A candy cane 20 cm high has a circumference of 3 cm. The red “ribbon” makes exactly 4 complete turns round the candy cane while its two ends touch the cane's top and bottom. How long is the ribbon in cm?
Visualize the candy cane made in the form of a paper cylinder.
Cut the cylinder open, you will see that the "ribbon" around the candy cane or cylinder consists of 4 diagonals for 4 different rectangles.
the height of each of those rectangles is 20/4 or 5 cm
and the width is 3 cm
So the diagonal or hypotenuse is ..
√(5^2 + 3^2) = √34
So the length of the "ribbon" is 4√34
To find the length of the ribbon, we need to determine the total length covered by 4 complete turns around the candy cane.
First, let's calculate the circumference of the candy cane using the given information about its height and circumference. The circumference of a cylinder is given by the formula:
Circumference = 2πr
where π is a mathematical constant approximately equal to 3.14159 and r is the radius of the cylinder.
Since the candy cane is in the shape of a cylinder, we can find its radius by dividing the circumference by 2π:
Radius = Circumference / (2π)
= 3 cm / (2 * 3.14159)
≈ 0.477 cm (rounded to three decimal places)
Now, we know the radius of the candy cane, and we need to find the length of the ribbon when it makes 4 complete turns around the candy cane.
The length covered by one complete turn around the candy cane is equal to the circumference of a circle with a radius equal to the height of the candy cane. In this case, that's 20 cm.
Length of one complete turn = Circumference of circle with radius 20 cm
= 2π(20)
≈ 125.66 cm (rounded to two decimal places)
Since the ribbon makes 4 complete turns, we can simply multiply the length of one complete turn by 4 to get the total length of the ribbon:
Total length of the ribbon = Length of one complete turn * 4
≈ 125.66 cm * 4
≈ 502.64 cm (rounded to two decimal places)
Therefore, the length of the ribbon around the candy cane is approximately 502.64 cm.