Calculate the value of the symbol in brackets: 31. $$ T=a r^{-1}(n) $$ if $$ T=204, a=8 $$ and $$ r=0,6 $$

Question

UpStudy AI · Accepted Answer

To solve for $$ n $$ in the equation:

$$
T = a \cdot r^{-n}
$$

given:
- $$ T = 204 $$
- $$ a = 8 $$
- $$ r = 0.6 $$

**Step 1: Plug in the known values**

$$
204 = 8 \cdot (0.6)^{-n}
$$

**Step 2: Isolate the exponential term**

$$
\frac{204}{8} = (0.6)^{-n}
$$

$$
25.5 = (0.6)^{-n}
$$

**Step 3: Take the natural logarithm of both sides**

$$
\ln(25.5) = \ln\left((0.6)^{-n}\right)
$$

$$
\ln(25.5) = -n \cdot \ln(0.6)
$$

**Step 4: Solve for $$ n $$**

$$
n = -\frac{\ln(25.5)}{\ln(0.6)}
$$

**Step 5: Calculate the values**

$$
\ln(25.5) \approx 3.238
$$

$$
\ln(0.6) \approx -0.511
$$

$$
n = -\frac{3.238}{-0.511} \approx 6.35
$$

**Final Answer:**

The value of $$ n $$ is approximately **6.35**.
The value of $$ n $$ is approximately **6.35**.

Calculate the value of the symbol in brackets: 31. \( T=a r^{-1}(n) \) if \( T=204, a=8 \) and \( r=0,6 \)