Sum of radii of two spherical ball is 22 cm and the diff of their volumes is 3050 2/3. Find radii. In this question if i r+R= 22 (eq 1)and Pi(r2-R2)=3050 2/3 (eq 2) and subs value of r from eq 1 in eq 2; my answer wont match with textbook. What should i do???
To solve this problem, the best approach would be to use a system of equations to find the radii of the two balls.
Let's call the radii of the two balls r and R, where r is the radius of the smaller ball and R is the radius of the larger ball.
From the given information, we have two equations:
1) r + R = 22 (Equation 1)
2) π(r^2 - R^2) = 3050 2/3 (Equation 2)
To solve this system of equations, we can start by rearranging Equation 1 to solve for r:
r = 22 - R
Now substitute this value of r in Equation 2:
π((22 - R)^2 - R^2) = 3050 2/3
Simplifying further:
(484 - 44R + R^2 - R^2)π = 3050 2/3
44Rπ = 3050 2/3 - 484π
44Rπ = 3050 2/3 - (484π / 1)
44Rπ = (3050 2/3 * 3 - 484π) / 3
44Rπ = (9150 - 484π) / 3
Now, solve for R:
R = (9150 - 484π) / (3 * 44π)
And once you have the value of R, substitute it back into Equation 1 to find r:
r = 22 - R
Calculating these values using a calculator will give you the exact radii of the two balls.
If your answer does not match the textbook, please double-check your calculations and make sure you have correctly substituted the values. Math errors or rounding errors can sometimes lead to different results.
To solve the problem correctly, let's go over the steps again step by step:
Let's assume the radius of the first ball is 'r' and the radius of the second ball is 'R'.
1. According to the given information, we can write the equation: r + R = 22 (equation 1), which is the sum of the radii.
2. The difference in volumes between the two spheres can be expressed as: (4/3) * π * r^3 - (4/3) * π * R^3 = 3050 2/3 (equation 2).
Now, to solve the problem:
a. Substitute the value of 'r' from equation 1 into equation 2:
(4/3) * π * (22 - R)^3 - (4/3) * π * R^3 = 3050 2/3
b. Simplify equation 2 further:
(4/3) * π * [(22^3 - 3 * 22^2 * R + 3 * 22 * R^2 - R^3) - R^3] = 3050 2/3
c. Expand the expression in the brackets:
(4/3) * π * (22^3 - 3 * 22^2 * R + 3 * 22 * R^2 - 2 * R^3) - (4/3) * π * R^3 = 3050 2/3
d. Simplify and combine like terms:
(4/3) * π * (22^3 - 3 * 22^2 * R + 3 * 22 * R^2 - 2 * R^3 - R^3) = 3050 2/3
e. Continue simplifying:
(4/3) * π * (22^3 - 3 * 22^2 * R + 66 * R^2 - 4 * R^3) = 3050 2/3
f. Finally, simplify further and solve for R by equating the expression to 3050 2/3:
(4/3) * π * (22^3 - 3 * 22^2 * R + 66 * R^2 - 4 * R^3) = 3050 2/3
Now, you can solve this equation to find the value of R. Use this value of R in equation 1, r + R = 22, to solve for r.