Answer :
Answer:
a) 0.4
b) 0.133
c) [tex]L \geq 99[/tex]
Step-by-step explanation:
We are given the following information in the question:
The load is said to be uniformly distributed over that part of the beam between 90 and 105 pounds per linear foot.
a = 90 and b = 105
Thus, the probability distribution function is given by
[tex]f(x) = \displaystyle\frac{1}{b-a} = \frac{1}{105-90} = \frac{1}{15},\\\\90 \leq x \leq 105[/tex]
a) P( beam load exceeds 99 pounds per linear foot)
P( x > 99)
[tex]=\displaystyle\int_{99}^{105} f(x) dx\\\\=\displaystyle\int_{99}^{105} \frac{1}{15} dx\\\\=\frac{1}{15}[x]_{99}^{105} = \frac{1}{15}(105-99) = 0.4[/tex]
b) P( beam load less than 92 pounds per linear foot)
P( x < 92)
[tex]=\displaystyle\int_{90}^{92} f(x) dx\\\\=\displaystyle\int_{90}^{92} \frac{1}{15} dx\\\\=\frac{1}{15}[x]_{90}^{92} = \frac{1}{15}(92-90) = 0.133[/tex]
c) We have to find L such that
[tex]\displaystyle\int_{L}^{105} f(x) dx\\\\=\displaystyle\int_{L}^{105} \frac{1}{15} dx\\\\=\frac{1}{15}[x]_{L}^{105} = \frac{1}{15}(105-L) = 0.4\\\\\Rightarrow L = 99[/tex]
The beam load should be greater than or equal to 99 such that the probability that the beam load exceeds L is 0.4.
Final answer:
The probability that the load exceeds 99 pounds per linear foot is 40%. There's a 13.33% chance that the load is less than 92 pounds per linear foot. The value L such that there is a 40% chance the load exceeds it is 99 pounds per linear foot.
Explanation:
To solve problems related to uniformly distributed loads on beams, we use the principles of probabilities. Since the load is uniformly distributed between 90 and 105 pounds per linear foot, each value within this range is equally likely.
a. Probability of exceeding 99 pounds per linear foot
The range of distribution is 105 - 90 = 15 pounds per linear foot. To find the probability that the load exceeds 99 pounds per linear foot, we calculate:
Probability = (Upper limit - Desired value) / Total range
Probability = (105 - 99) / 15 = 6 / 15 = 0.4
Thus, there's a 40% chance that the load exceeds 99 pounds per linear foot.
b. Probability of being less than 92 pounds per linear foot
Similarly, to find the probability that the load is less than 92 pounds:
Probability = (Desired value - Lower limit) / Total range
Probability = (92 - 90) / 15 = 2 / 15 ≈ 0.1333
Therefore, there's a roughly 13.33% chance that the load is less than 92 pounds per linear foot.
c. Finding value L for a 0.4 probability of exceeding the load
To find the value L such that the probability of exceeding it is 0.4, we work backward:
0.4 = (105 - L) / 15
L = 105 - (0.4 * 15)
L = 105 - 6
L = 99 pounds per linear foot
So, the value of L for which there is a 40% chance that the load exceeds it is 99 pounds per linear foot.
