Cylinder A has radius 4 times the radius of cylinder B, and height 4 times the height of cylinder B.
What is the ratio of the lateral area of A to the lateral area of B?
The surface areas of two similar shapes are proportional to
the squares of their corresponding sides
S(Cylinder 1)/S(cylinder 2) = 4^2/1^2 = 16
So A's surface area is 16 times that of B's surface
or, the long way ...
cylinder B: radius -- r, height --- h
cylinder A: radius = 4r, height = 4h
surface area of A = 2πr^2 + 2πrh = 2(πr^2 + πrh)
surface area of B = 2π(16r^2) + 2π(4r)(4h)
= 32πr^2 + 32πrh = 32(πr^2 + πrh) or 16 times that of A
Ah, the world of cylinders, where circles go for a spin! Now, let me calculate the ratio of the lateral area of Cylinder A to Cylinder B for you.
The lateral area of a cylinder is given by the formula 2πrh, where r is the radius and h is the height.
So, let's consider Cylinder A first. If the radius of Cylinder B is r, then the radius of Cylinder A would be 4r. Similarly, if the height of Cylinder B is h, then the height of Cylinder A would be 4h.
Now, let's plug in these values into the formula for the lateral area of both cylinders.
Lateral area of Cylinder A = 2π(4r)(4h) = 32πrh
Lateral area of Cylinder B = 2π(r)(h) = 2πrh
To find the ratio of the lateral area of A to B, we divide the lateral area of A by the lateral area of B:
(32πrh) / (2πrh) = 32/2 = 16/1
Voila! The ratio of the lateral area of Cylinder A to Cylinder B is 16:1. That's quite the ratio, isn't it?
To find the ratio of the lateral area of cylinder A to the lateral area of cylinder B, we need to compare their surface areas.
The surface area of a cylinder is given by the formula: 2πrh, where r is the radius and h is the height.
Let's denote the radius of cylinder B as rB and the height of cylinder B as hB.
According to the problem, the radius of cylinder A is 4 times the radius of cylinder B. So, the radius of cylinder A is 4rB.
Similarly, the height of cylinder A is 4 times the height of cylinder B. So, the height of cylinder A is 4hB.
Now, let's calculate the lateral area of cylinder A and cylinder B.
Lateral area of cylinder A = 2π(4rB)(4hB) = 32πrBhB
Lateral area of cylinder B = 2π(rB)(hB) = 2πrBhB
Therefore, the ratio of the lateral area of cylinder A to the lateral area of cylinder B is:
(32πrBhB) / (2πrBhB) = 16
So, the ratio of the lateral area of cylinder A to the lateral area of cylinder B is 16:1.
To find the ratio of the lateral area of cylinder A to the lateral area of cylinder B, we first need to understand what the lateral area of a cylinder is.
The lateral area of a cylinder is the sum of the areas of all the sides, excluding the top and bottom surfaces. It can be calculated using the formula:
Lateral Area = circumference of base × height of cylinder
Now, let's calculate the lateral area of each cylinder.
Cylinder A:
Given that the radius of cylinder A is 4 times the radius of cylinder B, and the height of cylinder A is 4 times the height of cylinder B, we can say that if the radius of cylinder B is 'r' and the height of cylinder B is 'h', then the radius of cylinder A is 4r and the height of cylinder A is 4h.
The circumference of the base of cylinder A is given by:
Circumference of base of A = 2π(radius of A) = 2π(4r) = 8πr
Therefore, the lateral area of cylinder A is:
Lateral Area of A = circumference of base of A × height of A = 8πr * 4h = 32πrh
Cylinder B:
The circumference of the base of cylinder B is given by:
Circumference of base of B = 2π(radius of B) = 2πr
Therefore, the lateral area of cylinder B is:
Lateral Area of B = circumference of base of B × height of B = 2πr * h = 2πrh
Now, to find the ratio of the lateral area of A to the lateral area of B, we divide the lateral area of A by the lateral area of B:
Ratio = (Lateral Area of A) / (Lateral Area of B)
Ratio = (32πrh) / (2πrh)
Ratio = 16
Therefore, the ratio of the lateral area of cylinder A to the lateral area of cylinder B is 16:1.