the ratio of total surface area to the curved surface area of a right circular cylinder is 3:2 find the volume if its total surface area is 14784 cm2
To solve this problem, we need to find the formulas for the total surface area (TSA) and curved surface area (CSA) of a right circular cylinder.
The formula for the total surface area of a right circular cylinder is:
TSA = 2πrh + 2πr²,
where "r" represents the radius of the base of the cylinder, and "h" represents the height of the cylinder.
The formula for the curved surface area of a right circular cylinder is:
CSA = 2πrh.
Let's assume that the curved surface area is represented by the ratio "3x," and the total surface area is represented by the ratio "2x."
According to the given information, the ratio of the total surface area to the curved surface area is 3:2, which can be written as:
TSA : CSA = 2x : 3x.
Now, substituting the formulas for TSA and CSA, we have:
(2πrh + 2πr²) : (2πrh) = 2x : 3x.
Simplifying the formula, we get:
(2rh + 2r²) : (2rh) = 2 : 3.
Next, let's simplify the equation further by dividing both sides by 2:
(rh + r²) : rh = 2 : 3.
We can multiply both sides by rh to get rid of the denominators:
rh + r² = (2/3)rh.
Now, let's rearrange the equation to isolate the term with the variable "r":
r² - (2/3)rh + rh = 0.
Combining like terms:
r² - (2/3)rh + (3/3)rh = 0.
r² + (rh/3) = 0.
Factoring out the common factor "r," we have:
r(r + h/3) = 0.
Since "r" cannot be zero, we can ignore the first factor "r" and focus on the second factor:
r + h/3 = 0.
Next, we solve for "r" in terms of "h":
r = -h/3.
Since the radius cannot be negative in this context, we discard this solution.
Now, let's find the value of "x" by substituting the value of "r" we found into the equation for the total surface area:
TSA = 2πrh + 2πr².
14784 = 2πrh + 2πr².
Substituting r = -h/3:
14784 = 2π(-h/3)h + 2π(-h/3)².
14784 = -2πh²/3 - 2πh²/9.
Multiplying both sides by -1 to eliminate the negative sign:
-14784 = 2πh²/3 + 2πh²/9.
Adding the fractions:
-14784 = (6πh² + 2πh²)/9.
Combining like terms:
-14784 = 8πh²/9.
Multiplying both sides by 9/8π:
h² = -14784 * (9/8π).
Taking the square root of both sides:
h = sqrt(-14784 * (9/8π)).
Since the height cannot be negative, we discard this solution as well.
Therefore, there is no valid solution for the given problem.
total SA = 2πr^2 + 2πrh ---> top + bottom + side
curved SA = 2πrh
(2πr^2 + 2πrh)/2πrh = 3/2
4πr^2 + 4πrh = 6πrh
4πr^2 = 2πrh
divide by 2πr
2r = h
2πr^2 + 2πrh = 14784
2πr^2 + 2πr(2r) = 14784
6πr^2 = 14784
r^2 = 2464/π
r = appr 28.01 cm
h = appr 56.01 cm