a cylinder.

The formula is versatile and can be applied to any cylinder dimensions given the known radius and height.

Solving Geometry Problems with Cylinder Formulas

Geometry problems often involve understanding and using specific formulas to find unknown values based on given measurements. In this exercise, we're tackling the problem of finding the volume of a cylinder.
To break it down:

• Identify the Shape: Recognize that a cylinder has two parallel circular bases with a certain height connecting them.
• Know Your Dimensions: Spot the measurements given: radius and height. For this cylinder, the radius is 2.56 cm, and the height is 56.32 cm.
• Apply the Formula: Use the volume formula \( V = \pi r^2 h \).
• Calculate Order: Compute in steps: square the radius, multiply by π, then by height. This systematic approach helps avoid errors.

Using these steps helps unlock the solution to the geometry problem, ensuring you follow a logical path to find the volume efficiently and accurately.