a cylinder of wooden block has base radius 'r' cm height 'h' cm and a total surface area of 32pi cm^2
1)show that r^2+hr=16
2)given that its height is 4cm more than its base radius, find the value of 'r' and of 'h'
the surface area is 2πr^2 + 2πrh
= 2π(r^2+hr)
Now it's clear that r^2+hr=16...
Now, using h=r+4, you have
2πr(r+r+4) = 32π
To solve these problems, we need to use the formula for the total surface area of a cylinder. The formula is given by 2πr² + 2πrh, where r is the base radius and h is the height of the cylinder.
1) To show that r² + hr = 16, we will use the given information that the total surface area is 32π cm².
Total Surface Area = 2πr² + 2πrh
Given: Total Surface Area = 32π cm²
Equating these two equations, we get:
2πr² + 2πrh = 32π
Dividing both sides by 2π, we get:
r² + rh = 16
This is the required equation r² + hr = 16.
2) Given that the height is 4 cm more than the base radius, we can express this relationship as:
h = r + 4
Substituting this value of h in the equation r² + hr = 16, we get:
r² + r(r + 4) = 16
Expanding the equation, we have:
r² + r² + 4r = 16
Combining like terms, we obtain:
2r² + 4r - 16 = 0
Dividing the equation by 2, we get:
r² + 2r - 8 = 0
Now, we can solve this quadratic equation using factorization, completing the square, or by using the quadratic formula. Factoring the equation gives us:
(r + 4)(r - 2) = 0
Setting each factor equal to zero, we can solve for r:
r + 4 = 0 or r - 2 = 0
If we solve these equations, we find:
r = -4 or r = 2
Since the radius of a cylinder cannot be negative, we can discard the solution r = -4. Therefore, the value of r is 2 cm.
Finally, we can find the value of h by substituting the value of r into the given relationship h = r + 4:
h = 2 + 4
h = 6
So, the value of r is 2 cm, and the value of h is 6 cm.