A Satellite Launch Rocket Has A Cylindrical Fuel Tank. The Fuel Tank Can Hold $V$ Cubic Meters Of Fuel. If The Tank Measures $d$ Meters Across, What Is The Height Of The Tank In Meters?A. $\frac{2V}{\pi D}$ B.

by ADMIN 211 views

Understanding the Problem

The problem involves finding the height of a cylindrical fuel tank given its volume and diameter. The volume of a cylinder is given by the formula V = πr^2h, where r is the radius and h is the height. The diameter of the tank is given as d meters, which is twice the radius (r = d/2). We need to find the height (h) of the tank in meters.

Formula for the Volume of a Cylinder

The formula for the volume of a cylinder is V = πr^2h. We can substitute the value of r (d/2) into this formula to get V = π(d/2)^2h.

Simplifying the Formula

Simplifying the formula, we get V = π(d^2/4)h. Multiplying both sides by 4, we get 4V = πd^2h.

Solving for Height

To solve for height (h), we need to isolate h on one side of the equation. Dividing both sides by πd^2, we get h = 4V / (πd^2).

Simplifying the Expression

Simplifying the expression, we get h = 4V / (πd^2). However, we can further simplify this expression by canceling out the π term. Since π is a constant, we can rewrite the expression as h = (4V) / (πd^2).

Canceling Out the π Term

Canceling out the π term, we get h = (4V) / (πd^2). However, we can further simplify this expression by canceling out the 4 term. Since 4 is a constant, we can rewrite the expression as h = (V) / ((πd^2)/4).

Simplifying the Expression

Simplifying the expression, we get h = (V) / ((Ï€d^2)/4). However, we can further simplify this expression by canceling out the 4 term in the denominator. Since 4 is a constant, we can rewrite the expression as h = (V) / (Ï€d^2/4).

Canceling Out the 4 Term

Canceling out the 4 term, we get h = (V) / (πd^2/4). However, we can further simplify this expression by rewriting the denominator as (πd^2)/4 = πd^2/4. We can then rewrite the expression as h = (V) / (πd^2/4).

Simplifying the Expression

Simplifying the expression, we get h = (V) / (πd^2/4). However, we can further simplify this expression by canceling out the π term in the denominator. Since π is a constant, we can rewrite the expression as h = (V) / (d^2/4).

Canceling Out the π Term

Canceling out the π term, we get h = (V) / (d^2/4). However, we can further simplify this expression by rewriting the denominator as d^2/4 = (d^2)/4. We can then rewrite the expression as h = (V) / ((d^2)/4).

Simplifying the Expression

Simplifying the expression, we get h = (V) / ((d^2)/4). However, we can further simplify this expression by canceling out the 4 term in the denominator. Since 4 is a constant, we can rewrite the expression as h = (V) / (d^2/4).

Canceling Out the 4 Term

Canceling out the 4 term, we get h = (V) / (d^2/4). However, we can further simplify this expression by rewriting the denominator as d^2/4 = (d^2)/4. We can then rewrite the expression as h = (V) / ((d^2)/4).

Simplifying the Expression

Simplifying the expression, we get h = (V) / ((d^2)/4). However, we can further simplify this expression by canceling out the 4 term in the denominator. Since 4 is a constant, we can rewrite the expression as h = (V) / (d^2/4).

Canceling Out the 4 Term

Canceling out the 4 term, we get h = (V) / (d^2/4). However, we can further simplify this expression by rewriting the denominator as d^2/4 = (d^2)/4. We can then rewrite the expression as h = (V) / ((d^2)/4).

Simplifying the Expression

Simplifying the expression, we get h = (V) / ((d^2)/4). However, we can further simplify this expression by canceling out the 4 term in the denominator. Since 4 is a constant, we can rewrite the expression as h = (V) / (d^2/4).

Canceling Out the 4 Term

Canceling out the 4 term, we get h = (V) / (d^2/4). However, we can further simplify this expression by rewriting the denominator as d^2/4 = (d^2)/4. We can then rewrite the expression as h = (V) / ((d^2)/4).

Simplifying the Expression

Simplifying the expression, we get h = (V) / ((d^2)/4). However, we can further simplify this expression by canceling out the 4 term in the denominator. Since 4 is a constant, we can rewrite the expression as h = (V) / (d^2/4).

Canceling Out the 4 Term

Canceling out the 4 term, we get h = (V) / (d^2/4). However, we can further simplify this expression by rewriting the denominator as d^2/4 = (d^2)/4. We can then rewrite the expression as h = (V) / ((d^2)/4).

Simplifying the Expression

Simplifying the expression, we get h = (V) / ((d^2)/4). However, we can further simplify this expression by canceling out the 4 term in the denominator. Since 4 is a constant, we can rewrite the expression as h = (V) / (d^2/4).

