Ranger was given this expression to simplify. 4(2x – 5) What advice to simplify the expression would you give Ranger? Check all

Answer: Option C - Construction Y, as point E serves as the circumcenter of triangle LMN.

Point E is ideal for the warehouse since it maintains equal distances from the three stores located at L, M, and N.

Step-by-step explanation:

To effectively tackle an algebra problem, establishing a geometric illustration frequently proves beneficial. Geometry states that three points define a circle; meaning, there is one unique circle encompassing all three non-collinear points. Identifying the point equidistant from these three points corresponds to finding the circle's center that passes through them (as all points on the circle share equal distance from the center).

Considering our points L, M, and N, draw the segments LM, LN, and MN to outline a triangle. Construct the perpendicular bisectors of any two line segments, and their intersection, point E, will indicate the center of this circle.

As illustrated in Construction Y, E represents the circumcenter of triangle LMN.

This makes it the most suitable place for the warehouse, being equidistant from all three stores at L, M, and N.

QED!

Response:

a) The average is 10 and the variance is 0.0625.

b) 0.6826 = 68.26% likelihood that the average time of visitors falls within 15 seconds of 10 minutes.

c) 10.58 minutes.

Step-by-step clarification:

To figure this out, we must comprehend the normal probability distribution alongside the central limit theorem.

Normal Probability Distribution

Issues involving normal distributions can be resolved using the z-score formula.

For a data set with mean and standard deviation , the z-score related to a measure X is defined as:

The z-score illustrates how many standard deviations the measure is positioned from the mean. Once the z-score is calculated, we refer to the z-score table to find the p-value matching this z-score. This p-value reflects the likelihood that the measurement is less than X, which means it indicates the percentile of X. To determine the chance that the measure exceeds X, we subtract the p-value from 1.

Central Limit Theorem

The Central Limit Theorem states that for a normally distributed random variable X, defined by mean and standard deviation , the sampling distribution of sample means of size n can be approximated as a normal distribution with mean and standard deviation .

For non-normal distributions, this theorem applies if n is at least 30.

In this case, the distribution has a mean of 10 minutes and a standard deviation of 2 minutes.

This indicates that

Assuming 64 visitors independently access the site.

This implies that

a. The expected value and variance of the average time of the visitors.

b. The chance that the mean visit duration is within 15 seconds of 10 minutes.

15 seconds = 15/60 = 0.25 minutes, thus between 9.75 and 10.25 seconds, which corresponds to the p-value of z when X = 10.25 minus the p-value of z when X = 9.75.

X = 10.25

Through the Central Limit Theorem

has a p-value of 0.8413.

X = 9.75

has a p-value of 0.1587.

0.6826 = 68.26% chance that the average visitor time is within 15 seconds of 10 minutes.

This corresponds to the 99th percentile, denoting X when Z equals a p-value of 0.99, hence X when Z = 2.327.

Thus, the result is 10.58 minutes.

1. 95 miles: 15 miles = 6.333 gallons for Kevin

2. 125 miles: 40 miles = 3.125 gallons for Amanda

3. 6.333 - 3.125 = 3.208 additional gallons for Kevin