If rope length eighteen cm is use to form a sector of a cycle of radius c.2 cm on a school sport ground calculate the size of the angle of the sector?
This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Identify the given values and the formula for the perimeter of a sector.
The total rope length represents the perimeter of the sector.
The perimeter $P$ of a sector is given by the formula $P = 2r + L$, where $r$ is the radius and $L$ is the arc length.
Given:
Perimeter $P = 18$ cm
Radius $r = 2$ cm (interpreting "c.2 cm" as 2 cm)
Step 2: Calculate the arc length $L$.
Substitute the given values into the perimeter formula:
$$18 = 2(2) + L$$
$$18 = 4 + L$$
Subtract 4 from both sides to find $L$:
$$L = 18 - 4$$
$$L = 14 \text{ cm}$$
Step 3: Calculate the angle of the sector in radians.
The formula for arc length is $L = r\theta$, where $\theta$ is the angle in radians.
Substitute the values for $L$ and $r$:
$$14 = 2\theta$$
Divide by 2 to find $\theta$:
$$\theta = \frac{14}{2}$$
$$\theta = 7 \text{ radians}$$
The size of the angle of the sector is $\boxed{\text{7 radians}}$.
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