The maintenance department at the main campus of a large state university receives daily requests to replace fluorecent lightbul
bs. The distribution of the number of daily requests is bell-shaped and has a mean of 40 and a standard deviation of 4. Using the 68-95-99.7 rule, what is the approximate percentage of lightbulb replacement requests numbering between 40 and 48?
1 answer:
Answer: 47.5%
Step-by-step explanation:
Given : The distribution of the number of daily requests is bell-shaped and has a mean of and a standard deviation of 40
.
To find : The approximate percentage of lightbulb replacement requests numbering between 40 and 48.
We can see that i.e. 48 is two standard deviations ( to the right) from the mean. (1)
And 40 is the mean value () itself . It means the lower limit lies at the center of the normal curve (at z=0).
It means we need only right the region which is 2 standard deviations from the mean (at z=2).
According to the 68-95-99.7 rule, 95% of the population falls within two deviation from the mean in both left and right side of the normal curve.
Then, the required area will be :
Hence, the approximate percentage of lightbulb replacement requests numbering between 40 and 48 = 47.5%
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Answer:
The room dimensions that will minimize the cost of the paint are 12 ft x 12 ft x 11 ft.
Step-by-step explanation:
We can find first the volume equation using the formula of the volume of a box.
Thus we get the constraint function
Then since we are asked to minimize the cost, we can write the cost function which is the area of each one of the walls and ceiling multiplied by the painting cost.
Lagrange Multipliers to find minimum cost.
We can continue finding the partial derivatives to build the system of equations required for Lagrange Multipliers method.
And the constraint function
So we get
We can multiply each side of each equation by the dimension which is missing to get the full volume on the right side.
Then we can set each the equations equal to each other, so from the first one and the second equation we get
We can subtract 0.11xy from both sides.
And we can divide both sides by 0.12z to get
We can repeat the process by setting the first and third equation equal to each other.
We can subtract 0.12 xz from both sides
And we can solve by z
So if we replace that as well y = x on the constraint for the volume euqation we get
We can then solve for x
And taking the cube root
So then we can use the equations we have found for y and z in terms of x
And
Then the dimensions of the room that will minimize the cost are 12ft x 12 ft x 11 ft. Since you have to enter using commas you can write 12, 12, 11, please check as well if you have to insert the units that are feet for each.