Question

Triana has taken a job refinishing a large ice cream cone that hangs outside the Soda Spot. The sign is made of a half sphere of radius 1 foot atop a pointed cone with a top that has the same radius and that is 6 feet high. How many square feet of surface does Triana need to cover with new paint? A. 12.56 square feet B. 19.1 square feet C. 25.4 square feet D. 32.6 square feet

Short answer

C. 25.4 square feet

Step-by-step solution

- Calculate the Surface Area of the Half Sphere

The formula for the surface area of a sphere is \[ 4 \pi r^2 \]. Since the sign is a half sphere, the surface area of the half sphere (excluding the base) is half of that. With a radius \left( r \right) of 1 foot: \left( 4 \pi \left(1\right)^2 \right) \div 2 = 2 \pi \text{ square feet}.

- Calculate the Surface Area of the Cone

The formula for the lateral surface area of a cone is \[ \pi r l \], where \left( r \right) is the radius and \left( l \right) is the slant height. First, find the slant height \left( l \right). Using the Pythagorean theorem for a right triangle formed by the radius, height of cone, and slant height: \[ l = \sqrt{r^2 + h^2} = \sqrt{1^2 + 6^2} = \sqrt{37} \approx 6.08 \]. Then, the lateral surface area becomes \[ \pi \left(1 \right) \left(6.08 \right) \approx 19.1 \text{ square feet} \].

- Add the Surface Areas

Add the surface area of the half sphere and the surface area of the cone: \[ 2 \pi + 19.1 \approx 25.4 \text{ square feet} \. \]

Select the Answer Choice

With the total surface area calculated to be approximately 25.4 square feet, the correct answer is choice C: 25.4 square feet.

Key concepts

geometry

Geometry helps us understand the shapes and sizes of objects around us. It is a branch of mathematics that deals with properties and relations of points, lines, surfaces, and solids. In this exercise, we are calculating the surface area of a complex shape made up of a half-sphere and a cone. To solve this, we use different geometric formulas suited for these specific shapes.

Understanding geometry is crucial for solving real-world problems, such as determining the amount of paint needed to cover a sign.

surface area of a sphere

The surface area of a sphere can be calculated using the formula \(\text{Surface Area} = 4 \pi r^2\).

Here, \(r\) is the radius of the sphere. However, in this problem, the sign is made of a half-sphere.

So, we just need half the surface area, \(\frac{1}{2} \left( 4 \pi r^2 \right) = 2 \pi r^2\).

For a radius of 1 foot: \( 2 \pi (1)^2 = 2 \pi \text{ square feet} \).

This tells us how much surface area the half-sphere contributes to the sign.

surface area of a cone

To find the surface area of a cone, we need to calculate the lateral (side) surface area using the formula: \( \pi r l \).

Here, \(r\) is the radius, and \(l\) is the slant height of the cone.

First, calculate the slant height using the Pythagorean theorem: \( l = \sqrt{r^2 + h^2}\).

For a radius of 1 foot and height of 6 feet: \( l = \sqrt{1^2 + 6^2} = \sqrt{37} \approx 6.08 \).

Now we can find the lateral surface area: \(\text{Lateral Surface Area} = \pi \cdot 1 \cdot 6.08 \approx 19.1\) square feet.

Pythagorean theorem

The Pythagorean theorem is a fundamental concept in geometry. It states that for a right-angled triangle: \( a^2 + b^2 = c^2 \), where \(a\) and \(b\) are the lengths of the legs, and \(c\) is the length of the hypotenuse (longest side).

In this exercise, we use it to find the slant height \(l\) of the cone.

By considering the cone's radius (1 foot) and height (6 feet) as the two legs of the right triangle, we calculate: \( l = \sqrt{1^2 + 6^2} = \sqrt{37} \approx 6.08 \).

Understanding this step is essential to accurately determine the cone's surface area.