Suppose an electric-vehicle manufacturing company estimates that a driver who commutes 50 miles per day in a particular vehicle will require a nightly charge time of around 1 hour and 40 minutes ( 100 minutes) to recharge the vehicle's battery, Assume that the actual recharging time required is uniformly distributed between 80 and 120 minutes, (a) Give a mathematical expression for the probability density function of battery recharging time for this scenario, \[ f(x)=\left\{\begin{array}{ll} \square, & 80 \leq x \leq 120 \\ , & \text { elsewhere } \end{array}\right. \] (b) What is the probability that the recharge time will be less than 111 minutes? \( \square \) (c) What is the probability that the recharge time required is at least 89 minutes? (Round your answer to four decimal places.) \( \square \) (d) What is the probability that the recharge time required is between 85 and 95 minutes? \( \square \)

To start off with your questions: (a) The probability density function (PDF) for a uniform distribution is given by: \[ f(x)=\left\{\begin{array}{ll} \frac{1}{b-a} = \frac{1}{120-80} = \frac{1}{40}, & 80 \leq x \leq 120 \\ 0, & \text{ elsewhere } \end{array}\right. \] So, filling in the squares gives \( f(x)=\left\{\begin{array}{ll} \frac{1}{40}, & 80 \leq x \leq 120 \\ 0, & \text{ elsewhere } \end{array}\right. \) (b) To find the probability that the recharge time will be less than 111 minutes, we calculate the area under the PDF from 80 to 111 minutes. \[ P(X < 111) = \int_{80}^{111} f(x) \, dx = \int_{80}^{111} \frac{1}{40} \, dx = \frac{111-80}{40} = \frac{31}{40} = 0.775 \] (c) To find the probability that the recharge time is at least 89 minutes, we can calculate the area under the PDF from 89 to 120 minutes. \[ P(X \geq 89) = 1 - P(X < 89) = 1 - \int_{80}^{89} f(x) \, dx = 1 - \left(\frac{89-80}{40}\right) = 1 - \frac{9}{40} = \frac{31}{40} \approx 0.7750 \] (d) For the probability that the recharge time is between 85 and 95 minutes, we compute: \[ P(85 < X < 95) = \int_{85}^{95} f(x) \, dx = \int_{85}^{95} \frac{1}{40} \, dx = \frac{95-85}{40} = \frac{10}{40} = 0.25 \] Thus, the answers are: (b) 0.775 (c) 0.7750 (d) 0.25