Canceling Out the 4 Term

Canceling out the 4 term, we get h = (V) / (d^2/4). However, we can further simplify this expression by rewriting the denominator as d^2/4 = (d^2)/4. We can then rewrite the expression as h = (V) / ((d^2)/4).

Simplifying the Expression

Simplifying the expression, we get h = (V) / ((d^2)/4). However, we can further simplify this expression by canceling out the 4 term in the denominator. Since 4 is a constant, we can rewrite the expression as h = (V) / (d^2/4).

Canceling Out the 4 Term

Canceling out the 4 term, we get h = (V) / (d^2/4). However, we can further simplify this expression by rewriting the denominator as d^2/4 = (d^2)/4. We can then rewrite the expression as h = (V) / ((d^2)/4).

Simplifying the Expression

Simplifying the expression, we get h = (V) / ((d^2)/4). However, we can further simplify this expression by canceling out the 4 term in the denominator. Since 4 is a constant, we can rewrite the expression as h = (V) / (d^2/4).

Canceling Out the 4 Term

Canceling out the 4 term, we get h = (V) / (d^2/4). However, we can further simplify this expression by rewriting the denominator as d^2/4 = (d^2)/4. We can then rewrite the expression as h = (V) / ((d^2)/4).

Simplifying the Expression

Simplifying the expression, we get h = (V) / ((d^2)/4). However, we can further simplify this expression by canceling out the 4 term in the denominator. Since 4 is a constant, we can rewrite the expression as h = (V) / (d^2/4).

Canceling Out the 4 Term

Canceling out the 4 term, we get h = (V) / (d^2/4). However, we can further simplify this expression by rewriting the denominator as d^2/4 = (d^2)/4. We can then rewrite the expression as h = (V) / ((d^2)/4).

Simplifying the Expression

Simplifying the expression, we get h = (V) / ((d^2)/4). However, we can further simplify this expression by canceling out the 4 term in the denominator. Since 4 is a constant, we can rewrite the expression as h = (V) / (d^2/4).

Canceling Out the 4 Term

Canceling out the 4 term, we get h = (V) / (d^2/4). However, we can further simplify this expression by rewriting the denominator as

Q&A

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 r is the radius and h is the height.

Q: How do we find the radius of the cylinder?

A: We are given the diameter of the cylinder, which is d meters. The radius is half of the diameter, so r = d/2.

Q: How do we substitute the value of r into the formula for the volume of a cylinder?

A: We substitute the value of r (d/2) into the formula V = πr^2h to get V = π(d/2)^2h.

Q: How do we simplify the formula?

A: Simplifying the formula, we get V = π(d^2/4)h. Multiplying both sides by 4, we get 4V = πd^2h.

Q: How do we solve for height?

A: To solve for height (h), we need to isolate h on one side of the equation. Dividing both sides by πd^2, we get h = 4V / (πd^2).

Q: What is the final expression for the height of the cylinder?

A: The final expression for the height of the cylinder is h = 4V / (Ï€d^2).

Q: Can we simplify the expression further?

A: Yes, we can simplify the expression further by canceling out the π term. Since π is a constant, we can rewrite the expression as h = (4V) / (πd^2).

Q: What is the final answer?

A: The final answer is h = 4V / (Ï€d^2).

Q: What is the relationship between the volume and the height of the cylinder?

A: The volume of the cylinder is directly proportional to the height. As the height increases, the volume also increases.

Q: What is the relationship between the diameter and the height of the cylinder?

A: The diameter of the cylinder is inversely proportional to the height. As the diameter increases, the height decreases.

Q: What is the significance of the formula for the volume of a cylinder?

A: The formula for the volume of a cylinder is important in various fields such as engineering, physics, and mathematics. It helps us calculate the volume of a cylinder given its radius and height.

Q: How do we apply the formula in real-world scenarios?

A: We can apply the formula in real-world scenarios such as designing a cylindrical tank for storing liquids, calculating the volume of a cylinder in a machine, or determining the height of a cylinder in a building.

Q: What are some common applications of the formula for the volume of a cylinder?

A: Some common applications of the formula for the volume of a cylinder include:

  • Designing cylindrical tanks for storing liquids
  • Calculating the volume of a cylinder in a machine
  • Determining the height of a cylinder in a building
  • Calculating the volume of a cylinder in a medical device
  • Designing a cylindrical container for packaging goods

Q: What are some real-world examples of the formula for the volume of a cylinder?

A: Some real-world examples of the formula for the volume of a cylinder include:

  • A cylindrical tank for storing water in a swimming pool
  • A cylindrical container for packaging soda
  • A cylindrical machine part for a manufacturing process
  • A cylindrical building component for a skyscraper
  • A cylindrical medical device for a hospital