Given that the area of the coloured part in the figure is 155 square foot, find the value of 𝑥 to the nearest tenth.
Let’s recall what we actually remember about the area of a circular sector. For a circular sector with a radius 𝑟 and an angle 𝜃 radians, its area can be found using the formula, a half 𝑟 squared 𝜃. We actually know that the area of our circular sector is 155 square feet. And we’re looking to find the length of the radius.
We’re going to substitute what we know about our sector into the formula for sector area. However, before we do, we need to convert the angle 239 degrees into radians. To do this, we recall that two 𝜋 radians is equal to 360 degrees. We can then divide through by 360. And that tells us that one degree is equal to two 𝜋 divided by 360, which is equal to 𝜋 by 180 radians. Once we have this, we can multiply through by 239. And that tells us that 239 degrees is equal to 239𝜋 over 180 radians.
Substituting what we now know about our circular sector into the formula for the area tells us that the area is a half 𝑥 squared multiplied by 239𝜋 over 180. And of course, we know that the area is 155. So we equate this expression to 155. At this point, we could multiply by two and then by 180 to begin solving the equation. Instead though, we choose to do that in one step and multiply both sides by 360. 155 multiplied by 360 is 55800. So we see that 55800 is equal to 239𝜋𝑥 squared. We then divide through by 239𝜋. That gives us 74.316 and so on is equal to 𝑥 squared.
Now we won’t round this number just yet. Instead to solve, we’re going to find the square root of both sides of our equation. So we’ll square root this number in its unrounded form. That’s 8.6207 and so no. Now, when we square root, we should recall that there are two solutions, a positive and a negative. In this case though, we can immediately disregard the negative solution as 𝑥 is a length. And we can’t have a negative length.
Finally, we round this number to the nearest tenth. And that tells us that 𝑥 is equal to 8.6. We don’t need to give any units here since we’re told that 𝑥 is the number of feet. So 𝑥 is simply a number. It is 8.6.