Question

Высота цилиндра на 10 см больше радиуса основания,а полная поверхность равна 144П см.Определить радиус основания и высоту.

Answer 1

H=r+10
114π=2*πr^2+2πr*(r+10)
57=r^2+r^2+10
2r^2=47
r=√47
h=√47+10

Answer 2

Problem Analysis

We are given that the height of a cylinder is 10 cm greater than the radius of its base, and the total surface area of the cylinder is 144π cm². We need to determine the radius of the base and the height of the cylinder.

Solution

Let's assume that the radius of the base of the cylinder is r cm. According to the given information, the height of the cylinder is r + 10 cm.

The total surface area of a cylinder is given by the formula: S = 2πr² + 2πrh

Substituting the values we have: 144π = 2πr² + 2πr(r + 10)

Simplifying the equation: 144 = 2r² + 2r(r + 10) 144 = 2r² + 2r² + 20r 144 = 4r² + 20r 4r² + 20r - 144 = 0

Now we can solve this quadratic equation to find the value of r.

Solving the Quadratic Equation

We can solve the quadratic equation using the quadratic formula: r = (-b ± √(b² - 4ac)) / (2a)

In this case, the equation is in the form ar² + br + c = 0, where: a = 4 b = 20 c = -144

Substituting the values into the quadratic formula: r = (-20 ± √(20² - 4 * 4 * -144)) / (2 * 4)

Simplifying the equation: r = (-20 ± √(400 + 2304)) / 8 r = (-20 ± √(2704)) / 8 r = (-20 ± 52) / 8

We have two possible solutions for r: 1. r = (-20 + 52) / 8 = 32 / 8 = 4 2. r = (-20 - 52) / 8 = -72 / 8 = -9

Since the radius cannot be negative, we discard the second solution.

Therefore, the radius of the base of the cylinder is 4 cm.

To find the height of the cylinder, we can substitute the value of r into the equation for the height: height = r + 10 = 4 + 10 = 14 cm

Answer

The radius of the base of the cylinder is 4 cm and the height of the cylinder is 14 cm.