79): Find the area of the sector of the circle. Radius: 2.5 feet. Central Angle: 225 degrees.
I multiplied (1/2)(25/4)(5pi/4), and I got 100pi/16, which reduced to 50pi/8 when I halved it. I reduced that to 25pi/4 or 19.63. But the book's answer is 12.27. How'd I mess up?
Area of sector of a circle with radius r for sector given in degrees (theta):
A = (1/2)(r^2)(theta)((pi)/(180)
Left off the trailing parenthesis...
Area of sector of a circle with radius r for sector given in degrees (theta):
A = (1/2)(r^2)(theta)((pi)/(180))
Well, you mentioned multiplying (1/2)(25/4)(5pi/4), but it seems like you made a little mistake. Let's break it down for you:
The formula for finding the area of a sector is (1/2) * r^2 * θ, where r is the radius and θ is the central angle in radians.
So in your case, with a radius of 2.5 feet and a central angle of 225 degrees (which is equal to 225 * pi/180 radians), we can substitute those values into the formula:
Area = (1/2) * (2.5)^2 * (225 * π/ 180)
Now let's simplify that:
Area = (1/2) * 6.25 * (225π / 180)
= 3.125 * (225π / 180)
= (3.125 * 225π) / 180
= 7.03125π
So the area of the sector is approximately 7.03125π square feet.
It seems like you got confused somewhere after multiplying, but don't worry, math can be a bit tricky sometimes. Just keep practicing, and I'm sure you'll get the hang of it. Keep clowning around, my friend!
To find the area of the sector of a circle, you can use the formula:
\[ \text{Area} = \frac{\text{Central Angle}}{360} \times \pi \times \text{Radius}^2 \]
In this case, the radius is given as 2.5 feet and the central angle is 225 degrees. Let's calculate the area step by step:
1. Convert the central angle from degrees to radians. Since 1 degree is equal to π/180 radians, the central angle of 225 degrees is equal to (225π/180) radians, which simplifies to (5π/4) radians.
2. Plug the values into the formula:
\[ \text{Area} = \frac{225\pi/180}{360} \times \pi \times (2.5)^2 \]
3. Simplify the fraction:
\[ \text{Area} = \frac{5\pi}{8} \times \pi \times 6.25 \]
4. Multiply the fractions and simplify:
\[ \text{Area} = \frac{25\pi}{8} \times 6.25 = \frac{156.25\pi}{8} \]
Now, let's calculate the approximate value:
\[ \text{Area} \approx \frac{156.25 \times 3.14}{8} \approx 60.92 \]
Therefore, the correct approximate value for the area of the sector is 60.92, not 12.27. It seems like you made some mistakes during the calculation.
To find the area of a sector of a circle, you need to use the formula:
Area of sector = (1/2) * r^2 * θ
where r is the radius of the circle and θ is the central angle in radians.
Let's go through the calculation step-by-step to find where you might have made a mistake.
Given:
Radius (r) = 2.5 feet
Central Angle (θ) = 225 degrees
First, convert the central angle from degrees to radians:
θ (in radians) = θ (in degrees) * π/180
θ (in radians) = 225 * π/180
θ (in radians) = 5π/4
Next, substitute the values into the area of sector formula:
Area of sector = (1/2) * (2.5^2) * (5π/4)
Area of sector = (1/2) * 6.25 * (5π/4)
Area of sector = 3.125 * (5π/4)
Area of sector = (15.625π)/4
Now, let's simplify this result:
Area of sector ≈ 3.91π
It appears that you made a mistake in reducing the fraction (15.625π)/4. When you simplified it to 25π/4, you actually doubled the numerator. The correct simplified answer is 3.91π, which is approximately 12.27 (rounded to two decimal places).
So, the book's answer of 12.27 is correct.