What is the ratio of the surface area for a pair of similar cylinders when the ratio of the heights is 1:3?
A) 1:4 B) 1:27 C) 1:9 D) 1:1
My answer is C) 1:9. Is this correct? Thanks.
I am not certain what you mean by similar?
Same diameter, different heights?
Same ratio changing height and diameter?
Are you considering surface area to include the ends?
If the same ratio is changing height and diameter is 1:3, then
SA=2PIr^2 + 2PIr(h) and if r and h are increased by three, then SA changes by a factor of nine.
So then the answer is it suppose to be 1:27 or 1:9?
Thanks.
To find the ratio of the surface areas of two similar cylinders, we need to consider that the ratio of their heights is given as 1:3.
The surface area of a cylinder consists of two bases and the lateral surface area. The base of the cylinder is a circle, and the area of a circle is given by the formula A = πr^2, where r is the radius of the circle. The lateral surface area of a cylinder is given by the formula A = 2πrh, where r is the radius and h is the height.
Let's assume that the two cylinders have radii R and r, and heights H and h, respectively. Since the ratio of their heights is 1:3, we can express h as h = (1/3)H.
For the first cylinder:
Base area = πR^2
Lateral surface area = 2πRH
For the second cylinder:
Base area = πr^2
Lateral surface area = 2πrh = 2π(1/3)Hr = (2/3)πHR
To find the ratio of the surface areas, we add the base area and the lateral surface area for each cylinder:
First cylinder:
Surface area = Base area + Lateral surface area = πR^2 + 2πRH
Second cylinder:
Surface area = Base area + Lateral surface area = πr^2 + (2/3)πHR
To compare the two surface areas, we can divide the surface area of the first cylinder by the surface area of the second cylinder:
(πR^2 + 2πRH) / (πr^2 + (2/3)πHR)
Notice that we have common factors of π, which cancel out:
(R^2 + 2RH) / (r^2 + (2/3)HR)
We know that h = (1/3)H, so we substitute that in:
(R^2 + 2R(1/3)H) / (r^2 + (2/3)Hr)
Simplifying further:
(R^2 + (2/3)RH) / (r^2 + (2/3)Hr)
Since the question asks for the ratio of the surface areas, we can divide the numerator and denominator by H:
(R^2/H + (2/3)R) / (r^2/H + (2/3)r)
We have a common factor of (2/3) in both terms of the numerator:
[(2/3)(R^2/H + R)] / [(2/3)(r^2/H + r)]
Cancelling (2/3) from numerator and denominator:
(R^2/H + R) / (r^2/H + r)
Observe that this ratio simplifies to:
(R + R^2/H) / (r + r^2/H)
Since we know that h = (1/3)H, we can substitute:
(R + R^2/(1/3)H) / (r + r^2/(1/3)H)
Simplifying within the parentheses:
(R + 3R^2/H) / (r + 3r^2/H)
The ratio of the surface areas can then be written as:
(R + 3R^2/H) : (r + 3r^2/H)
We are given that the ratio of heights is 1:3, so we can substitute h = (1/3)H:
(R + 3(R^2/(1/3)H)) : (r + 3(r^2/(1/3)H))
Further simplifying:
(R + 9R^2/H) : (r + 9r^2/H)
Now, we can see that the ratio of the surface area is (R + 9R^2/H):(r + 9r^2/H). From this expression, it is evident that the ratio of the surface areas is not 1:9.
Thus, the correct answer is not C) 1:9.