Answer :
We start with the given period function for the trapeze artist:
[tex]$$
t = 2\sqrt{\frac{r}{32}}.
$$[/tex]
### Step 1. Solve for the Rope Length
First, we isolate the square root by dividing both sides by 2:
[tex]$$
\sqrt{\frac{r}{32}} = \frac{t}{2}.
$$[/tex]
Next, square both sides to remove the square root:
[tex]$$
\frac{r}{32} = \left(\frac{t}{2}\right)^2,
$$[/tex]
and then solve for [tex]$r$[/tex]:
[tex]$$
r = 32\left(\frac{t}{2}\right)^2.
$$[/tex]
### Step 2. Find the Rope Length for [tex]$t = 6$[/tex] Seconds
Substitute [tex]$t = 6$[/tex] into the equation:
[tex]$$
r_1 = 32\left(\frac{6}{2}\right)^2.
$$[/tex]
Since [tex]$\frac{6}{2} = 3$[/tex], we have:
[tex]$$
r_1 = 32 \times 3^2 = 32 \times 9 = 288 \text{ feet}.
$$[/tex]
### Step 3. Find the Rope Length for [tex]$t = 4$[/tex] Seconds
Now substitute [tex]$t = 4$[/tex] into the equation:
[tex]$$
r_2 = 32\left(\frac{4}{2}\right)^2.
$$[/tex]
Since [tex]$\frac{4}{2} = 2$[/tex], we get:
[tex]$$
r_2 = 32 \times 2^2 = 32 \times 4 = 128 \text{ feet}.
$$[/tex]
### Step 4. Compare the Rope Lengths
We need to check if the rope length with a 6-second period is [tex]$\frac{3}{2}$[/tex] times as long as the rope length with a 4-second period. Compute the ratio:
[tex]$$
\frac{r_1}{r_2} = \frac{288}{128}.
$$[/tex]
Simplify the fraction by dividing both the numerator and denominator by 32:
[tex]$$
\frac{r_1}{r_2} = \frac{288 \div 32}{128 \div 32} = \frac{9}{4} = 2.25.
$$[/tex]
### Step 5. Analyze the Result
The problem asks whether the rope is [tex]$\frac{3}{2}$[/tex] (which is [tex]$1.5$[/tex]) times as long. However, we found that the ratio is [tex]$\frac{9}{4}$[/tex] (which is [tex]$2.25$[/tex]).
### Conclusion
Since [tex]$\frac{9}{4} \neq \frac{3}{2}$[/tex], the rope used when it takes 6 seconds to swing is not [tex]$\frac{3}{2}$[/tex] times as long as the rope used when it takes 4 seconds to swing.
[tex]$$
t = 2\sqrt{\frac{r}{32}}.
$$[/tex]
### Step 1. Solve for the Rope Length
First, we isolate the square root by dividing both sides by 2:
[tex]$$
\sqrt{\frac{r}{32}} = \frac{t}{2}.
$$[/tex]
Next, square both sides to remove the square root:
[tex]$$
\frac{r}{32} = \left(\frac{t}{2}\right)^2,
$$[/tex]
and then solve for [tex]$r$[/tex]:
[tex]$$
r = 32\left(\frac{t}{2}\right)^2.
$$[/tex]
### Step 2. Find the Rope Length for [tex]$t = 6$[/tex] Seconds
Substitute [tex]$t = 6$[/tex] into the equation:
[tex]$$
r_1 = 32\left(\frac{6}{2}\right)^2.
$$[/tex]
Since [tex]$\frac{6}{2} = 3$[/tex], we have:
[tex]$$
r_1 = 32 \times 3^2 = 32 \times 9 = 288 \text{ feet}.
$$[/tex]
### Step 3. Find the Rope Length for [tex]$t = 4$[/tex] Seconds
Now substitute [tex]$t = 4$[/tex] into the equation:
[tex]$$
r_2 = 32\left(\frac{4}{2}\right)^2.
$$[/tex]
Since [tex]$\frac{4}{2} = 2$[/tex], we get:
[tex]$$
r_2 = 32 \times 2^2 = 32 \times 4 = 128 \text{ feet}.
$$[/tex]
### Step 4. Compare the Rope Lengths
We need to check if the rope length with a 6-second period is [tex]$\frac{3}{2}$[/tex] times as long as the rope length with a 4-second period. Compute the ratio:
[tex]$$
\frac{r_1}{r_2} = \frac{288}{128}.
$$[/tex]
Simplify the fraction by dividing both the numerator and denominator by 32:
[tex]$$
\frac{r_1}{r_2} = \frac{288 \div 32}{128 \div 32} = \frac{9}{4} = 2.25.
$$[/tex]
### Step 5. Analyze the Result
The problem asks whether the rope is [tex]$\frac{3}{2}$[/tex] (which is [tex]$1.5$[/tex]) times as long. However, we found that the ratio is [tex]$\frac{9}{4}$[/tex] (which is [tex]$2.25$[/tex]).
### Conclusion
Since [tex]$\frac{9}{4} \neq \frac{3}{2}$[/tex], the rope used when it takes 6 seconds to swing is not [tex]$\frac{3}{2}$[/tex] times as long as the rope used when it takes 4 seconds to swing.
