Math / Geometry

QuestionThe perimeter of a semicircle is 20.56 kilometers. What is the semicircle's area? Use $\pi \approx 3.14$ and round your answer to the nearest hundredth.

Studdy Solution

STEP 1

1. The perimeter of the semicircle includes both the curved part (half of the circumference of a full circle) and the diameter.
2. The formula for the circumference of a full circle is $C = 2\pi r$ , where $r$ is the radius of the circle.
3. The formula for the area of a semicircle is half the area of a full circle, which is $\frac{1}{2}\pi r^2$ .
4. The value of $\pi$ is approximated as $3.14$ .

STEP 2

1. Express the perimeter of the semicircle in terms of its radius.
2. Solve for the radius using the given perimeter.
3. Calculate the area of the semicircle using the radius obtained.
4. Round the answer to the nearest hundredth.

STEP 3

Express the perimeter $P$ of the semicircle in terms of its radius $r$ .
The perimeter of the semicircle includes the diameter and the curved part: $P = \pi r + 2r$ Given that $P = 20.56$ km, we have: $20.56 = \pi r + 2r$

STEP 4

Substitute the value of $\pi \approx 3.14$ into the equation.
$20.56 = 3.14r + 2r$

STEP 5

Combine like terms to solve for the radius $r$ .
$20.56 = 5.14r$ $r = \frac{20.56}{5.14}$

STEP 6

Calculate the value of $r$ .
$r = \frac{20.56}{5.14} \approx 4 \text{ km}$

STEP 7

Calculate the area $A$ of the semicircle using the formula $A = \frac{1}{2}\pi r^2$ .
$A = \frac{1}{2} \pi r^2$ Substitute $\pi \approx 3.14$ and $r = 4$ : $A = \frac{1}{2} \times 3.14 \times 4^2$

STEP 8

Simplify the expression to find $A$ .
$A = \frac{1}{2} \times 3.14 \times 16$ $A = 1.57 \times 16$ $A \approx 25.12 \text{ km}^2$

STEP 9

Round the answer to the nearest hundredth.
The area of the semicircle is approximately $25.12 \text{ km}^2$ .
Solution: The area of the semicircle is $25.12 \text{ km}^2$ .

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QuestionThe perimeter of a semicircle is 20.56 kilometers. What is the semicircle's area? Use $\pi \approx 3.14$ and round your answer to the nearest hundredth.

Studdy Solution

STEP 1

STEP 2

1. Express the perimeter of the semicircle in terms of its radius.
2. Solve for the radius using the given perimeter.
3. Calculate the area of the semicircle using the radius obtained.
4. Round the answer to the nearest hundredth.

STEP 3

Express the perimeter $P$ of the semicircle in terms of its radius $r$ .
The perimeter of the semicircle includes the diameter and the curved part: $P = \pi r + 2r$ Given that $P = 20.56$ km, we have: $20.56 = \pi r + 2r$

STEP 4

Substitute the value of $\pi \approx 3.14$ into the equation.
$20.56 = 3.14r + 2r$

STEP 5

Combine like terms to solve for the radius $r$ .
$20.56 = 5.14r$ $r = \frac{20.56}{5.14}$

STEP 6

Calculate the value of $r$ .
$r = \frac{20.56}{5.14} \approx 4 \text{ km}$

STEP 7

Calculate the area $A$ of the semicircle using the formula $A = \frac{1}{2}\pi r^2$ .
$A = \frac{1}{2} \pi r^2$ Substitute $\pi \approx 3.14$ and $r = 4$ : $A = \frac{1}{2} \times 3.14 \times 4^2$

STEP 8

Simplify the expression to find $A$ .
$A = \frac{1}{2} \times 3.14 \times 16$ $A = 1.57 \times 16$ $A \approx 25.12 \text{ km}^2$

STEP 9

Round the answer to the nearest hundredth.
The area of the semicircle is approximately $25.12 \text{ km}^2$ .
Solution: The area of the semicircle is $25.12 \text{ km}^2$ .

