Video Transcript
In the given figure, the circle has a center 𝑂, and a radius of nine. If the length of arc 𝑀𝑁 is between four and five, what is one possible integer value of 𝑥? Note, the figure is not drawn to scale.
As the figure is not drawn to scale, we cannot measure our answer with a protractor. We’re told in the question that the radius of the circle is nine. This means that the lengths 𝑂𝑁 and 𝑂𝑀 are both equal to nine. The arc length of any sector is equal to 𝜃 divided by 360 multiplied by 𝜋𝑑, where 𝜃 is the angle of the sector and 𝑑 is the diameter.
As the radius of our circle is equal to nine, the diameter will be equal to two multiplied by nine, as the diameter is double the radius. Two multiplied by nine is equal to 18. Therefore, the diameter of the circle is 18. Substituting in our value for the diameter and our unknown angle 𝑥, the arc length is equal to 𝑥 divided by 360 multiplied by 18𝜋. 18 and 360 are both divisible by 18. 18 divided by 18 is one. And 360 divided by 18 is equal to 20. This is because 36 is double 18. The arc length 𝑀𝑁 simplifies to 𝜋𝑥 over 20.
We’re told in the question that the arc length is between four and five. We can write this as an inequality so that 𝜋𝑥 over 20 is greater than four but less than five. We can then multiply all three parts of the inequality by 20. Four multiplied by 20 is equal to 80. Five multiplied by 20 is equal to 100. This means that 𝜋𝑥 is greater than 80 but less than 100. As we’re trying to find values for 𝑥, we can divide the inequality by 𝜋. This means that 𝑥 is greater than 80 over 𝜋 but less than 100 over 𝜋.
80 divided by 𝜋 is equal to 25.46 etcetera. 100 divided by 𝜋 is equal to 31.83 etcetera. This means that 𝑥 is great to than 25.46 and less than 31.83. We have been asked to find integer values of 𝑥. An integer is a whole number, either positive or negative. The whole numbers that lie between 25.46 and 31.83 are 26, 27, 28, 29, 30, and 31. The question asked us to pick one possible integer. Therefore, we can select any one of these six answers. If the length of the arc 𝑀𝑁 is between four and five, then the value of the angle 𝑥 in degrees could be 26, 27, 28, 29, 30, or 31.
