Question

$AB is an arc of a circle, with centre 'O', and radius 6 cm. \\The length of arc AB is 4\pi cm. \\The area of sector AOB is k\pi cm^{2}.\\Find the value of k.$4

Answer 1

Given :

• AB is arc of circle.
• Radius of circle = 6 cm
• Length of Arc AB = 4πcm
• Area of sector = kπ cm²

To Find :

• Value of k

Solution :

We have, the area of sector, radius and length of arc, so using the formula and using the available we can find k.

Formula :

$\large{\boxed{\rm{\red{Area\:of\:sector\:=\:{\dfrac{Length\:of\:arc\:\times\:radius}{2}}}}}}$

Put in the available data,

$\longrightarrow$ $\mathtt{k\:\pi\:=\:{\dfrac{4\:\pi\:\times\:6}{2}}}$

$\longrightarrow$ $\mathtt{2k\pi\:=\:4\:\pi\:6}$

$\longrightarrow$ $\mathtt{2k\:\pi\:=\:24\:\pi}$

$\longrightarrow$ $\mathtt{k\:=\:{\dfrac{24\:\pi}{2\:\pi}}}$

$\longrightarrow$ $\mathtt{k\:=\:{\dfrac{24}{2}}}$

$\longrightarrow$ $\mathtt{k\:=\:12}$

$\large{\boxed{\sf{\blue{Value\:of\:k\:=\:12}}}}$

VeRiFiCaTiOn :

We have k = 12. So putting the value of k in the formula we can verify the solution.

Let's begin.

$\longrightarrow$ $\mathtt{12\:\pi\:=\:{\dfrac{4\:\pi\:\times\:6}{2}}}$

$\longrightarrow$ $\mathtt{12\:\pi\:=\:{\dfrac{24\:\pi}{2}}}$

$\longrightarrow$ $\mathtt{12\:\pi\:=\:12\:\pi}$

$\longrightarrow$ $\mathtt{12\:=\:12}$

Since, LHS equals the RHS of the equation, we conclude that the value of k = 12 is appropriate.

Answer 2

Answer:

★Value of K will be 12 cm ★

Step-by-step explanation:

Given:

• AB is an arc of a circle , with centre 'O'
• Radius of circle is 6 cm
• Length of the Arc AB is 4πcm²

To Find: The value of 'k'

Solution: As we know that the formula for finding the value of k is

→ Kπ = Length of arc x Radius / 2

So, Putting the values,

$\small\implies{\sf }$ Kπ = 6π x 4 / 2

$\small\implies{\sf }$ Kπ = 24π/2

$\small\implies{\sf }$ Kπ = 12π

$\small\implies{\sf }$ K = 12

Hence, We got the value of 'K' = 12 cm