1. (a) Find the equation of a plane π that contains the line in the intersection of the planes x+y+3z =

   2andx−y+z=1andtheorigin. Howmanysuchplanesarethere?

2. (b) What if instead of the origin we ask for the plane containing the same line and the point

   (0, −1/4, 3/4)? What changes?

3. (c) Back to the plane π you found in item a, let Rπ be the reflection across π. Pick an explicit

identify the equation . If it is a parabola, give its vertex, focus, and directrix; if an ellipse, give its centre, vertices, and foci ; if a hyperbola, give its centre, vertices, foci, and asymptotes.

(x + 1)^2/4 - y^2/9−=1

Profit = Revenue - Cost

P(x) = R(x) - C(x)

Given: R(x) = x^2 - 30x

Given: C(x) = 5x + 100

X is hundreds of items sold / P, R, C are in hundreds of dollars

(1) Determine the Initial Cost?

(2) Determine the maximum Profit and number of items required for that profit?

(3) Determine the maximum Revenue and number of items required for that revenue?

(4) Find the break even points, P(x) = 0. What do the values represent in relation to the business?

Top Toys is planning a new radio and TV advertising campaign. A radio commerical costs $300 and a TV ad costs $2000. A total budget of $20,000 is allocated to the campaign. However, to ensure that each medium will have at least one radio commercial and one TV ad, the most that can be allocated to either medium cannot exceed 80% of the total budget. It is estimated that the first radio commerical will reach 5000 people, with each additional commercial reaching only 2000 more people. For TV, the first ad will reach 4500 people, and each additional ad an additional 3000 people. How should the budgeted amount be allocated between radio and TV?

solve each equation find all solutions in the interval 0 2π) leave your answers in the exact form.

a) sin θ = cos(2θ)

b) sin (2θ) + cos(2θ) = √2/2

c) cos^3 + cos^2 - 3 cosθ - 3 = 0

d) sin 5x - sin 3x = cos 4x

e)sin(3x) + sin^2(x) + cos^2(x) = tan^2(x) - sec^2(x)

1

(a) Given Δ ABC , construct equilateral triangles Δ BCD , Δ CAE , and Δ ABF outside of Δ ABC . Prove that AD = BE=CF .

(b)Let ABCD be a convex quadrilateral. Show that the sum of the two diagonals of ABCD is less than the perimeter P of ABCD, but more than the semiperimeter P 2 of ABCD.

We are investigating two different retirment plans that we can invest our money in. Plan A invests $45,000 with an annual interest rate of 4.8% compounded continuously, and plan B invests $50,000 with an annual interest rate of 4.5%

Suppose we plan on retiring in 5 years. how much will plan A be worth at that time?

How much will Plan B be worth in 5 years?

Suppose instead we need at least $100,000 in this account we retire, and we don't plan on contributing any more to this account. (note that you actually need quite a bit more to retire, we'll assume that we have money in other accounts as well). How long will it take plan A to reach a balance of 100,000?

How long will it take plan B to reach a balance of $100,000?

Using your graphing calculator to graph the compound interest function for Plan A and the compund interest for Plan B using the window setting [0, 45] by [0, 400000]. Find the point of intersection (rounded to two decimal places) between the two functions, and interpert it.

when would it make more sense to choose plan A? (hint: When would plan A be worth more then plan B?)

When would it make more sense to choose Plan B?

Fine the T value of the point of interesction that you found in (what would it make more sense to choose Plan B) Algebraically. Show all work and incomplete solutions recieve no credit do not round answer til the end of this problem.

Please show all work for each one please Thank you so much!

The populations of termites and spiders in a certain house are growing exponentially. The house contains 80 termites the day you move in. After four days, the house contains 150 termites. Three days after moving in, there are two times as many termites as spiders. Eight days after moving in, there were four times as many termites as spiders.

How long (in days) does it take the population of spiders to triple?

Show that among all collections with 2n-1 natural numbers in them there are
exactly n numbers whose sum is divisible by n.

1- A carpenter purchased 50 ft of redwood and 70 ft of pine for a total cost of $259. A second purchase, at the same prices, included 80 ft of redwood and 50 ft of pine for a total cost of $340. Find the cost per foot of redwood and of pine.

2- An investment club placed $27,000 into two simple interest accounts. On one account, the annual simple interest rate is 6.5%. On the other, the annual simple interest rate is 2.5%. How much should be invested in each account so that both accounts earn the same annual interest?

3-Two investments earn an annual income of $422. One investment earns an annual simple interest rate of 8.9%, and the other investment earns an annual simple interest rate of 6.1%. The total amount invested is $6000. How much is invested in each account?

1. An engineer collected the data below showing the speed, s, of a Toyota Camry and its average miles per gallon, M

a. Plot the data above, treating speed as the independent variabl (x). What type of relation, linear or quadratic, appears to exist between speed and miles per gallon?

b. Based on your response above, find either a linear or a quadratic model that describes the relationship between the two variables.

c. Use your model to predict the mpg of a Toyota Camry that is travelling 63 mph

d. Does this model have a maximum value? if so, what is it? if the model does not have a maximum value, find the y-intercept. Does this intercept make sense in the context of the problem

The cities of Abnarca and Bonipto have populations that are growing exponentially. In 1980, Abnarca had a population of 23,000 people. In 1990, its population was 28,000. Bonipto had a population of 35,000 in 1980. The population of Bonipto doubles every 55 years.

(a) How long does it take the population of Abnarca to double?

(b) How long will it take for Abnarca's population to equal that of Bonipto?

1. A manufacturing company manufactures a cardboard box with a square base and a height of 15 inches. Suppose the equation x² +60x- 7,200 = 0 can be used to find the length and width of the base of the box, each measuring x inches.

Write the equation in factored form.

Use the zero product property to solve the equation. Show all the steps needed to find both answers.

Explain how the solution relates to this situation

2. City engineers decide to build a rectangular dog park that has an area of 3,600 square yards, where the length of the park is 10 more yards than twice its width. The equation x² +5x-1,800 = 0 can be used to find the width of the dog park.

Write the equation in factored form

Use the zero product property to solve the equation. Show all the steps needed to find both answers.

Explain how the solution relates to this situation.

3. A graphic designer uses a photo editing program to increase both the height and width of a square image by 3 inches. Suppose the equation x² +6x-55 = 0 can be used to find the height and width of the original image.

Write the equation in factored form.

Use the zero product property to solve the equation. Show all the steps needed to find both answers.

Explain how the solution relates to this situation.

Which of the following statements are TRUE or False. (Explain why it’s true or false?)

Subtracting a positive number from a negative number always gives you a negative number.

Subtracting two negative numbers always gives you a negative number.

Subtracting a - b is the same as adding a + (-b).

A positive number minus a negative number is always a positive number.

The difference of a number and its opposite gives you zero.

Zero minus a number is the same as a number minus zero.