Answer :
Let's solve each part of the question in detail:
### Question 1:
#### What is the length of the side of the side panel opposite the obtuse angle? Round your answer to the nearest hundredth.
Unfortunately, without specific dimensions or angles given for the side panel, we cannot determine the length of the side opposite the obtuse angle. Additional information about the triangle or its angles is needed.
Answer: Insufficient information to determine the length.
### Question 2:
#### What is the area of the side panel? Round your answer to the nearest hundredth.
Similar to Question 1, without specific dimensions or additional specifics about the side panel, it is impossible to determine the area with the current information.
Answer: Insufficient information to determine the area.
### Question 3:
#### On the side of the panel, a bright yellow triangle is painted whose sides measure 2 feet, 3.5 feet, and 5 feet. What is the area of the bright yellow triangle? Round your answer to the nearest hundredth.
To find the area of a triangle when the lengths of all three sides are known, we can use Heron's formula. The steps are as follows:
1. Calculate the semi-perimeter [tex]\(s\)[/tex]:
[tex]\[ s = \frac{a + b + c}{2} \][/tex]
where [tex]\(a = 2\)[/tex] feet, [tex]\(b = 3.5\)[/tex] feet, and [tex]\(c = 5\)[/tex] feet.
[tex]\[ s = \frac{2 + 3.5 + 5}{2} = \frac{10.5}{2} = 5.25 \text{ feet} \][/tex]
2. Use Heron's formula to find the area [tex]\(A\)[/tex]:
[tex]\[ A = \sqrt{s(s-a)(s-b)(s-c)} \][/tex]
Substitute the values:
[tex]\[ A = \sqrt{5.25(5.25 - 2)(5.25 - 3.5)(5.25 - 5)} = \sqrt{5.25 \times 3.25 \times 1.75 \times 0.25} \][/tex]
3. Calculate the expression inside the square root:
[tex]\[ 5.25 \times 3.25 \times 1.75 \times 0.25 \approx 7.4766 \][/tex]
4. Take the square root:
[tex]\[ A = \sqrt{7.4766} \approx 2.73 \text{ square feet} \][/tex]
Answer: 2.73 square feet.
### Question 4:
#### The front wheels on DeMarius' car are divided into sectors of equal area. The radius of each wheel is 8 inches. The area painted blue is twice the area painted green. The area painted green is half the area painted red. What is the area painted red on one of the front wheels? Round your answer to the nearest hundredth.
Given:
- The radius of the wheel [tex]\(r = 8\)[/tex] inches.
1. Calculate the total area of the wheel:
[tex]\[ \text{Total area} = \pi r^2 = \pi (8^2) = 64\pi \text{ square inches} \][/tex]
2. Determine the ratios:
- Let the area painted red be [tex]\(R\)[/tex].
- The area painted green is [tex]\(0.5R\)[/tex].
- The area painted blue is [tex]\(2 \times 0.5R = R\)[/tex].
3. Sum of areas of sectors:
Since the wheel is divided into equal sectors and the areas are proportional, we can sum up as follows:
[tex]\[ R + 0.5R + R = 2.5R \][/tex]
And given that the wheel is divided into equal sectors with known areas:
[tex]\[ 2.5R = 64\pi \][/tex]
4. Solve for [tex]\(R\)[/tex]:
[tex]\[ R = \frac{64\pi}{2.5} = \frac{64 \times 3.1416}{2.5} \approx 80.42 \text{ square inches} \][/tex]
Answer: 80.42 square inches.
### Question 5:
#### The rear wheels of DeMarius' car complete 5 rotations for every full rotation of a front wheel. What is the radius, in feet, of a rear wheel on the car? Write your answer as a simplified fraction.
Given:
- The radius of the front wheel is 8 inches.
- 1 foot = 12 inches, so the radius of the front wheel in feet is:
[tex]\[ \text{Front wheel radius} = \frac{8}{12} = \frac{2}{3} \text{ feet} \][/tex]
Since the rear wheel completes 5 rotations for every full rotation of the front wheel, the rear wheel travels the same distance in 5 rotations that the front wheel travels in 1 rotation.
1. Understand the relationship:
- [tex]\( \text{Circumference of the rear wheel} = \text{Circumference of the front wheel} \div 5 \)[/tex]
- The circumference of the front wheel is [tex]\(2\pi \times \frac{2}{3} = \frac{4\pi}{3} \text{ feet}\)[/tex].
2. Solve for the radius of the rear wheel:
- Let the radius of the rear wheel be [tex]\(r\)[/tex].
- The circumference of the rear wheel [tex]\(= 2\pi r \div 5 = \frac{4\pi}{3}\)[/tex].
Equating the circumferences:
[tex]\[ 2\pi r = 5 \cdot \frac{4\pi}{3} \][/tex]
Simplifying for [tex]\(r\)[/tex]:
[tex]\[ r = \frac{20\pi}{6\pi} = \frac{20}{6} = \frac{10}{3} \text{ feet} \][/tex]
Answer: [tex]\(\frac{10}{3}\)[/tex] feet.
