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. 80 sxS 120 f(x) = elsewhere (b) What is the probability that the recharge time will be less than 111 minutes? (c) What is the probability that the recharge time required is at least 89 minutes? (Round your answer to four decimal places.) (d) What is the probability that the recharge time required is between 85 and 115 minutes?
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. 80 sxS 120 f(x) = elsewhere (b) What is the probability that the recharge time will be less than 111 minutes? (c) What is the probability that the recharge time required is at least 89 minutes? (Round your answer to four decimal places.) (d) What is the probability that the recharge time required is between 85 and 115 minutes?
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![**Title: Understanding Uniform Distribution for EV Charging Times**
**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) Mathematical Expression for the Probability Density Function (PDF):**
Define the function \( f(x) \) for the battery recharging time:
\[
f(x) =
\begin{cases}
\frac{1}{40}, & 80 \leq x \leq 120 \\
0, & \text{elsewhere}
\end{cases}
\]
*Explanation: This uniform distribution assumes all recharge times between 80 and 120 minutes are equally likely, thereby having a constant probability density \( \frac{1}{40} \).*
---
**(b) Probability for Recharge Time Less than 111 Minutes:**
*Question:* What is the probability that the recharge time will be less than 111 minutes?
*Explanation: To find this, calculate the area under the PDF curve from 80 to 111 minutes.*
---
**(c) Probability for Recharge Time of At Least 89 Minutes:**
*Question:* What is the probability that the recharge time required is at least 89 minutes? (Round your answer to four decimal places.)
*Explanation: This calculation involves finding the area under the PDF curve from 89 to 120 minutes.*
---
**(d) Probability for Recharge Time Between 85 and 115 Minutes:**
*Question:* What is the probability that the recharge time required is between 85 and 115 minutes?
*Explanation: You will calculate the area under the PDF curve between these two times.*
---
In this scenario, understanding how to calculate probabilities using a uniform distribution aids in predicting the likely charge times for an EV, allowing for better planning and efficient vehicle use.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F96e0847f-4973-4b99-ac1e-7e94417321c7%2Faca5232e-6005-4ecb-b073-69330d04dedf%2Fy9x5s9p_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Understanding Uniform Distribution for EV Charging Times**
**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) Mathematical Expression for the Probability Density Function (PDF):**
Define the function \( f(x) \) for the battery recharging time:
\[
f(x) =
\begin{cases}
\frac{1}{40}, & 80 \leq x \leq 120 \\
0, & \text{elsewhere}
\end{cases}
\]
*Explanation: This uniform distribution assumes all recharge times between 80 and 120 minutes are equally likely, thereby having a constant probability density \( \frac{1}{40} \).*
---
**(b) Probability for Recharge Time Less than 111 Minutes:**
*Question:* What is the probability that the recharge time will be less than 111 minutes?
*Explanation: To find this, calculate the area under the PDF curve from 80 to 111 minutes.*
---
**(c) Probability for Recharge Time of At Least 89 Minutes:**
*Question:* What is the probability that the recharge time required is at least 89 minutes? (Round your answer to four decimal places.)
*Explanation: This calculation involves finding the area under the PDF curve from 89 to 120 minutes.*
---
**(d) Probability for Recharge Time Between 85 and 115 Minutes:**
*Question:* What is the probability that the recharge time required is between 85 and 115 minutes?
*Explanation: You will calculate the area under the PDF curve between these two times.*
---
In this scenario, understanding how to calculate probabilities using a uniform distribution aids in predicting the likely charge times for an EV, allowing for better planning and efficient vehicle use.
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