To find the actual height of the pole when a measurement is made with a certain percentage error, we can use the concept of percentage error to relate the measured value and the actual value.
The formula for percentage error is given by:
\[ \text{Percentage Error} = \left( \frac{\text{Measured Value} - \text{Actual Value}}{\text{Actual Value}} \right) \times 100 \]
In this case, we know the following:
- Measured Value = 5.98 m
- Percentage Error = 5%
Since the measured value is smaller than the actual value, we can rearrange the formula to find the actual value:
Let \( x \) be the actual height of the pole. According to the given percentage error:
\[ 5 = \left( \frac{5.98 - x}{x} \right) \times 100 \]
Now, convert the percentage to a decimal by dividing by 100:
\[ 0.05 = \frac{5.98 - x}{x} \]
To eliminate the fraction, multiply both sides by \( x \):
\[ 0.05x = 5.98 - x \]
Next, add \( x \) to both sides:
\[ 0.05x + x = 5.98 \]
This simplifies to:
\[ 1.05x = 5.98 \]
Now, solve for \( x \) by dividing both sides by 1.05:
\[ x = \frac{5.98}{1.05} \approx 5.69857 \]
Thus, the actual height of the pole is approximately:
\[ x \approx 5.70 \text{ m} \] (rounded to two decimal places).
So, the actual height of the pole is approximately 5.70 meters.