### Question 1:
#### What is the length of the side of the side panel opposite the obtuse angle? Round your answer to the nearest hundredth.
Unfortunately, without specific dimensions or angles given for the side panel, we cannot determine the length of the side opposite the obtuse angle. Additional information about the triangle or its angles is needed.
Answer: Insufficient information to determine the length.
### Question 2:
#### What is the area of the side panel? Round your answer to the nearest hundredth.
Similar to Question 1, without specific dimensions or additional specifics about the side panel, it is impossible to determine the area with the current information.
Answer: Insufficient information to determine the area.
### Question 3:
#### On the side of the panel, a bright yellow triangle is painted whose sides measure 2 feet, 3.5 feet, and 5 feet. What is the area of the bright yellow triangle? Round your answer to the nearest hundredth.
To find the area of a triangle when the lengths of all three sides are known, we can use Heron's formula. The steps are as follows:
1. Calculate the semi-perimeter [tex]\(s\)[/tex]:
[tex]\[ s = \frac{a + b + c}{2} \][/tex]
where [tex]\(a = 2\)[/tex] feet, [tex]\(b = 3.5\)[/tex] feet, and [tex]\(c = 5\)[/tex] feet.
[tex]\[ s = \frac{2 + 3.5 + 5}{2} = \frac{10.5}{2} = 5.25 \text{ feet} \][/tex]
2. Use Heron's formula to find the area [tex]\(A\)[/tex]:
[tex]\[ A = \sqrt{s(s-a)(s-b)(s-c)} \][/tex]
Substitute the values:
[tex]\[ A = \sqrt{5.25(5.25 - 2)(5.25 - 3.5)(5.25 - 5)} = \sqrt{5.25 \times 3.25 \times 1.75 \times 0.25} \][/tex]
3. Calculate the expression inside the square root:
[tex]\[ 5.25 \times 3.25 \times 1.75 \times 0.25 \approx 7.4766 \][/tex]
4. Take the square root:
[tex]\[ A = \sqrt{7.4766} \approx 2.73 \text{ square feet} \][/tex]
Answer: 2.73 square feet.
### Question 4:
#### The front wheels on DeMarius' car are divided into sectors of equal area. The radius of each wheel is 8 inches. The area painted blue is twice the area painted green. The area painted green is half the area painted red. What is the area painted red on one of the front wheels? Round your answer to the nearest hundredth.
Given:
- The radius of the wheel [tex]\(r = 8\)[/tex] inches.
1. Calculate the total area of the wheel:
[tex]\[ \text{Total area} = \pi r^2 = \pi (8^2) = 64\pi \text{ square inches} \][/tex]
2. Determine the ratios:
- Let the area painted red be [tex]\(R\)[/tex].
- The area painted green is [tex]\(0.5R\)[/tex].
- The area painted blue is [tex]\(2 \times 0.5R = R\)[/tex].
3. Sum of areas of sectors:
Since the wheel is divided into equal sectors and the areas are proportional, we can sum up as follows:
[tex]\[ R + 0.5R + R = 2.5R \][/tex]
And given that the wheel is divided into equal sectors with known areas:
[tex]\[ 2.5R = 64\pi \][/tex]
4. Solve for [tex]\(R\)[/tex]:
[tex]\[ R = \frac{64\pi}{2.5} = \frac{64 \times 3.1416}{2.5} \approx 80.42 \text{ square inches} \][/tex]
Answer: 80.42 square inches.
### Question 5:
#### The rear wheels of DeMarius' car complete 5 rotations for every full rotation of a front wheel. What is the radius, in feet, of a rear wheel on the car? Write your answer as a simplified fraction.
Given:
- The radius of the front wheel is 8 inches.
- 1 foot = 12 inches, so the radius of the front wheel in feet is:
[tex]\[ \text{Front wheel radius} = \frac{8}{12} = \frac{2}{3} \text{ feet} \][/tex]
Since the rear wheel completes 5 rotations for every full rotation of the front wheel, the rear wheel travels the same distance in 5 rotations that the front wheel travels in 1 rotation.
1. Understand the relationship:
- [tex]\( \text{Circumference of the rear wheel} = \text{Circumference of the front wheel} \div 5 \)[/tex]
- The circumference of the front wheel is [tex]\(2\pi \times \frac{2}{3} = \frac{4\pi}{3} \text{ feet}\)[/tex].
2. Solve for the radius of the rear wheel:
- Let the radius of the rear wheel be [tex]\(r\)[/tex].
- The circumference of the rear wheel [tex]\(= 2\pi r \div 5 = \frac{4\pi}{3}\)[/tex].
Equating the circumferences:
[tex]\[ 2\pi r = 5 \cdot \frac{4\pi}{3} \][/tex]
Simplifying for [tex]\(r\)[/tex]:
[tex]\[ r = \frac{20\pi}{6\pi} = \frac{20}{6} = \frac{10}{3} \text{ feet} \][/tex]
Answer: [tex]\(\frac{10}{3}\)[/tex] feet.
