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Answer :
Sure! Let's solve each part of the exercise one step at a time.
1. Find the constant of proportionality:
We know that [tex]\( y \propto \frac{1}{x} \)[/tex], which implies [tex]\( y = \frac{k}{x} \)[/tex] where [tex]\( k \)[/tex] is the constant of proportionality.
Given [tex]\( y = 6 \)[/tex] when [tex]\( x = 4 \)[/tex]:
[tex]\[
6 = \frac{k}{4}
\][/tex]
Solving for [tex]\( k \)[/tex], we get:
[tex]\[
k = 6 \times 4 = 24
\][/tex]
2. Find the value of [tex]\( y \)[/tex] when [tex]\( x = 10 \)[/tex]:
Since [tex]\( y \)[/tex] is inversely proportional to [tex]\( x \)[/tex], we have [tex]\( y \times x = \text{constant} \)[/tex].
Given [tex]\( x = 25 \)[/tex] and [tex]\( y = 8 \)[/tex]:
[tex]\[
y \times 25 = 8 \times 25 = 200
\][/tex]
Now, find [tex]\( y \)[/tex] when [tex]\( x = 10 \)[/tex]:
[tex]\[
y \times 10 = 200 \implies y = \frac{200}{10} = 20
\][/tex]
3. Which table shows an inverse proportional relationship?
For an inverse proportional relationship, the product [tex]\( x \times y \)[/tex] should be constant.
- Table (a):
[tex]\[
\begin{aligned}
2 \times 36 & = 72, \\
4 \times 18 & = 72, \\
8 \times 9 & = 72.
\end{aligned}
\][/tex]
Constants are equal, so Table (a) shows an inverse proportional relationship.
- Table (b):
[tex]\[
\begin{aligned}
5 \times 20 & = 100, \\
10 \times 10 & = 100, \\
15 \times 5 & = 75.
\end{aligned}
\][/tex]
Constants are not equal, so Table (b) does not.
4. How long will 20 laborers take to harvest coffee?
Using the inverse relationship between laborers and days:
[tex]\[
\text{Days taken by 20 laborers} = \frac{35 \times 8}{20} = 14 \text{ days}
\][/tex]
5. Find [tex]\( y \)[/tex] when [tex]\( x = 30 \)[/tex]:
Given [tex]\( x = 25 \)[/tex] and [tex]\( y = 15 \)[/tex]:
[tex]\[
25 \times 15 = 375
\][/tex]
New [tex]\( x = 30 \)[/tex]:
[tex]\[
y \times 30 = 375 \implies y = \frac{375}{30} = 12.5
\][/tex]
6. How long will 50 and 150 men take?
- 50 men:
[tex]\[
\text{Days needed} = \frac{100 \times 10}{50} = 20 \text{ days}
\][/tex]
- 150 men:
[tex]\[
\text{Days needed} = \frac{100 \times 10}{150} \approx 6.67 \text{ days}
\][/tex]
7. How many days will 16 workers take?
Using the inverse relation for workers and days:
[tex]\[
\text{Days for 16 workers} = \frac{36 \times 12}{16} = 27 \text{ days}
\][/tex]
And that's how you can solve each part of the exercise step by step!
1. Find the constant of proportionality:
We know that [tex]\( y \propto \frac{1}{x} \)[/tex], which implies [tex]\( y = \frac{k}{x} \)[/tex] where [tex]\( k \)[/tex] is the constant of proportionality.
Given [tex]\( y = 6 \)[/tex] when [tex]\( x = 4 \)[/tex]:
[tex]\[
6 = \frac{k}{4}
\][/tex]
Solving for [tex]\( k \)[/tex], we get:
[tex]\[
k = 6 \times 4 = 24
\][/tex]
2. Find the value of [tex]\( y \)[/tex] when [tex]\( x = 10 \)[/tex]:
Since [tex]\( y \)[/tex] is inversely proportional to [tex]\( x \)[/tex], we have [tex]\( y \times x = \text{constant} \)[/tex].
Given [tex]\( x = 25 \)[/tex] and [tex]\( y = 8 \)[/tex]:
[tex]\[
y \times 25 = 8 \times 25 = 200
\][/tex]
Now, find [tex]\( y \)[/tex] when [tex]\( x = 10 \)[/tex]:
[tex]\[
y \times 10 = 200 \implies y = \frac{200}{10} = 20
\][/tex]
3. Which table shows an inverse proportional relationship?
For an inverse proportional relationship, the product [tex]\( x \times y \)[/tex] should be constant.
- Table (a):
[tex]\[
\begin{aligned}
2 \times 36 & = 72, \\
4 \times 18 & = 72, \\
8 \times 9 & = 72.
\end{aligned}
\][/tex]
Constants are equal, so Table (a) shows an inverse proportional relationship.
- Table (b):
[tex]\[
\begin{aligned}
5 \times 20 & = 100, \\
10 \times 10 & = 100, \\
15 \times 5 & = 75.
\end{aligned}
\][/tex]
Constants are not equal, so Table (b) does not.
4. How long will 20 laborers take to harvest coffee?
Using the inverse relationship between laborers and days:
[tex]\[
\text{Days taken by 20 laborers} = \frac{35 \times 8}{20} = 14 \text{ days}
\][/tex]
5. Find [tex]\( y \)[/tex] when [tex]\( x = 30 \)[/tex]:
Given [tex]\( x = 25 \)[/tex] and [tex]\( y = 15 \)[/tex]:
[tex]\[
25 \times 15 = 375
\][/tex]
New [tex]\( x = 30 \)[/tex]:
[tex]\[
y \times 30 = 375 \implies y = \frac{375}{30} = 12.5
\][/tex]
6. How long will 50 and 150 men take?
- 50 men:
[tex]\[
\text{Days needed} = \frac{100 \times 10}{50} = 20 \text{ days}
\][/tex]
- 150 men:
[tex]\[
\text{Days needed} = \frac{100 \times 10}{150} \approx 6.67 \text{ days}
\][/tex]
7. How many days will 16 workers take?
Using the inverse relation for workers and days:
[tex]\[
\text{Days for 16 workers} = \frac{36 \times 12}{16} = 27 \text{ days}
\][/tex]
And that's how you can solve each part of the exercise step by step!
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