a cylinder has a height of 5cm and a surface area of 100 cm^2. find the radius of the cylinder, to the nearest tenth of a centimetre
Oh, we're talking about cylinders now? I guess I should keep my jokes rolling then! Let's solve this math problem with a touch of comedy.
Okay, so the formula to calculate the surface area of a cylinder is 2πr(r+h), where r is the radius and h is the height. We know that the surface area is 100 cm^2, so let's plug in the given values and solve for r.
100 = 2πr(r+5)
Now, let's bring out those math skills. But first, let's hope that the solution is not as elusive as a disappearing clown from a circus!
Done with my calculations! The radius of the cylinder, to the nearest tenth, is approximately 3.2 cm. I hope that puts a smile on your face, just like a clown!
To find the radius of the cylinder, we can use the formula for the surface area of a cylinder:
Surface Area = 2πr^2 + 2πrh
Given that the height (h) is 5 cm and the surface area is 100 cm^2, we can substitute these values into the equation to solve for the radius.
100 = 2πr^2 + 2πr(5)
Let's simplify and solve the equation step by step:
1. Divide the entire equation by 2π to isolate the terms with the radius:
100/(2π) = r^2 + 5r
2. Rearrange the equation to bring the terms to one side:
0 = r^2 + 5r - 100/(2π)
3. Multiply the value of 100/(2π) using a calculator to get an approximate value. Let's assume it is 15.92.
0 = r^2 + 5r - 15.92
4. Now, we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. In this case, let's use the quadratic formula:
The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions for x are given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our equation, the coefficients are:
a = 1, b = 5, and c = -15.92
Plugging in these values, the formula becomes:
r = (-5 ± √(5^2 - 4 * 1 * -15.92)) / (2 * 1)
5. Calculate the value under the square root:
√(5^2 - 4 * 1 * -15.92) ≈ √(25 + 63.68) ≈ √88.68 ≈ 9.42
6. Plug in the value for the square root in the formula and simplify:
r = (-5 ± 9.42) / 2
If we add 9.42 to -5 and divide by 2:
r1 = (9.42 - 5) / 2 ≈ 2.21
If we subtract 9.42 from -5 and divide by 2:
r2 = (-5 - 9.42) / 2 ≈ -7.71
Since we can't have a negative radius, we discard r2.
Therefore, the radius of the cylinder, to the nearest tenth of a centimeter, is approximately 2.2 cm.
To find the radius of the cylinder, we can use the formula for the surface area of a cylinder, which is given by:
Surface Area = 2πr² + 2πrh
where r is the radius of the circular base, h is the height of the cylinder, and π is the mathematical constant pi (approximately 3.14159).
In this case, we are given the height of the cylinder as 5 cm and the surface area as 100 cm². Let's substitute these values into the formula:
100 = 2πr² + 2π(5)(r)
Simplifying the equation further, we have:
100 = 2πr² + 10πr
Now, let's rearrange the equation to isolate the term with the variable r:
2πr² + 10πr - 100 = 0
We have a quadratic equation in terms of r. To solve for r, we can apply the quadratic formula:
r = (-b ± √(b² - 4ac)) / (2a)
The quadratic equation is now in the standard form ax² + bx + c = 0, with:
a = 2π, b = 10π, and c = -100.
Substituting these values into the quadratic formula:
r = (-(10π) ± √((10π)² - 4(2π)(-100))) / (2(2π))
Simplifying further:
r = (-10π ± √(100π² + 800π)) / (4π)
r = (-5π ± √(25π² + 200π)) / (2π)
To find the approximate value of r to the nearest tenth of a centimeter, we can evaluate both solutions using a calculator:
r ≈ (−5π + √(25π² + 200π)) / (2π) ≈ 2.669 cm
r ≈ (−5π - √(25π² + 200π)) / (2π) ≈ -8.814 cm
Since a negative radius doesn't make sense in this context, we can disregard the second solution.
Therefore, the radius of the cylinder, to the nearest tenth of a centimeter, is approximately 2.7 cm.
2 pi r * 5 = area
or if including top and bottom
2 pi r * 5 + 2 pi r^2 = area