Question

An elevated storage tank is shaped like a sphere and needs to be recoated. If the diameter of the tank is 65 feet, what is the surface area?

Short answer

The surface area is approximately 13272.9 square feet.

Step-by-step solution

Identify the formula for the surface area of a sphere

The surface area of a sphere can be found using the formula: \[ A = 4 \pi r^2 \] where \( A \) is the surface area and \( r \) is the radius of the sphere.

Determine the radius of the sphere

The diameter of the spherical tank is given as 65 feet. The radius is half of the diameter: \[ r = \frac{65}{2} = 32.5 \text{ feet} \]

Substitute the radius into the formula

Using the radius calculated in Step 2, substitute \( r = 32.5 \) feet into the formula: \[ A = 4 \pi (32.5)^2 \]

Calculate the surface area

First, square the radius: \( (32.5)^2 = 1056.25 \). Then, use the value of \( \pi \approx 3.14159 \) to find the surface area: \[ A = 4 \pi \cdot 1056.25 = 4 \cdot 3.14159 \cdot 1056.25 \approx 13272.9 \text{ square feet} \]

Key concepts

geometry

Geometry is a branch of mathematics that deals with shapes, sizes, and properties of space. Understanding geometry is crucial when working with three-dimensional objects like spheres.
In this exercise, you are dealing with a spherical tank. The sphere is a perfect example of a geometric shape that we encounter in real life.
Knowing how to calculate dimensions such as surface area helps in tasks like recoating the tank or estimating materials needed. Grasping the basic geometric properties of a sphere will make these calculations easier and more intuitive.

sphere calculations

Sphere calculations involve finding properties like volume and surface area. Let’s focus on surface area here.
To perform these calculations, you often need specific values such as radius or diameter. Knowing the relationships between these values lets you solve different problems efficiently.
For surface area, you use the radius, and from the radius, you can derive other properties of the sphere.
Remember to convert all measurements to the same units before calculating, and always double-check your math to avoid simple mistakes.

surface area formula

The surface area of a sphere can be determined using the formula: \( A = 4 \pi r^2 \). Here, 'A' represents the surface area, and 'r' is the radius.
This formula tells us that the surface area is directly proportional to the square of the radius. If you know the radius, you can easily find the surface area by plugging it into the formula.
In our example, substituting the radius (32.5 feet) into the formula, we get: \[ A = 4 \pi (32.5)^2 \]First, calculate \( (32.5)^2 \) which equals 1056.25. Subsequently, multiply by \( 4 \text{ and } \pi (\approx 3.14159) \)\[ A = 4 \cdot 3.14159 \cdot 1056.25 \approx 13272.9 \text{ square feet} \]This is the surface area that needs recoating.

radius and diameter relationship

Understanding the relationship between the radius and diameter is vital for any sphere-related calculations.
The diameter is the straight line passing through the center of the sphere, touching both sides. The radius is half of the diameter.

• Formula: \( d = 2r \)
• To find the radius: \( r = \frac{d}{2} \)

For example, if the diameter of the tank is 65 feet, the radius would be: \[ r = \frac{65}{2} = 32.5 \text{ feet} \]This fundamental relationship helps in breaking down complex problems into manageable steps and is used directly in the surface area formula.