The question aims to find the **probability** that a package can weigh more than **50 pounds** and how much chemical is contained in **90%** of the package.

The question depends on the concept of **Probability** **Density Function (PDF).** The **PDF** is the probability function that represents the likelihood of all the **values** of the **continuous random variable.**

A **probability density function** or **PDF** is used in probability theory to describe the **chance** of a random variable staying within a particular specific **range** of values. These functions describe the **probability** density function of normal distribution and how there exists **mean** and **deviation.**

## Expert Answer

The **probability density function** of the **net weight** in **pounds** for all the packaged **chemical herbicides** is given as:

\[ f(x) = 2.2 \hspace{0.2in} 49.8 \lt x \lt 50.2\ lbs \]

**a)** To calculate the **probability** that a **package** of **chemical herbicides** will weigh more than **50 pounds**, we can integrate the probability density function. It is given as:

\[ P ( X \gt 50 ) = \int_{50}^{50.2} 2.2 \, dx \]

\[ P ( X \gt 50 ) = 2.2 \big[ x \big]_{50}^{50.2} \]

\[ P ( X \gt 50 ) = 2.2 \big[ 50.2\ -\ 50 \big] \]

\[ P ( X \gt 50 ) = 2.2 \times 0.2 \]

\[ P ( X \gt 50 ) = 0.44 \]

**b)** To calculate how much **chemical** is contained in **90%** of all the packages of **herbicide,** let us use the same formula as above. The only difference from the above equation is that we have the **final probability. **We need to find the **chemical quantity** that yields that **probability.** The equation is given as:

\[ P ( X \gt x ) = \int_{x}^{50.2} 2.2 \, dx \]

\[ P ( X \gt x ) = 2.2 \big[ x \big]_{x}^{50.2} \]

\[ P ( X \gt x ) = 2.2 \big[ 50.2\ -\ x \big] \]

\[ P ( X \gt x ) = 110.44\ -\ 2.2x \]

\[ 0.90 = 110.44\ -\ 2.2x \]

\[ x = \dfrac{ 110.44\ -\ 0.90 }{ 2.2 } \]

\[ x = 49.79 \]

## Numerical Result

**a)** The **probability** that a package of **chemical herbicide** will weigh more than** 50 pounds** is calculated to be:

\[ P ( X \gt 50 ) = 0.44 \]

**b)** The **chemical** in **90%** of all the packages of **herbicide** is calculated to be:

\[ x = 49.79 \]

## Example

The **probability density function** of package **weight** in **kilograms** is given below. Find the **probability** that will weigh more than **10 kg.**

\[ f(x) = 1.7 \hspace{0.3in} 9.8 \lt x \lt 10.27 kg \]

The **probability** that a package will weigh more than **10 kg** is given as:

\[ P ( X \gt 10 ) = \int_{10}^{10.27} 1.7 \, dx \]

\[ P ( X \gt 10 ) = 1.7 \big[ x \big]_{10}^{10.27} \]

\[ P ( X \gt 10 ) = 1.7 \big[ 10.27\ -\ 10 \big] \]

\[ P ( X \gt 10 ) = 1.7 \times 0.27 \]

\[ P ( X \gt 10 ) = 0.459 \]