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QuestionThe perimeter of a semicircle is 20.56 kilometers. What is the semicircle's area? Use π≈3.14\pi \approx 3.14π≈3.14 and round your answer to the nearest hundredth.

Studdy Solution

STEP 1

STEP 2

1. Express the perimeter of the semicircle in terms of its radius.2. Solve for the radius using the given perimeter.3. Calculate the area of the semicircle using the radius obtained.4. Round the answer to the nearest hundredth.

STEP 3

Express the perimeter P P P of the semicircle in terms of its radius r r r.The perimeter of the semicircle includes the diameter and the curved part: P=πr+2r P = \pi r + 2r P=πr+2r Given that P=20.56 P = 20.56 P=20.56 km, we have: 20.56=πr+2r 20.56 = \pi r + 2r 20.56=πr+2r

STEP 4

Substitute the value of π≈3.14\pi \approx 3.14π≈3.14 into the equation.20.56=3.14r+2r 20.56 = 3.14r + 2r 20.56=3.14r+2r

STEP 5

Combine like terms to solve for the radius r r r.20.56=5.14r 20.56 = 5.14r 20.56=5.14r r=20.565.14 r = \frac{20.56}{5.14} r=5.1420.56​

STEP 6

Calculate the value of r r r.r=20.565.14≈4 km r = \frac{20.56}{5.14} \approx 4 \text{ km} r=5.1420.56​≈4 km

STEP 7

Calculate the area A A A of the semicircle using the formula A=12πr2 A = \frac{1}{2}\pi r^2 A=21​πr2.A=12πr2 A = \frac{1}{2} \pi r^2 A=21​πr2 Substitute π≈3.14\pi \approx 3.14π≈3.14 and r=4 r = 4 r=4: A=12×3.14×42 A = \frac{1}{2} \times 3.14 \times 4^2 A=21​×3.14×42

STEP 8

Simplify the expression to find A A A.A=12×3.14×16 A = \frac{1}{2} \times 3.14 \times 16 A=21​×3.14×16 A=1.57×16 A = 1.57 \times 16 A=1.57×16 A≈25.12 km2 A \approx 25.12 \text{ km}^2 A≈25.12 km2

STEP 9

Round the answer to the nearest hundredth.The area of the semicircle is approximately 25.12 km2 25.12 \text{ km}^2 25.12 km2.Solution: The area of the semicircle is 25.12 km2 25.12 \text{ km}^2 25.12 km2.

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QuestionThe perimeter of a semicircle is 20.56 kilometers. What is the semicircle's area? Use $\pi \approx 3.14$ and round your answer to the nearest hundredth.

1. Express the perimeter of the semicircle in terms of its radius.
2. Solve for the radius using the given perimeter.
3. Calculate the area of the semicircle using the radius obtained.
4. Round the answer to the nearest hundredth.

Express the perimeter $P$ of the semicircle in terms of its radius $r$ .
The perimeter of the semicircle includes the diameter and the curved part: $P = \pi r + 2r$ Given that $P = 20.56$ km, we have: $20.56 = \pi r + 2r$

Substitute the value of $\pi \approx 3.14$ into the equation.
$20.56 = 3.14r + 2r$

Combine like terms to solve for the radius $r$ .
$20.56 = 5.14r$ $r = \frac{20.56}{5.14}$

Calculate the value of $r$ .
$r = \frac{20.56}{5.14} \approx 4 \text{ km}$

Calculate the area $A$ of the semicircle using the formula $A = \frac{1}{2}\pi r^2$ .
$A = \frac{1}{2} \pi r^2$ Substitute $\pi \approx 3.14$ and $r = 4$ : $A = \frac{1}{2} \times 3.14 \times 4^2$

Simplify the expression to find $A$ .
$A = \frac{1}{2} \times 3.14 \times 16$ $A = 1.57 \times 16$ $A \approx 25.12 \text{ km}^2$

Round the answer to the nearest hundredth.
The area of the semicircle is approximately $25.12 \text{ km}^2$ .
Solution: The area of the semicircle is $25.12 \text{ km}^2$ .