A satellite launch rocket has a cylindrical fuel tank. The fuel tank can hold [tex]$V$[/tex] cubic meters of fuel. If the tank measures [tex]$d$[/tex] meters across, what is the height of the tank in meters?
A. [tex]$\frac{2 V}{\pi d^2}$[/tex]
B. [tex]$\frac{4 V}{d^2}$[/tex]
C. [tex]$\frac{V}{\pi d^2}$[/tex]
D. [tex]$\frac{4 V}{\pi d^2}$[/tex]
E. [tex]$\frac{8 V}{\pi d^2}$[/tex]
Answer (1)
The problem provides the volume V and diameter d of a cylindrical tank.
Calculate the radius r as half of the diameter: r = 2 d .
Substitute the radius into the cylinder volume formula: V = π r 2 h = π ( 2 d ) 2 h .
Solve for the height h : h = π d 2 4 V .
The height of the cylindrical tank is: π d 2 4 V
Explanation
Problem Analysis The problem states that a cylindrical fuel tank has a volume of V cubic meters and a diameter of d meters. We need to find the height of the tank.
Volume of a Cylinder The volume of a cylinder is given by the formula: V = π r 2 h where:
V is the volume,
r is the radius,
h is the height.
Radius Calculation The radius is half of the diameter, so we have: r = 2 d
Substitute Radius into Volume Formula Substitute the radius into the volume formula: V = π ( 2 d ) 2 h V = π 4 d 2 h
Solve for Height Now, solve for the height h :
h = π 4 d 2 V h = π d 2 4 V
Final Answer The height of the tank is π d 2 4 V meters. Therefore, the correct answer is D.
Examples
Understanding the volume of cylinders is crucial in many real-world applications. For instance, when designing storage tanks for liquids or gases, engineers need to calculate the height of the tank based on its desired volume and diameter. This ensures that the tank can hold the required amount of substance, optimizing space and resources. Similarly, in architecture, calculating the volume of cylindrical columns helps determine the amount of material needed for construction, ensuring structural integrity and cost-effectiveness.
