Find, to the nearest hundredth, the area of the circle 𝑥 plus nine squared plus 𝑦 minus three squared equals 49.0.
The area of a circle can be found using the formula 𝜋𝑟 squared, where 𝑟 represents the radius of the circle. So in order to answer this question, we need to find the radius of the circle whose equation we’ve been given. To do so, we can recall the center-radius form of the equation of a circle. If a circle has center with coordinates ℎ, 𝑘 and a radius of 𝑟 units, then the center-radius form of its equation is 𝑥 minus ℎ squared plus 𝑦 minus 𝑘 squared equals 𝑟 squared.
We can see then that our circle is in center-radius form. And comparing the right side of the two equations, we have that 𝑟 squared is equal to 49.0. Now, we haven’t actually been asked to give the coordinates of the center of the circle. But just for completeness, if we compare the two forms, we can see that 𝑘 is equal to three because we have 𝑦 minus 𝑘 in the second equation and 𝑦 minus three in the first.
The value of ℎ, we have to think about slightly more carefully because we have negative ℎ in the general form but plus nine in our circle. So we have that negative ℎ is equal to nine. And dividing through by negative one, we see that ℎ is equal to negative nine. This means that the center of our circle is the point with coordinates negative nine, three. Although, as we already said, we haven’t actually been asked to work this out. Let’s return to calculating the radius of our circle.
We have that 𝑟 squared is equal to 49.0. So to find the value of 𝑟, we need to take the square root of both sides of the equation. Given 𝑟 is equal to the square root of 49.0, which is exactly seven, we use the positive square root as the radius needs to be a positive value. Now, we didn’t actually need to perform the last steps and work out the value of 𝑟 exactly because our formula for the area of the circle uses 𝑟 squared not 𝑟. So once we’d determine that 𝑟 squared is equal to 49.0 or just 49, we could in fact just substitute this value for 𝑟 squared into our area formula. We have then that the area of this circle is equal to 𝜋 multiplied by 49.
Now, if we were asked to give an exact answer or if we didn’t have access to a calculator, we could leave our answer as a multiple of 𝜋. It’s 49𝜋. But in this question, we’ve been asked to give our answer to the nearest hundredth. So we evaluate 49𝜋 on our calculator and it gives 153.93804, and the decimal continues. The digit in the next column to the right of the hundredth column, that’s the thousandth column, is an eight. And as this is greater than five, this tells us that we’re rounding up. So the area of the circle, to the nearest hundredth, is 153.94. The units for this answer are just general square units.