How can the distance formula be derived from the pythagorean theorem?

We start by placing a right triangle in the coordinate plane as seen below. Notice that we have given the vertices arbitrary x- and y-coordinates.

Notice that, although the coordinates are arbitrary, the y-coordinates of the vertices on the horizontal leg are the same since they will have the same y-value. Likewise, the x-coordinates of the vertices on the vertical leg are the same as the will have the same x-value.

The Pythagorean Theorem states that the sum of the squares of the length of the legs is equal to the square of the length of the hypotenuse.

To find the lengths of the legs in terms of their coordinates we will use the of the differences of their coordinates as seen in the figure below.

So, according to the Pythagorean Theorem

##(|x_2-x_1|)^2+(|y_2-y_1|)^2=d^2##

We can drop the absolute value symbols since we are squaring the differences in the coordinates (the result will be positive).

##(x_2-x_1)^2+(y_2-y_1)^2=d^2##

Taking the square root of both sides of the equation results in the .

##d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)##

MATH 221 Week 8 Final Exam1

This pack of MATH 221 Week 8 Final Exam1 contains answers on:

1. Find the equation of the regression line for the given data. then construct A SCATTER PLOT of the data and draw the regression line. (each pair of variables has a significant correlation.) then use the regression equation to predict the value of y for each of the given x- values, if meaningful. the caloric content and the sodium content(in milligrams) for 6 beef ho dogs are shown in the table below.

2. A humane society claims that less that 31% of US households own a dog. In a random sample of 395 us households, 153 say they own a dog. At a=0.01, is there evidence to support the society’s claim?

3. The altitude (in thousands of feet) and speed of a plane (in feet per second) for various altitudes is shown below. Complete parts (a) though (c).

4. A researcher wishes to estimate, with 99% confidence, the proportion of adults who have high speed internet access. Her estimate must be accurate within 3% of the true proportion.

b) No preliminary estimate is avaliable. Find the minimum sample size needed.

5. Determine whether the variable is qualitative or quantitative.

6. About 16% of the population of the large country is nervous around strangers. If two people are randomly selected, what is the probability that both are nervous around strangers? What is the possibility at least one is nervous around strangers?

7. The table below shows the number of male and female students enrolled in nursing at a university for a certain semester. A student is selected at random. Complete parts (a) through (d)

a Find the probability that the student is male or a nursing major

P (being male or a being a nursing major)

(Round to the nearest thousandth as needed)

b Find the probability that the student is female or not a nursing major

P(being female or not being a nursing major)

(Round to the nearest thousandth as needed)

c Find the probability that the student is not female or a nursing major

P (not being female or being a nursing major

(Round to the nearest thousandth as needed)

d Are the events “being male” and “being a nursing major” mutually exclusive? Explain

8. A quality control manager randomly selects 30 bottles of tomato sauce that were filled on October 11 to assess the calibration of the filling machine.

1. What is opulation in the study?

2. What is the sample in the study?

9. The state test scores for 12 randomly selected high school seniors are shown on the right.

Assume the population is normally distributed.

a) Find the sample mean

b) Find the sample standard deviation

c) Construct a 95% confidence interval for the population mean

MATH 221 Week 8 Final Exam2

This file of MATH 221 Week 8 Final Exam2 gives answers on:

a. Find the range of the data set.

b. Change 46.9 to 63.1 and find the range of the new data set.

3. A random sample of fifty-three 200-meters swims has a mean of 3.47 minutes and standard deviation of 0.06 minutes. Construct a 90% confidence interval for the population mean time. Interpret the results.

4. Write the null and alternative hypotheses. Indentify which is the claim. A study claims that the mean survival time for certain cancer patients treated immediately with chemotherapy and radiation is 16 points.

6. A soccer ball manufacturer wants to estimate the mean circumference of soccer balls within 0.05 in.

a. Determine the minimum sample size required to construct a 95% confidence interval for the population mean.

b. Repeat part (a) using a standard deviation of 0.30 in. which standard deviation requires a larger sample size?

7. For your study on the food consumption habits of teenage males, you randomly select 10 teenage males and ask each how many 12-ounce servings of soda he drinks each day. The results are listed below. At a=0.05, is there enough evidence to support the claim that the teenage males drink fewer than three 12-ounce servings of soda per day? Assume the population is normally distributed.

8. The times per week a student uses a lab computer are normally distributed, with a mean of 6.4 hours and a standard deviation of 1.1 hours. A student is randomly selected. Find the following probabilities.

a. Find the probability that the student uses a lab computer less than 5 hours per week.

b. Find the probability that the student uses a lab computer between 4 and 8 hours per week.

c. Find the probability that the student uses a lab computer more than 9 hours per week.

9. Ford wants to administer a satisfaction survey to its current customers. Using their customer database, the company randomly selects 90 customers and asks them about their level of satisfaction with the company. What type of sampling is used?

10. The graph to the right shows the number of cities with certain average annual rainfall amounts (in inches). Identify the level of measurements of the data listed on the horizontal axis in the graph.

11. Use technology to find the sample size, mean, median, medium data value, and maximum data value of the data. The data represents the amounts (in dollars) made by several families during a community yard sale.

12. A light bulb manufacturer guarantees that the mean life of a certain type of light bulb is at least 825 hours. A random sample of 44 light bulbs has a mean life of 806 hours with a standard deviation of 75 hours. Do you have enough evidence to reject the manufacturer’s claim? Use a = 0.03

13. 31% of adults say cashews are their favorite kind of nut. You randomly select 12 adults and ask each to name his or her favorite nut. Find the probability that the number who say cashews are their favorite nut is (a) exactly three, (b) at least four, and (c) most two. If convenient , use technology to find the probabilities.

MAT 117 Week 1 DQ 2 Additional Responses

This pack of MAT 117 Week 1 DQ 2 Additional Responses comprises:

How is dividing a polynomial by a binomial similar to or different from the long division you learned in elementary school? Show an example of the similarity from the text examples or problems.

Can understanding how to do one kind of division help you with understanding the other kind? What are some examples from real life in which you might use polynomial division?