Answer :
Outer diameter: 39 cm, inner diameter: 11 cm, solved by analyzing curved surface area and volume equations.
Define the variables
Let:
R = outer radius of the cylinder
r = inner radius of the cylinder
h = height of the cylinder (which is given as 14 cm)
Write down the equations
The difference between the outer and inner curved surface area of the cylinder is given by:
2π(Rh - rh) = 88 cm²
The volume of the metal used in making the cylinder is given by:
π(R² - r²)h = 176 cm³
Solve for the unknowns
We can solve the system of equations for R and r.
From the first equation, we can isolate h:
h = 88 cm² / 2π(R - r)
Substitute this expression for h in the second equation:
π(R² - r²) (88 cm² / 2π(R - r)) = 176 cm³
Simplify and solve for R² - r²:
R² - r² = 352 cm²
Now, we can use the Pythagorean theorem to relate R and r:
(R + r)² = R² + 2Rr + r²
Substitute R² - r² = 352 cm² into the equation:
(R + r)² = 352 cm² + 2Rr
(R + r)² = 352 cm² + 2R(352 cm²) / (R + r)
(R + r)² = 704 cm² + 2R²
This is a quadratic equation in R + r. We can solve for R + r using the quadratic formula:
R + r = (-b ± √(b² - 4ac)) / 2a
where a = 2, b = 0, and c = -704
R + r = ± √(0² - 4 * 2 * -704) / 2 * 2
R + r = ± √2816 / 4
R + r = ± 14
Since R + r must be positive, we take the positive solution:
R + r = 14
Now, we can use the equation R² - r² = 352 cm² to solve for R and r individually:
R² - r² = 352 cm²
(R + r)(R - r) = 352 cm²
(14)(R - r) = 352 cm²
R - r = 25 cm
Add the equations R + r = 14 and R - r = 25 to get:
2R = 39
R = 19.5 cm
Substitute R back into the equation R - r = 25 to solve for r:
19.5 cm - r = 25 cm
r = 5.5 cm
Find the diameters
The diameter of the outer cylinder is 2R, and the diameter of the inner cylinder is 2r:
Diameter of outer cylinder = 2 * 19.5 cm = 39 cm
Diameter of inner cylinder = 2 * 5.5 cm = 11 cm
Therefore, the outer diameter of the cylinder is 39 cm and the inner diameter of the cylinder is 11 cm.
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