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Answer :
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5. Gift Wrapping Problem
a. We need to find a function that models the relationship between the number of people (n) and the time (t) it takes to wrap the gifts. Looking at the given data, as the number of people increases, the time needed decreases, suggesting an inverse relationship. The formula for an inverse proportionality is:
[tex]\[
t = \frac{k}{n}
\][/tex]
where [tex]\( k \)[/tex] is a constant. To find [tex]\( k \)[/tex], we can use one of the data points provided. Using the first data point where 2 people take 100 minutes:
[tex]\[
k = n \times t = 2 \times 100 = 200
\][/tex]
So, the function that models this problem is:
[tex]\[
t = \frac{200}{n}
\][/tex]
b. To find how many people the department store needs so that it takes approximately 30 minutes to wrap gifts, we'll use the function:
[tex]\[
30 = \frac{200}{n}
\][/tex]
Rearranging to solve for [tex]\( n \)[/tex]:
[tex]\[
n = \frac{200}{30} \approx 6.67
\][/tex]
Since the number of people must be a whole number, the department store would need approximately 7 people to complete the work in about 30 minutes.
6. Direct Variation Problem
Given that [tex]\( y \)[/tex] varies directly as [tex]\( x \)[/tex], this can be written as:
[tex]\[
y = kx
\][/tex]
where [tex]\( k \)[/tex] is a constant. We know that [tex]\( y = -4 \)[/tex] when [tex]\( x = 2 \)[/tex]. So we can write:
[tex]\[
-4 = k \times 2 \quad \Rightarrow \quad k = \frac{-4}{2} = -2
\][/tex]
Now, to find [tex]\( y \)[/tex] when [tex]\( x = -6 \)[/tex], substitute back into the equation:
[tex]\[
y = -2 \times -6 = 12
\][/tex]
7. Direct Variation Insight
If [tex]\( x \)[/tex] and [tex]\( y \)[/tex] vary directly, as [tex]\( x \)[/tex] decreases, [tex]\( y \)[/tex] will also decrease. This is because they are proportional to each other by a constant factor.
8. Inverse Variation Insight
If [tex]\( x \)[/tex] and [tex]\( y \)[/tex] vary inversely, as [tex]\( y \)[/tex] increases, [tex]\( x \)[/tex] will decrease. This is because the product of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] is a constant, so when one increases, the other must decrease to keep the product the same.
5. Gift Wrapping Problem
a. We need to find a function that models the relationship between the number of people (n) and the time (t) it takes to wrap the gifts. Looking at the given data, as the number of people increases, the time needed decreases, suggesting an inverse relationship. The formula for an inverse proportionality is:
[tex]\[
t = \frac{k}{n}
\][/tex]
where [tex]\( k \)[/tex] is a constant. To find [tex]\( k \)[/tex], we can use one of the data points provided. Using the first data point where 2 people take 100 minutes:
[tex]\[
k = n \times t = 2 \times 100 = 200
\][/tex]
So, the function that models this problem is:
[tex]\[
t = \frac{200}{n}
\][/tex]
b. To find how many people the department store needs so that it takes approximately 30 minutes to wrap gifts, we'll use the function:
[tex]\[
30 = \frac{200}{n}
\][/tex]
Rearranging to solve for [tex]\( n \)[/tex]:
[tex]\[
n = \frac{200}{30} \approx 6.67
\][/tex]
Since the number of people must be a whole number, the department store would need approximately 7 people to complete the work in about 30 minutes.
6. Direct Variation Problem
Given that [tex]\( y \)[/tex] varies directly as [tex]\( x \)[/tex], this can be written as:
[tex]\[
y = kx
\][/tex]
where [tex]\( k \)[/tex] is a constant. We know that [tex]\( y = -4 \)[/tex] when [tex]\( x = 2 \)[/tex]. So we can write:
[tex]\[
-4 = k \times 2 \quad \Rightarrow \quad k = \frac{-4}{2} = -2
\][/tex]
Now, to find [tex]\( y \)[/tex] when [tex]\( x = -6 \)[/tex], substitute back into the equation:
[tex]\[
y = -2 \times -6 = 12
\][/tex]
7. Direct Variation Insight
If [tex]\( x \)[/tex] and [tex]\( y \)[/tex] vary directly, as [tex]\( x \)[/tex] decreases, [tex]\( y \)[/tex] will also decrease. This is because they are proportional to each other by a constant factor.
8. Inverse Variation Insight
If [tex]\( x \)[/tex] and [tex]\( y \)[/tex] vary inversely, as [tex]\( y \)[/tex] increases, [tex]\( x \)[/tex] will decrease. This is because the product of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] is a constant, so when one increases, the other must decrease to keep the product the same.
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