A container is to be made in the shape of a cylinder with a conical top. The lateral surface areas of the cylinder and cone are S1 = 2(pi)rh and S2 = 2(pi)r√(r^2 + h^2). The surface area of the base of the container is B= (pi)r^2. The height of the cylinder and cone are equal. The radius of the container is 5 inches and its total surface area is 275(pi) in^2. Find the total height of the container.
Height equals 24 inches
Let's solve step-by-step to find the total height of the container.
Step 1: Write down the given information:
- Radius of the container (r) = 5 inches
- Total surface area of the container = 275(pi) square inches
Step 2: Calculate the surface areas of the cylinder and cone:
- Lateral surface area of the cylinder (S1) = 2(pi)rh
- Surface area of the cone (S2) = 2(pi)r√(r^2 + h^2)
Since the height of the cylinder and cone are equal, let's denote the height as h.
Step 3: Use the given surface area equations:
- S1 = 2(pi)rh
- S2 = 2(pi)r√(r^2 + h^2)
Step 4: Use the surface area formulas to create an equation for the total surface area of the container:
- Total surface area (A) = S1 + S2 + B
Since the base surface area (B) of the container is (pi)r^2, we can substitute B into the equation:
- A = S1 + S2 + (pi)r^2
Step 5: Substitute the values we know into the equation for the total surface area:
- A = 275(pi)
- r = 5
A = 275(pi) --> S1 + S2 + (pi)r^2 = 275(pi)
Step 6: Substitute the formulas for S1 and S2 into the equation:
- S1 = 2(pi)rh
- S2 = 2(pi)r√(r^2 + h^2)
2(pi)rh + 2(pi)r√(r^2 + h^2) + (pi)r^2 = 275(pi)
Step 7: Simplify the equation by canceling out (pi) and rearrange the equation:
2rh + 2r√(r^2 + h^2) + r^2 = 275
Step 8: Divide both sides of the equation by r:
2h + 2√(r^2 + h^2) + r = 275/r
Step 9: Substitute the value of r, which is 5, into the equation:
2h + 2√(25 + h^2) + 5 = 275/5
Step 10: Simplify the equation further:
2h + 2√(25 + h^2) + 5 = 55
Step 11: Subtract 5 from both sides of the equation:
2h + 2√(25 + h^2) = 55 - 5
2h + 2√(25 + h^2) = 50
Step 12: Divide both sides of the equation by 2:
h + √(25 + h^2) = 25
Step 13: Square both sides of the equation to eliminate the square root:
(h + √(25 + h^2))^2 = 25^2
h^2 + 2h√(25 + h^2) + 25 + h^2 = 625
2h^2 + 2h√(25 + h^2) - 600 = 0
Step 14: Rearrange the equation to the form of a quadratic equation:
2h^2 + 2h√(25 + h^2) - 600 = 0
h^2 + h√(25 + h^2) - 300 = 0
Step 15: Solve this quadratic equation using the quadratic formula or factoring. Unfortunately, the equation does not factor nicely, so we will use the quadratic formula:
h = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 1, b = 1, and c = -300. Plugging those values into the quadratic formula:
h = (-1 ± √(1^2 - 4(1)(-300))) / (2(1))
h = (-1 ± √(1 + 1200)) / (2)
h = (-1 ± √(1201)) / 2
Since the height cannot be negative, we take the positive root:
h ≈ (√(1201) - 1) / 2
So, the total height of the container is approximately (√(1201) - 1) / 2 inches.
To find the total height of the container, we need to find the respective heights of the cylinder and the cone. Let's break down the steps to solve this problem:
Step 1: Understand the problem and identify the given information.
- We have a container in the shape of a cylinder with a conical top.
- The radius of the container is given as 5 inches.
- The total surface area of the container is given as 275(pi) square inches.
- The height of the cylinder and the cone are equal.
Step 2: Set up the equation for the total surface area of the container.
The total surface area (TSA) of the container is the sum of the lateral surface area of the cylinder, the lateral surface area of the cone, and the surface area of the base. We can write this as an equation:
TSA = S1 + S2 + B
Given the formulas for the surface areas of the cylinder and cone, we can substitute the corresponding values:
275(pi) = 2(pi)rh + 2(pi)r√(r^2 + h^2) + (pi)r^2
Step 3: Simplify the equation.
To simplify, we can divide the entire equation by (pi) to get:
275 = 2rh + 2r√(r^2 + h^2) + r^2
Step 4: Substitute the radius value.
The problem states that the radius of the container is 5 inches, so we can substitute r = 5 into the equation:
275 = 2(5)h + 2(5)√(5^2 + h^2) + (5)^2
Simplifying further:
275 = 10h + 10√(25 + h^2) + 25
Step 5: Solve the equation.
To solve this equation, we need to isolate the variable h. Let's rearrange the equation:
10h + 10√(25 + h^2) = 275 - 25
10h + 10√(25 + h^2) = 250
Now divide both sides by 10 to isolate the square root term:
h + √(25 + h^2) = 25
Next, let's square both sides of the equation to remove the square root:
(h + √(25 + h^2))^2 = 25^2
Expanding the left side of the equation:
h^2 + 2h√(25 + h^2) + 25 + h^2 = 625
Simplifying further:
2h√(25 + h^2) + 2h^2 = 600
Rearranging the terms:
2h(√(25 + h^2) + h) = 600
Now divide both sides by 2h:
√(25 + h^2) + h = 300/h
Rearranging the equation:
√(25 + h^2) = 300/h - h
Square both sides of the equation to remove the square root:
25 + h^2 = (300/h - h)^2
Expand the right side of the equation:
25 + h^2 = (300/h)^2 - 2(300/h)h + h^2
Simplifying further:
25 = (300/h)^2 - 2(300) + h^2
25 + 2(300) = (300/h)^2
625 = (300/h)^2
Now take the square root of both sides:
25 = 300/h
Dividing both sides by 25:
h = 300/25
Simplifying:
h = 12
Step 6: Determine the total height of the container.
Since the height of the cylinder (h) is equal to the height of the cone, the total height (H) of the container is equal to twice the height of the cylinder:
H = 2h = 2(12) = 24 inches
Therefore, the total height of the container is 24 inches.