Which Is The Correct First Step In Finding The Height Of A Cylinder With A Volume Of $178 \pi$ Cubic Inches And A Radius Of 8 Inches?${ \begin{array}{c} V = \pi R^2 H \ 178 \pi = \pi (8)^2 H \end{array} }$
Introduction
In mathematics, finding the height of a cylinder is a common problem that involves using the formula for the volume of a cylinder. The formula for the volume of a cylinder is given by V = πr^2h, where V is the volume, π is a constant, r is the radius, and h is the height. In this article, we will discuss the correct first step in finding the height of a cylinder with a volume of 178π cubic inches and a radius of 8 inches.
Understanding the Formula
The formula for the volume of a cylinder is V = πr^2h. This formula can be rearranged to solve for the height (h) of the cylinder. To do this, we need to isolate the variable h on one side of the equation. We can start by dividing both sides of the equation by πr^2.
Step 1: Divide Both Sides by πr^2
To isolate the variable h, we need to divide both sides of the equation by πr^2. This will give us the equation:
Substituting the given values for V and r, we get:
Simplifying the Equation
Now that we have the equation in the form h = V / (πr^2), we can simplify it by canceling out the π terms. This will give us:
Calculating the Height
Now that we have the simplified equation, we can calculate the height of the cylinder. To do this, we need to evaluate the expression (8)^2, which is equal to 64. Then, we can divide 178 by 64 to get the height.
Evaluating the Expression
To evaluate the expression h = 178/64, we can use a calculator or perform the division manually. Either way, we get:
Conclusion
In conclusion, the correct first step in finding the height of a cylinder with a volume of 178π cubic inches and a radius of 8 inches is to divide both sides of the equation by πr^2. This will give us the equation h = V / (πr^2), which can be simplified to h = 178 / (8)^2. Finally, we can calculate the height by evaluating the expression h = 178/64.
Additional Tips and Tricks
- When working with the formula for the volume of a cylinder, make sure to use the correct units for the variables. In this case, the volume is given in cubic inches, and the radius is given in inches.
- When simplifying the equation, make sure to cancel out any common factors between the numerator and denominator.
- When evaluating the expression, make sure to use a calculator or perform the division manually to get an accurate result.
Real-World Applications
The formula for the volume of a cylinder has many real-world applications. For example, it can be used to calculate the volume of a cylindrical tank or a cylindrical pipe. It can also be used to calculate the volume of a cylindrical container or a cylindrical vessel.
Common Mistakes to Avoid
- One common mistake to avoid is to forget to cancel out the π terms when simplifying the equation.
- Another common mistake to avoid is to forget to evaluate the expression correctly when calculating the height.
- Finally, make sure to use the correct units for the variables when working with the formula for the volume of a cylinder.
Conclusion
Q: What is the formula for the volume of a cylinder?
A: The formula for the volume of a cylinder is V = πr^2h, where V is the volume, π is a constant, r is the radius, and h is the height.
Q: How do I find the height of a cylinder if I know the volume and radius?
A: To find the height of a cylinder, you need to rearrange the formula V = πr^2h to solve for h. This can be done by dividing both sides of the equation by πr^2, which gives you the equation h = V / (πr^2).
Q: What if I have a volume of 178Ï€ cubic inches and a radius of 8 inches? How do I find the height?
A: To find the height of the cylinder, you can substitute the given values into the equation h = V / (Ï€r^2). This gives you the equation h = 178Ï€ / (Ï€(8)^2). Simplifying this equation, you get h = 178 / (8)^2, which is equal to 2.78125.
Q: What are some common mistakes to avoid when finding the height of a cylinder?
A: Some common mistakes to avoid when finding the height of a cylinder include forgetting to cancel out the π terms when simplifying the equation, forgetting to evaluate the expression correctly when calculating the height, and using the wrong units for the variables.
Q: What are some real-world applications of the formula for the volume of a cylinder?
A: The formula for the volume of a cylinder has many real-world applications, including calculating the volume of a cylindrical tank or pipe, calculating the volume of a cylindrical container or vessel, and calculating the volume of a cylindrical object.
Q: How do I use the formula for the volume of a cylinder to calculate the volume of a cylindrical tank?
A: To calculate the volume of a cylindrical tank, you need to know the radius and height of the tank. You can use the formula V = πr^2h to calculate the volume, where V is the volume, π is a constant, r is the radius, and h is the height.
Q: What if I have a cylindrical tank with a radius of 10 inches and a height of 20 inches? How do I calculate the volume?
A: To calculate the volume of the cylindrical tank, you can substitute the given values into the formula V = πr^2h. This gives you the equation V = π(10)^2(20), which is equal to 12566.37 cubic inches.
Q: What are some tips for working with the formula for the volume of a cylinder?
A: Some tips for working with the formula for the volume of a cylinder include using the correct units for the variables, canceling out common factors between the numerator and denominator, and evaluating expressions correctly when calculating the height.
Q: How do I use the formula for the volume of a cylinder to calculate the volume of a cylindrical pipe?
A: To calculate the volume of a cylindrical pipe, you need to know the radius and height of the pipe. You can use the formula V = πr^2h to calculate the volume, where V is the volume, π is a constant, r is the radius, and h is the height.
Q: What if I have a cylindrical pipe with a radius of 5 inches and a height of 15 inches? How do I calculate the volume?
A: To calculate the volume of the cylindrical pipe, you can substitute the given values into the formula V = πr^2h. This gives you the equation V = π(5)^2(15), which is equal to 1766.97 cubic inches.
Conclusion
In conclusion, finding the height of a cylinder involves using the formula for the volume of a cylinder. By following the steps outlined in this article, you can accurately find the height of a cylinder. Additionally, the formula for the volume of a cylinder has many real-world applications, including calculating the volume of a cylindrical tank or pipe, calculating the volume of a cylindrical container or vessel, and calculating the volume of a cylindrical object.