Trucks in a delivery fleet travel a mean of 110
 miles per day with a standard deviation of 35
 miles per day. The mileage per day is distributed normally. Find the probability that a truck drives between 89
 and 169
 miles in a day. Round your answer to four decimal places.

To solve this problem, we use the properties of the normal distribution. We have:

- Mean \(\mu = 110\) miles per day
- Standard deviation \(\sigma = 35\) miles per day

We need to find the probability that a truck drives between 89 and 169 miles in a day. This can be expressed as:

\[
P(89 \leq X \leq 169)
\]

where \(X\) follows a normal distribution \(X \sim N(110, 35^2)\).

### Step 1: Standardize the Values
To find the probability, we convert the values 89 and 169 to their respective z-scores using the formula:

\[
z = \frac{X - \mu}{\sigma}
\]

1. For \(X = 89\):

\[
z_1 = \frac{89 - 110}{35} = \frac{-21}{35} \approx -0.6
\]

2. For \(X = 169\):

\[
z_2 = \frac{169 - 110}{35} = \frac{59}{35} \approx 1.6857
\]

### Step 2: Find the Corresponding Probabilities
Next, we use the standard normal distribution table to find \(P(Z \leq z_1)\) and \(P(Z \leq z_2)\).

1. \(P(Z \leq -0.6)\) ≈ 0.2743
2. \(P(Z \leq 1.6857)\) ≈ 0.9539

### Step 3: Calculate the Probability
To find the probability that \(X\) is between 89 and 169, subtract the probability of \(Z \leq -0.6\) from the probability of \(Z \leq 1.6857\):

\[
P(89 \leq X \leq 169) = P(Z \leq 1.6857) - P(Z \leq -0.6)
\]

\[
P(89 \leq X \leq 169) = 0.9539 - 0.2743 = 0.6796
\]

### Final Answer
The probability that a truck drives between 89 and 169 miles in a day is approximately:

\[
\boxed{0.6796}
\]

Would you like more details on the steps or any further explanations?

### Related Questions
1. What is the probability that a truck drives more than 150 miles in a day?
2. What is the probability that a truck drives less than 100 miles in a day?
3. If the standard deviation changed to 40 miles, how would the probability change for driving between 89 and 169 miles?
4. What is the z-score if a truck drives exactly 200 miles in a day?
5. How would you calculate the 90th percentile for miles driven in this delivery fleet?

**Tip:** When working with normal distributions, always standardize using z-scores for easier calculations.

Trucks in a delivery fleet travel a mean of 110 miles per day with a standard deviation of 35 miles per day. The mileage per day is distributed normally. Find the probability that a truck drives between 89 and 169 miles in a day. Round your answer to four decimal places.

This problem involves calculating the probability that a truck in a delivery fleet drives between 89 and 169 miles in a day, given a mean of 110 miles and a standard deviation of 35 miles. Learn how to use the normal distribution and z-scores to solve the problem. Suitable for Grades 10-12.

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