The sum of the radii of two concentric circles is 40. A chord with length 36 of the larger circle is trisected by the smaller circle. The area of the smaller circle is k*pi. Find the value of k. Express your answer as an exact decimal.
Please show work. Thank you very much.
r + R = 40
so
r = 40 - R
pi r^2 = k pi
so
k = r^2
right triangles from center to middle of chord
r^2 - 6^2 = R^2 - 18^2
R^2 -r^2 = 324-36 = 288
so we have two equations
R = 40 - r
R^2 -r^2 = 288
then
(40-r)^2 -r^2 = 288
1600 -80 r +r^2-r^2 = 288
80 r = 1312
r = 16.4
r^2 = 268.96
but k = r^2 = 268.96
To solve this problem, we need to find the radius of the smaller circle. Let's denote the radius of the smaller circle as r and the radius of the larger circle as R.
Since the sum of the radii of the two concentric circles is 40, we can write the following equation:
r + R = 40 -- (1)
We are given that a chord with a length of 36 of the larger circle is trisected by the smaller circle. This means that the two smaller segments of the chord (formed by the smaller circle) are equal in length.
Let's draw a diagram to visualize the situation:
---------------
\ | /
\ |α /
\ | /
O <-- Center of circles
/ | \
/ |β \
/_____|_____\
In the diagram, O represents the center of the circles, and the chord of the larger circle is divided into three segments: α, β, and α.
Since we have trisection, we can deduce that α + β + α = 36.
Simplifying, we find:
2α + β = 36 -- (2)
Now, we need to find a relation between the lengths of the α segment and the β segment. Let's use the Power of a Point theorem. According to this theorem, if two chords intersect inside a circle, the product of the lengths of their segments is equal. In our case, α * (α + β) = (r * r) = r².
Substituting the value of α + β from equation (2), we get:
α * (2α + β) = r²
2α² + αβ = r² -- (3)
We need to find a relation between R and r. Let's consider the chord of the larger circle where α and β segments are formed. By the Pythagorean theorem, we can write the following equation:
(α + β)² + R² = (2R)²
α² + β² + 2αβ + R² = 4R²
2αβ + R² = 3R² - α² - β²
2αβ = 3R² - α² - β² - R²
2αβ = 2R² - α² - β² -- (4)
Now, we can solve equations (3) and (4) simultaneously. Substituting equation (3) into equation (4), we get:
2(2α² + αβ) = 2R² - α² - β²
4α² + 2αβ = 2R² - α² - β²
5α² + 2αβ = 2R² -- (5)
To solve equations (4) and (5), we will express α² and β² in terms of α and R using equation (1):
(R - r)² + (r + α)² = r²
R² - 2Rr + r² + r² + 2rα + α² = r²
R² - 2Rr + 2rα + α² = 0 -- (6)
To solve equations (5) and (6), we will substitute α² from equation (6) into equation (5):
5(-R² + 2Rr - 2rα) + 2α(-R) = 2R²
-5R² + 10Rr - 10rα - 2Rα = 2R²
-5R² + 10Rr - 10rα - 2Rα = 2R²
-7R² + 2Rr - 10rα = 0 -- (7)
Now, we will solve equations (1) and (7) together to find the values of r and R.
We can use the substitution method to solve these equations. From equation (1), we have:
r = 40 - R
Substituting this value into equation (7), we get:
-7R² + 2R(40 - R) - 10(40 - R)α = 0
-7R² + 80R - 2R² - 400α + 10Rα = 0
-9R² + 70R + 400α = 0
Solving this quadratic equation for R, we get:
R² - (70/9)R - (400α/9) = 0
Using the quadratic formula, the value of R (radius of the larger circle) can be found as:
R = (70/18) + sqrt((70/9)² + (400α/9))
Since the chord of the larger circle is trisected, we know that α = β = 12. Substituting this value in the equation for R, we get:
R = (70/18) + sqrt((70/9)² + (400 * 12/9))
R = (70/18) + sqrt(4900/81 + 1600/3)
R = (70/18) + sqrt((4900 + 4800)/81)
R = (70/18) + sqrt(9700/81)
R = (70/18) + sqrt(112.35)
R ≈ 54.658
Now, we can substitute the value of R into equation (1) to find the value of r:
r + 54.658 = 40
r ≈ 40 - 54.658
r ≈ -14.658
However, since we're dealing with lengths, the negative value doesn't make sense in this context. Therefore, we discard r = -14.658 and conclude that the smaller radius is approximately 14.658.
Finally, we can find the area of the smaller circle using the formula for the area of a circle:
Area = π * r²
Substituting the value of r, we get:
Area = π * (14.658)²
Area ≈ π * 214.658
Area ≈ 668.965π
Therefore, the value of k is approximately 668.965.
Thus, k ≈ 668.965.