A designer is designing a cylindrical punching bag. He wants to use a maximum of 1.3m^2 of material. He has determined that the diameter of bag should be 36cm. Determine the maximum possible height of the bag to nearest tenth if a centimeter.
Please help and show steps on how you obtained thus answer. I am stuck!! Thank you for your help and time.
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To determine the maximum possible height of the cylindrical punching bag, we can start by finding the area of the bag's material that will be used.
The area of a cylindrical punching bag is given by the formula:
A = 2πrh + πr^2,
where A is the total area, r is the radius, and h is the height.
We are given that the diameter of the bag is 36 cm, so the radius (r) can be calculated by dividing the diameter by 2:
r = 36 cm / 2 = 18 cm.
Now, we can substitute this value of r into the formula and set the total area to be equal to the maximum area of 1.3 m^2:
1.3 m^2 = 2π(18 cm)(h) + π(18 cm)^2.
Next, we need to convert the units to maintain consistency. Since 1 m is equal to 100 cm, we can convert m^2 to cm^2 by multiplying by 10,000:
1.3 m^2 = 1.3 * 10,000 cm^2 = 13,000 cm^2.
Now, we can rewrite the equation with the converted units:
13,000 cm^2 = 2π(18 cm)(h) + π(18 cm)^2.
Simplifying this equation by evaluating the terms and rearranging it to solve for h, we get:
13,000 = 36πh + 324π.
Next, we can divide both sides of the equation by 36π to isolate h:
(13,000 - 324π) / (36π) = h.
Using a calculator to approximate the value of π as 3.14, we can evaluate the right side of the equation:
(13,000 - 324π) / (36π) ≈ 14.53 cm.
Therefore, the maximum possible height of the cylindrical punching bag to the nearest tenth of a centimeter is approximately 14.53 cm.
To determine the maximum possible height of the cylindrical punching bag, we need to consider the surface area of the material available and the diameter of the bag.
Step 1: Find the radius of the bag
The formula to find the radius (r) of a circle given the diameter (d) is r = d/2. In this case, the diameter of the bag is 36 cm, so the radius is 36 cm / 2 = 18 cm.
Step 2: Calculate the surface area of the material
The surface area of a cylinder consists of two circles (top and bottom) and the curved surface. The formula to calculate the surface area of a cylinder is A = 2πr² + 2πrh, where r is the radius and h is the height of the cylinder.
In this case, we want to use a maximum of 1.3 m² of material, which is equal to 13000 cm². We need to solve the equation A = 13000 cm² for h.
13000 = 2π(18²) + 2π(18h)
13000 = 2π(324) + 36πh
13000 = 648π + 36πh
13000 = 684πh
h = 13000 / (684π)
h ≈ 5.95 cm
The maximum possible height of the cylindrical punching bag to the nearest tenth of a centimeter is approximately 5.95 cm.