Sure, let's tackle the problem step-by-step.
### Part (a)
First, we need to establish the relationship between the distance [tex]\( d \)[/tex] from the pivot and the weight [tex]\( w \)[/tex] of the person. The problem states that [tex]\( d \)[/tex] is inversely proportional to [tex]\( w \)[/tex]. When two quantities are inversely proportional, their product is a constant, denoted by [tex]\( k \)[/tex]. This relationship can be expressed as:
[tex]\[ d \cdot w = k \][/tex]
or equivalently,
[tex]\[ d = \frac{k}{w} \][/tex]
This equation represents the inverse proportionality between the distance and the weight.
### Part (b)
We are given that a person weighs 145 pounds and is 6 feet from the pivot. Using the inverse proportionality equation:
[tex]\[ d_1 \cdot w_1 = k \][/tex]
Substitute the given values:
[tex]\[ 6 \cdot 145 = k \][/tex]
Calculate [tex]\( k \)[/tex]:
[tex]\[ k = 870 \][/tex]
Now, we need to find the weight of a person who is 5 feet from the pivot. Let this weight be [tex]\( w_2 \)[/tex]. Using the relationship [tex]\( d \cdot w = k \)[/tex]:
[tex]\[ 5 \cdot w_2 = 870 \][/tex]
Solving for [tex]\( w_2 \)[/tex]:
[tex]\[ w_2 = \frac{870}{5} = 174 \][/tex]
So, the weight of the person who is 5 feet from the pivot is 174 pounds.
### Part (c)
Next, we want to find the distance from the pivot for a person who weighs 87 pounds. Let this distance be [tex]\( d_3 \)[/tex]. Using the relationship [tex]\( d \cdot w = k \)[/tex]:
[tex]\[ d_3 \cdot 87 = 870 \][/tex]
Solving for [tex]\( d_3 \)[/tex]:
[tex]\[ d_3 = \frac{870}{87} = 10 \][/tex]
So, the distance from the pivot for a person who weighs 87 pounds is 10 feet.
### Summary
To summarize:
- The equation representing the relationship is [tex]\( d \cdot w = k \)[/tex].
- Given a person weighs 145 lbs and is 6 ft from the pivot, a person who is 5 ft from the pivot would weigh 174 lbs.
- For a person weighing 87 lbs, the distance from the pivot would be 10 ft.
