Tristan has found a good job as a bookkeeping clerk after finishing his associates degree at the local community college. He is

Answer:

The variable x lies within the interval of all positive real numbers less than 5 cm.

Detailed solution:

Problem statement:

Determine the volume of the open-topped box as a function of the side length x (in centimeters) of the square cutouts.

Refer to the provided diagram for clarity.

Define:

x → length in centimeters of each square cutout side

The volume of the box with open top can be written as:

Given this, we have:

By substitution:

Determine the domain of x:

Because:

Therefore:

Domain is the interval (0,5)

That means all real numbers strictly greater than zero and less than 5 cm are valid for x.

Hence, the volume V as a function of x is:

Answer:

a) The likelihood that none of the sampled employees are from the Hawaii plant is 1.74%.

b) The chance that exactly 1 employee from the sample is found working in the Hawaii plant is 8.70%.

c) There is an 89.56% chance that 2 or more employees in the sample are from the Hawaii plant.

d) The probability that 9 employees from the sample are working at the Texas plant is 8.70%.

Step-by-step explanation:

Each employee has two potential employment locations: either Texas or Hawaii. Thus, the binomial probability distribution can be utilized to solve this scenario.

Binomial probability distribution

This distribution defines the probability of achieving exactly x successes in n trials where there are only two outcomes.

Here, denotes the number of ways to choose x objects from a set of n, represented by the subsequent formula.

And p is the probability of success occurring.

In this context, we know:

The sample comprises 10 employees, therefore .

a. Calculate the probability that none of the sampled employees are from the Hawaii plant (to 4 decimals)?

Given that 20 out of 60 employees are based in Hawaii:

We aim to find P(X = 0).

Thus, the likelihood that none in the sample are from Hawaii stands at 1.74%.

b. Calculate the probability that 1 employee from the sample is from the Hawaii plant?

This is represented as P(X = 1).

Therefore, there is an 8.70% possibility that 1 employee in the sample comes from Hawaii.

c. Calculate the probability that 2 or more employees in the sample are from the Hawaii plant?

We can observe two scenarios: either fewer than 2 employees are from Hawaii or 2 and beyond. The combined probabilities equal decimal 1. So:

We seek to find .

From problems a and b, we possess values for both probabilities.

Accordingly, the chance that 2 or more employees in this sample operate at the Hawaii plant is 89.56%.

d. Calculate the likelihood that 9 employees in the sample are working at the Texas plant?

This corresponds to the probability found in part b for 1 employee working in Hawaii.

Consequently, there is an 8.70% chance that 9 employees belong to the Texas plant.