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jeka94
3 years ago
9

A rectangle is inscribed in a circle.

Mathematics
1 answer:
sergey [27]3 years ago
4 0

Answer:

1. The area of the circle is 132.67 square centimeters

2. The area of the rectangle is 60 square centimeters

3. The area of the shaded region is 72.67 square centimeters

Step-by-step explanation:

* Lets explain how to solve the problem

- A rectangle is inscribed in a circle, that means the vertices of the

 rectangle lie on the circumference of the circle

∴ The diagonal of the rectangle = the diameter of the circle

- From the attached figure

∵ The diameter of the circle = 13 cm

∵ The radius of the circle is half the diameter

∴ The radius of the circle = 1/2 (13) = 6.5 cm

- The area of the circle = πr²

∵ π = 3.14

∴ The area of the circle = 3.14 × (6.5)²  = 132.67 cm²

1. The area of the circle is 132.67 square centimeters

- The area of any rectangle = length × width

∵ The length of the rectangle is 12 cm

∵ The width of the rectangle is 5 cm

∴ The area of the rectangle = 12 × 5 = 60 cm²

2. The area of the rectangle is 60 square centimeters

- The area of the shaded region is the difference between the

  area of the circle and the area of the rectangle

∵ The area of the circle = 132.67 cm²

∵ The area of the rectangle = 60 cm²

∵ The area of the shaded region = area of circle - area of rectangle

∴ The area of the shaded region = 132.67 - 60 = 72.67 cm²

3. The area of the shaded region is 72.67 square centimeters

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A fence must be built to enclose a rectangular area of 20,000 ft^2. Fencing material costs $4 per foot for the two sides facing
Marysya12 [62]

The cost of the least expensive fence is $3200

Step-by-step explanation:

The given is:

  • A fence must be built to enclose a rectangular area of 20,000 ft²
  • Fencing material costs $4 per foot for the two sides facing north and south and ​$8 per foot for the other two sides

We need to find the cost of the least expensive fence

Assume that the length of each side opposite to North or South is x feet and the length of each other sides  is y feet

∵ The length of the rectangle = x feet

∵ The width of the rectangle = y feet

∵ The rectangular area is 20,000 ft²

- Area of a rectangle = length × width

∴ x × y = 20,000

- Divide both sides by x to find y in terms of x

∴ y=\frac{20,000}{x}

The fence's length is equal to the perimeter of the rectangular area

∵ Perimeter of the rectangle = 2 length + 2 width

∴ Perimeter of the rectangle = 2x + 2y

∵ Fencing material costs $4 per foot for the two sides facing

   North and South

∴ x costs $4 per foot

∵ The cost of the other two sides is $8 per foot

∴ y costs $8 per feet

The cost of the fence is the sum of the products of 4 , 2x and 8 , 2y

∵ The cost of the fence (C) = 4(2x) + 8(2y)

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- Substitute y by its value above

∴ C=8x+16(\frac{20,000}{x})

∴ C=8x+\frac{320,000}{x}

To find the least expensive differentiate C with respect to x and equate the answer by 0 to find the value of x

∵ \frac{320,000}{x} can be written as 320,000x^{-1}

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∵ \frac{dC}{dx}=8(1)x^{1-1}+320,000(-1)x^{-1-1}

∴ \frac{dC}{dx}=8-320,000x^{-2}

- Equate \frac{dC}{dx} by zero

∴ 8-320,000x^{-2}=0

∵ -320,000x^{-2}=-\frac{320,000}{x^{2}}

∴ 8-\frac{320,000}{x^{2}}=0

- Subtract 8 from both sides

∴ -\frac{320,000}{x^{2}}=-8

- Multiply both sides by x²

∴ - 320,000 = - 8x²

- Divide both sides by -8

∴ 40,000 = x²

- Take √  for both sides

∴ 200 = x

Substitute x in the equation of C to find the least cost of fence

∵ C=8(200)+\frac{320,000}{(200)}

∴ C = 1600 + 1600

∴ C = 3200

The cost of the least expensive fence is $3200

Learn more:

You can learn more about the differentiation in brainly.com/question/4279146

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Hello!

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In this case, x_{1} and x_{1} is assigned to the ordered pair (1,3), while x_{2} and y_{2} is assigned to (3, 7). After assigning these points, you can substitute the pairs into the formula and simplify.

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OlgaM077 [116]

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             5.    16

Step-by-step explanation: am here to help

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