The amount of bread x (in hundreds os pounds ) that a bakery sells in a day has a P.d.f :
`f(x)={(kx","0lexlt5,),(k(10-x)",",5lexlt10),(0",",otherwise):}`
Then the probability that the amount of bread that will be sold tomorrow will be less than 500 pound is ..

Options

A. `1/4`
B. `1/3`
C. `1/2`
D. none of these

Answer

To solve the problem, we need to find the probability that the amount of bread sold tomorrow will be less than 500 pounds. Given the probability density function (P.d.f) for the amount of bread \( x \) (in hundreds of pounds), we can follow these steps:

### Step 1: Understand the P.d.f
The given P.d.f is:
\[ f(x) = \begin{cases} kx & \text{for } 0 < x < 5 \\ k(10 - x) & \text{for } 5 < x < 10 \\ 0 & \text{otherwise} \end{cases} \]
Here, \( x \) represents the amount of bread in hundreds of pounds.

### Step 2: Find the value of \( k \)
To find \( k \), we use the property that the total probability must equal 1. Thus, we need to integrate the P.d.f over its entire range:
\[ \int_{-\infty}^{\infty} f(x) \, dx = 1 \]
This simplifies to:
\[ \int_{0}^{5} kx \, dx + \int_{5}^{10} k(10 - x) \, dx = 1 \]

Calculating the first integral:
\[ \int_{0}^{5} kx \, dx = k \left[ \frac{x^2}{2} \right]_{0}^{5} = k \left( \frac{5^2}{2} - 0 \right) = \frac{25k}{2} \]

Calculating the second integral:
\[ \int_{5}^{10} k(10 - x) \, dx = k \left[ 10x - \frac{x^2}{2} \right]_{5}^{10} = k \left( (100 - 50) - (50 - \frac{25}{2}) \right) = k \left( 50 - (50 - \frac{25}{2}) \right) = k \left( \frac{25}{2} \right) \]

Combining both integrals:
\[ \frac{25k}{2} + \frac{25k}{2} = 1 \implies 25k = 1 \implies k = \frac{1}{25} \]

### Step 3: Calculate the Probability
Now we need to find the probability that the amount of bread sold is less than 500 pounds, which corresponds to \( x < 5 \):
\[ P(X < 5) = \int_{0}^{5} f(x) \, dx = \int_{0}^{5} \frac{1}{25} x \, dx \]

Calculating the integral:
\[ \int_{0}^{5} \frac{1}{25} x \, dx = \frac{1}{25} \left[ \frac{x^2}{2} \right]_{0}^{5} = \frac{1}{25} \left( \frac{5^2}{2} - 0 \right) = \frac{1}{25} \cdot \frac{25}{2} = \frac{1}{2} \]

### Conclusion
Thus, the probability that the amount of bread that will be sold tomorrow will be less than 500 pounds is:
\[ \boxed{\frac{1}{2}} \]