Problem 63 - Intermediate Algebra

Question

Use your calculator and evaluate each of the algebraic expressions for the indicated values. Express the final answers to the nearest tenth. $$ 2 \pi r^{2}+2 \pi r h, \quad \pi=3.14, r=7.8, \text { and } h=21.2 $$

Step-by-Step Solution

Answer

1423.7

Step 1: Identify the Formula

The given expression is \(2\pi r^2 + 2\pi rh\). It's a standard formula for calculating the surface area of a cylinder. The first part \(2\pi r^2\) represents the area of the cylinder's circles, while the second part \(2\pi rh\) is the area of the cylindrical side.

Step 2: Substitute the Given Values

Now, we replace \(\pi\) with 3.14, \(r\) with 7.8, and \(h\) with 21.2 in the expression: \[2(3.14)(7.8)^2 + 2(3.14)(7.8)(21.2)\]

Step 3: Calculate \(2\pi r^2\)

First, calculate \(7.8^2\), which equals 60.84. Then, multiply by \(2 \times 3.14\):\[2 \times 3.14 \times 60.84 = 382.872\]

Step 4: Calculate \(2\pi rh\)

Next, calculate \(2 \times 3.14 \times 7.8 \times 21.2\):\[2 \times 3.14 \times 7.8 \times 21.2 = 1040.8472\]

Step 5: Add the Results

Combine the results of the previous calculations to find the total:\[382.872 + 1040.8472 = 1423.7192\]

Step 6: Round to the Nearest Tenth

Round the final result 1423.7192 to the nearest tenth, which gives 1423.7.

Key Concepts

Surface Area of a Cylinder

When you think about a cylinder, picture a soup can. The surface area is like how much paper you need to wrap around the full surface of the can. There are two components to consider:

• The circles at the top and bottom.
• The curved side, like the label around the can.

The formula to find the surface area of a cylinder is: \[ S = 2\pi r^2 + 2\pi rh \]- **2\pi r^2**: This part calculates the area of the two circular bases. You multiply the circumference of one circle by the height, giving you the rectangle that wraps around.
- **2\pi rh**: This denotes the area of the rectangle created when unrolling the cylinder's side. Adding both parts gives the whole surface area. This formula helps us calculate how much material we need to cover the entire outside of the cylinder.

Substitution Method

Substitution is like putting puzzle pieces together. You replace variables in an equation with numbers. By doing this, you can calculate specific values.Here's how the substitution method works in our exercise:

• Look at the given equation: \(2\pi r^2 + 2\pi rh\).
• Identify the values you need to insert: \(\pi = 3.14\), \(r = 7.8\), and \(h = 21.2\).
• Substitute these values into the equation, replacing the corresponding variables.

By swapping out the symbols \(\pi\), \(r\), and \(h\) with their numerical values step by step, you slowly turn the expression into a straightforward calculation. This approach simplifies complex expressions, making them easier to handle and solve.

Rounding Numbers

Rounding numbers makes Math a little simpler and results easier to communicate. It involves adjusting a number to the nearest fixed point to make it simpler. Here's how you do it:

• Identify which place value you need to round to. In our case, it's the nearest tenth.
• Look at the digit right after your chosen place value—to decide whether to round up or not.
• If that digit is 5 or more, increase the target digit by one. Otherwise, leave it.

In the exercise, once we evaluate the surface area and get 1423.7192, you check the decimal part. With 9 following the target tenth place, it prompts us to round up to 1423.7. Rounding helps convey a cleaner, clearer number, especially helpful when precision to multiple decimals isn't crucial.