SOLUTION: Solve the given problem. The time required for an elevator to lift a weight varies jointly as the weight and distance through which it is to be lifted and inversely as the power

Question 1175841: Solve the given problem.
The time required for an elevator to lift a weight varies jointly as the weight and distance through which it is to be lifted and inversely as the power of the motor. It takes 30 seconds for a 10 horsepower motor to lift 100 pounds in 40 seconds through 40 feet?
1. What type of variation is involved in this problem?
2. What previously learned concepts are needed in solving the problem?
3. What is the mathematical equation of the given problem?
4. What is the unknown in the problem?
5. How will you solve the given problem involving combined variation?
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
this statement doesn't make much sense.

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It takes 30 seconds for a 10 horsepower motor to lift 100 pounds in 40 seconds through 40 feet?-
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i'm not exactly sure what you meant.
it looks like you combined two statements into one.

in variation type problems, you state what's given first and then solve for k.
once you solve for k, you can solve the problem, because k is tghe constant of variation and doesn't change with the same type of problem.

i will change your statement so you can see what i mean.

a revised staement would look like:

the time required for an elevator to lift a weight varies jointly as the weight and distance through which it is to be lifed and inversely as the power of the motor.

that statement leads to the following formula:

t = k * w * d / p

t = time
k = constant of variation
w = weight
d = distance
p = power of the motor

a restatement of your problem statement might be something like.

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It takes 30 seconds for a 10 horsepower motor to lift 100 pounds through 40 feet.
How far would it lift 100 pounds in 40 seconds?
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given this way, you would first solve for k.
t = k * w * d / p would becomes:
30 = k * 100 * 40 / 10
you would solve for k to get:
K = 30 * 10 / (100 * 40) = .075

now that you solved for k, you could solve the rest of the problem
the questions was how far could it lift 100 pounds in 40 seconds?

everything would be the same except for d because that's what you now want to solve for, and for t, because that's what was changed.
the formula becomes:
40 = .075 * 100 * d / 10
t was now 40.
k was now .075
w was still 100
d was what you were looking to solve for.
p was still 10.
you would solve for d to get:
d = 40 * 10 / (.075 * 100) = 53.33333 = 53 and 1/3 seconds.

you would confirm the calculations are corr4ect by usng that value of k in the original statement of the prolem.
you would get 30 = .075 * 100 * 40 / 10 becomes 30 = 30 which is true.
you would then do the problem again using that value of k.
you would get 40 = .075 * 100 * 53.33333... / 10 becomes 40 = 40 which is true.

the value of k was found successfully and the solution is confirmed to be good, using that value of k.

here's a reference on a combined variation type of problem you might find informative.

https://www.varsitytutors.com/hotmath/hotmath_help/topics/combined-variation#:~:text=Combined%20variation%20describes%20a%20situation,the%20variables%20are%20held%20constant).

here is a more in depth reference that includes; more problems as examples.

https://www.purplemath.com/modules/variatn.htm

answers to the question might be something like the following.

1. What type of variation is involved in this problem?

combined variation.

2. What previously learned concepts are needed in solving the problem?

concepts of direct variation and inverse variation.
direct variation is y = k * x
inverse variation is y = k / x
joint variation is y = k * x * y or y = k / (x * y), depending on the number of variables used.
combined variation is y = k * x * y / (z * w), depending on the number of variables used.

3. What is the mathematical equation of the given problem?

t = k * w * d / p
t is the time
k is the constant of variation
w is the weight
d is the distance
p is the power of the motor

4. What is the unknown in the problem?
the first unknown is the value of k.
once the value of k is found, it is ued to solve for d.

5. How will you solve the given problem involving combined variation?

first solve for k, then solve for d.

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