In a diagram O is the centre of the circle,if angle<POQ is x and diameter is 7cm while the area of the shaded portion is 27.5cm^2.find correct to the nearest degree the value of x.(take pie as 22/7)
To find the value of angle <POQ, we need to use the given information about the diameter and the shaded area. Let's break down the problem into steps:
Step 1: Find the radius of the circle.
Since the diameter is given as 7 cm, we can calculate the radius by dividing the diameter by 2.
radius = diameter / 2 = 7 cm / 2 = 3.5 cm
Step 2: Find the area of the whole circle.
The area of a circle can be calculated using the formula A = πr^2, where A is the area and r is the radius.
area_of_circle = π × (radius)^2 = (22/7) × (3.5 cm)^2 = 38.5 cm^2
Step 3: Find the area of the shaded portion.
The area of the shaded portion is given as 27.5 cm^2.
Step 4: Find the ratio of the shaded area to the total area.
ratio = area_of_shaded_portion / area_of_circle = 27.5 cm^2 / 38.5 cm^2 ≈ 0.7143
Step 5: Find the angle corresponding to the ratio.
Since the ratio is the ratio of the shaded area to the total area, it can also represent the ratio of the angle <POQ to 360 degrees (the full angle of the circle). So, we need to find x, which is the value of angle <POQ.
x / 360 degrees = ratio
x = 360 degrees × ratio
x = 360 degrees × 0.7143
x ≈ 257.14 degrees (rounded to the nearest degree)
Therefore, the value of angle <POQ is approximately 257 degrees.
To find the value of x, we need to use the given information and apply some mathematical formulas.
1. We know that the shaded portion is the difference between the area of the sector OPOQ and the area of the triangle OPQ. So, we can start by calculating the area of the sector.
2. The formula to calculate the area of a sector is A = (θ/360) * πr², where A is the area, θ is the central angle, and r is the radius of the circle.
3. In this case, the diameter of the circle is given as 7 cm, so the radius (r) is half of the diameter, which is 7/2 = 3.5 cm.
4. We need to find the central angle, x, in degrees. To do this, we can use the formula θ = (x/360) * 2πr, where θ is the central angle, x is the measure of the angle in degrees, and r is the radius.
5. Substituting the given values, we get θ = (x/360) * (2 * (22/7) * 3.5).
6. The area of the sector OPOQ can now be calculated as A = (θ/360) * πr².
7. Given that the area of the shaded portion is 27.5 cm², we can express this as the equation A = (θ/360) * πr² = 27.5.
Now, we can solve the equation for x.
I hope this helps! Let me know if you have any further questions.