A meteorologist predicts that the amount of rainfall (in inches) expected for a certain coastal community during a hurricane has the probability density function $f ( x ) = \frac { \pi } { 30 } \sin \frac { \pi x } { 15 },$ $0 \leq x \leq 15$ . Find and interpret the probability $P ( 0 \leq x \leq 5 )$ .
A)17% probability of receiving up to 5 inches of rain
B)25% probability of receiving up to 5 inches of rain
C)30% probability of receiving up to 5 inches of rain
D)75% probability of receiving up to 5 inches of rain
E)83% probability of receiving up to 5 inches of rain

Question

Quizplus · Accepted Answer

To find $P(0 \leq x \leq 5)$, we integrate the probability density function $f(x) = \frac{\pi}{30} \sin\left(\frac{\pi x}{15}\right)$ from 0 to 5. The integral gives the probability of receiving up to 5 inches of rain. The calculation is as follows:$$\int_{0}^{5} \frac{\pi}{30} \sin\left(\frac{\pi x}{15}\right) dx = \left[-\frac{1}{30} \cos\left(\frac{\pi x}{15}\right)\right]_{0}^{5} = -\frac{1}{30} \left[\cos\left(\frac{\pi \cdot 5}{15}\right) - \cos(0)\right]$$$$= -\frac{1}{30} \left[\cos\left(\frac{\pi}{3}\right) - 1\right] = -\frac{1}{30} \left[\frac{1}{2} - 1\right] = -\frac{1}{30} \left[-\frac{1}{2}\right] = \frac{1}{60}$$$$= \frac{1}{60} \times 2 = \frac{1}{30} = \frac{2}{60} = \frac{1}{30} \times 100\% = \frac{100}{30}\% = \frac{10}{3}\% \approx 33.33\%$$However, the correct calculation should involve correctly evaluating the integral, which seems to have been miscalculated in the steps provided. The correct integral evaluation should yield a result that matches one of the given options. Given the nature of the sine function and the limits of integration, the correct approach involves accurately integrating the function over the specified interval. The mistake in the calculation above suggests a misunderstanding in the integration process or a typographical error in the steps.The correct integral calculation should yield:$$\int_{0}^{5} \frac{\pi}{30} \sin\left(\frac{\pi x}{15}\right) dx = \left[-\cos\left(\frac{\pi x}{15}\right)\right]_{0}^{5} \times \frac{1}{2}$$$$= \frac{1}{2} \left(1 - \cos\left(\frac{\pi}{3}\right)\right) = \frac{1}{2} \left(1 - \frac{1}{2}\right) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$$Thus, the correct probability of receiving up to 5 inches of rain is 25%, which matches option B. The initial explanation contained errors in the integration process and the interpretation of the result. The correct interpretation is that there is a 25% probability of receiving up to 5 inches of rain in the given coastal community during the hurricane, based on the provided probability density function.

A Meteorologist Predicts That the Amount of Rainfall (In Inches) f(x)=π30sin⁡πx15,f ( x ) = \frac { \pi } { 30 } \sin \frac { \pi x } { 15 },f(x)=30π​sin15πx​,

A Meteorologist Predicts That the Amount of Rainfall (In Inches) $f ( x ) = \frac { \pi } { 30 } \sin \frac { \pi x } { 15 },$