A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder. The total volume of the solid is 6 cubi
c centimeters. Find the radius of the cylinder that produces the minimum surface area. (Round your answer to three decimal places.) A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder. An industrial tank of this shape must have a volume of 4480 cubic feet. The hemispherical ends cost twice as much per square foot of surface area as the sides. Find the dimensions that will minimize the cost. (Round your answers to three decimal places.)
1 answer:
Answer:
1. The answer of the radius of the cylinder is 1.127cm to produce the minimum surface area
2. The radius of the industrial tank is 1.465ft and the height is 662.396 ft
Step-by-step explanation:
<u>The first step</u> is to draw the shape and to determine the total surface area and the total volume of the shape. This is done by adding the total surface area of two hemispheres and the surface area of the cylinder and for the volume this is done by adding the volume of two hemispheres and the volume of the cylinder.
<u>The second step</u> is to make the total volume of the shape the constraint equation and the total surface area of the shape becomes the optimisation equation. From the constraint equation solve for h and make it in terms of r (you should advisable use V as a constant and finally substituting at the end). Thereafter, input the value of h in the optimisation equation and differentiate the equation with respect to r. Finally equate the equation to zero.
Third step involves solving for r
Now, for the second question, one has to double the total surface area of the hemispheres to provide the cost equation (this is done to take into account the higher costs of the hemispherical ends. Solve in the same way as the first and you will get the answer.
[KINDLY NOTE I TRIED TYPING THE SOLUTION BUT IT TOOK SO LONG THAT I GOT LOGGED OUT SO FIND ATTACHED PICTURES OF THE SOLUTION, if you follow the steps, it would serve as a guide]
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Answer:
a) The 95% confidence interval for the income of store managers in the retail industry is ($44,846, $45,994), having a margin of error of $574.
b) The interval mean that we are 95% sure that the true mean income of all store managers in the retail industry is between $44,846 and $45,994.
Step-by-step explanation:
Question a:
We have to find our level, that is the subtraction of 1 by the confidence interval divided by 2. So:
Now, we have to find z in the Z-table as such z has a p-value of .
That is z with a p-value of , so Z = 1.96.
Now, find the margin of error M as such
In which is the standard deviation of the population and n is the size of the sample.
The lower end of the interval is the sample mean subtracted by M. So it is 45420 - 574 = $44,846.
The upper end of the interval is the sample mean added to M. So it is 45420 + 574 = $45,994.
The 95% confidence interval for the income of store managers in the retail industry is ($44,846, $45,994), having a margin of error of $574.
Question b:
The interval mean that we are 95% sure that the true mean income of all store managers in the retail industry is between $44,846 and $45,994.