Math  /  Geometry

QuestionProblem Solving 11) A sector of a circle of radius 28 cm has perimeter P cmP \mathrm{~cm} and area A cm2A \mathrm{~cm}^{2}. Given that A=4PA=4 P, find the value of PP. 12) The percentage error for sinθ\sin \theta for a given positive value of θ\theta is 1%1 \%. Show that 100θ=101sinθ100 \theta=101 \sin \theta.
Answers

Studdy Solution

STEP 1

What is this asking? We're given a circle sector with a known radius, and a relationship between its area and perimeter.
We need to find the perimeter. Watch out! Don't mix up the formulas for area and perimeter of a sector!
Also, remember to convert between radians and degrees if needed.

STEP 2

1. Define variables and formulas
2. Set up the equation
3. Solve for the angle
4. Calculate the perimeter

STEP 3

Let rr be the **radius** of the circle, which is given as r=28r = 28 cm.
Let θ\theta be the **angle** of the sector in radians.

STEP 4

The **area** of the sector is given by A=12r2θA = \frac{1}{2} r^2 \theta.
The **perimeter** of the sector is given by P=2r+rθP = 2r + r\theta.
We are also given that A=4PA = 4P.

STEP 5

We know that A=4PA = 4P.
Let's **substitute** the formulas for AA and PP into this equation: 12r2θ=4(2r+rθ) \frac{1}{2} r^2 \theta = 4(2r + r\theta)

STEP 6

Let's **plug in** the value of r=28r = 28 cm: 12(282)θ=4(228+28θ) \frac{1}{2} (28^2) \theta = 4(2 \cdot 28 + 28\theta) 12784θ=4(56+28θ) \frac{1}{2} \cdot 784 \theta = 4(56 + 28\theta) 392θ=224+112θ 392 \theta = 224 + 112\theta

STEP 7

Now, let's **isolate** θ\theta by subtracting 112θ112\theta from both sides: 392θ112θ=224 392 \theta - 112 \theta = 224 280θ=224 280 \theta = 224

STEP 8

Now, **divide** both sides by 280 to find θ\theta: θ=224280=45 \theta = \frac{224}{280} = \frac{4}{5} So, θ=45\theta = \frac{4}{5} radians.

STEP 9

Now that we know θ\theta, we can **calculate** PP: P=2r+rθ=2(28)+2845=56+1125=56+22.4=78.4 P = 2r + r\theta = 2(28) + 28 \cdot \frac{4}{5} = 56 + \frac{112}{5} = 56 + 22.4 = 78.4

STEP 10

The perimeter of the sector is P=78.4P = \textbf{78.4} cm.

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