1. What is the ratio of the radii of two circles with area = 3 π square units and 75π square units?
b. What is the ratio of their circumferences?
2. What is the ratio of the sides of two trapezoids with area = 8 square units and 72 square units?
b. What is the ratio of their perimeters?
first circle: radius r1
π r1^2 = 3π
2nd circle: radius r2
πr2^2 = 75π
divide the 2nd by the first
π r2^2/(πr1^2) = 75π/3π
r2^2 / r1^2 = 25/1
r2 : r1 = 5 : 1
#2, assuming the two traps are similar,
ratio of areas = ratio of their corresponding sides
the answer for question 1 is wrong
To find the ratio of the radii of two circles, we first need to find the radii of the circles using the given areas.
1a. To find the radii of the circles, we can use the formula for the area of a circle, which is A = πr^2. We'll solve this equation for the radius (r).
For the first circle with an area of 3π square units:
3π = πr^2
Dividing both sides by π, we get:
r^2 = 3
Taking the square root of both sides, we find:
r = √3
For the second circle with an area of 75π square units:
75π = πr^2
Dividing both sides by π, we get:
r^2 = 75
Taking the square root of both sides, we find:
r = √75 = 5√3
Therefore, the ratio of the radii is:
(5√3) / √3 = 5
1b. To find the ratio of their circumferences (perimeters), we'll use the formula for the circumference of a circle, which is C = 2πr.
For the first circle:
C1 = 2π(√3) = 2√3π
For the second circle:
C2 = 2π(5√3) = 10√3π
Therefore, the ratio of their circumferences is:
10√3π / 2√3π = 5
Now let's move on to the trapezoids.
2a. To find the ratio of the sides of two trapezoids, we'll use the formula for the area of a trapezoid, which is A = (b1 + b2) * h / 2, where b1 and b2 are the lengths of the parallel sides and h is the height.
For the first trapezoid with an area of 8 square units:
8 = (b1 + b2) * h / 2
Multiplying both sides by 2 and dividing by h, we get:
16 / h = b1 + b2
For the second trapezoid with an area of 72 square units:
72 = (b1 + b2) * h / 2
Multiplying both sides by 2 and dividing by h, we get:
144 / h = b1 + b2
Therefore, the ratio of the lengths of the parallel sides is:
(144 / h) / (16 / h) = 144 / 16 = 9
2b. To find the ratio of their perimeters, we'll use the formula for the perimeter of a trapezoid, which is P = b1 + b2 + 2s, where s is the slant height.
For the first trapezoid, let's assume the slant height is s1:
P1 = b1 + b2 + 2s1
For the second trapezoid, let's assume the slant height is s2:
P2 = b1 + b2 + 2s2
Since we don't have enough information to determine the slant height, we cannot find the exact ratio of their perimeters.
I hope this explanation helps! Let me know if you have any further questions.