In a figure, the radius of the inner circle of the tire is twice the width of the tire itself. To the nearest integer, what percent of the entire area is the area of the tire?
My teacher told me the answer is 56%, by rounding the 55.5555...., but I don't understand how to do the problem!
***The figure is just a car tire, a circle inside a circle (thus, determining the inner radius and the width of the tire).
THANKS
If the tire has width w, then the inner radius is 2w, and the entire area has radius 3w.
So,
π(9w^2-4w^2)/π(9w^2) = 5/9 = .5555 or 56%
THANKS YOU SO MUCH!!!!!!!
To solve this problem, we need to understand the relationship between the radius of the inner circle and the width of the tire. Let's call the radius of the inner circle "r" and the width of the tire "w".
According to the problem, the radius of the inner circle (r) is twice the width of the tire (w). Mathematically, we can express this relationship as:
r = 2w
Now, to find the area of the entire circle, we can use the formula for the area of a circle, which is given by:
A = πr^2
Similarly, the area of the tire (excluding the inner circle) can be calculated using the formula:
A_tire = π(r + w)^2
Now, to calculate the percentage of the entire area that the area of the tire represents, we can find the ratio of the area of the tire to the area of the entire circle and multiply it by 100. Mathematically, this can be expressed as:
Percentage = (A_tire / A) * 100
Substituting the expressions for A_tire and A that we derived earlier, we get:
Percentage = ((π(r + w)^2) / (πr^2)) * 100
Simplifying the equation, we get:
Percentage = ((r + w)^2 / r^2) * 100
Now, we can substitute r = 2w (from the given relationship), which gives us:
Percentage = ((2w + w)^2 / (2w)^2) * 100
Simplifying further, we get:
Percentage = ((3w)^2 / (2w)^2) * 100
Percentage = (9w^2 / 4w^2) * 100
Percentage = 9/4 * 100
Percentage = 225%
To the nearest integer, 225% is equal to 225. Therefore, the area of the tire represents 225% of the entire area.
It seems like there might be an error in your teacher's statement of the answer. The correct percentage is 225%, not 56%.