# A manager estimates a band's profit p for a concert by using the function p(t)=-200t^2+2500t-c, where t is the price per ticket and c is the band's operation cost. The table shows the band's operating cost at three different concert locations. What range of ticket prices should the band charge at each location in order to make a profit of at least $1000 at each concert? Band's Costs Location Operating Cost Freemont Park$900 Saltillo Plaza $1500 Riverside Walk$2500

Question
A manager estimates a band's profit p for a concert by using the function p(t)=-200t^2+2500t-c, where t is the price per ticket and c is the band's operation cost. The table shows the band's operating cost at three different concert locations. What range of ticket prices should the band charge at each location in order to make a profit of at least $1000 at each concert? Band's Costs Location Operating Cost Freemont Park$900
Saltillo Plaza $1500 Riverside Walk$2500

2020-12-03
freemont park: c=900
p(t) = 1000 subsitute variables into equation
$$\displaystyle{p}{\left({t}\right)}=-{200}{\left({t}^{{2}}\right)}+{2500}{t}-{c}$$
$$\displaystyle{\left({1000}\right)}=-{200}{\left({t}^{{2}}\right)}+{2500}{t}-{\left({900}\right)}$$
$$\displaystyle{0}=-{200}{\left({t}^{{2}}\right)}+{2500}{t}-{1900}$$
$$\displaystyle{200}{t}^{{2}}-{2500}{t}+{1900}={0}$$
a = 200, b = -2500 c = 1900 quadratic equation
$$\displaystyle{t}=\frac{{{2500}\pm\sqrt{{{2500}^{{2}}-{4}{\left({200}\right)}{\left({1900}\right)}}}}}{{2}}{\left({200}\right)}$$
$$\displaystyle{t}=\frac{{{2500}\pm\sqrt{{{4730000}}}}}{{400}}$$
$$\displaystyle{t}=\frac{{25}}{{4}}+\frac{\sqrt{{{473}}}}{{4}}$$
$$\displaystyle{t}=\frac{{25}}{{4}}-\frac{\sqrt{{{473}}}}{{4}}$$
$$\displaystyle{t}={6.25}+{5.43714079273}=\{11.69}$$ or
$$\displaystyle{t}={6.25}-{5.43714079273}=\{0.81}$$
saltillo plaza: c = 1500
p(t) = 1000 subsitute variables into equation
$$\displaystyle{p}{\left({t}\right)}=-{200}{\left({t}^{{2}}\right)}+{2500}{t}-{c}$$
$$\displaystyle{\left({1000}\right)}=-{200}{\left({t}^{{2}}\right)}+{2500}{t}-{\left({1500}\right)}$$
$$\displaystyle{0}=-{200}{\left({t}^{{2}}\right)}+{2500}{t}-{2500}$$
$$\displaystyle{200}{t}^{{2}}-{2500}{t}+{2500}={0}$$
a = 200, b = -2500 c = 2500 quadratic equation
$$\displaystyle{t}=\frac{{{2500}\pm\sqrt{{{2500}^{{2}}-{4}{\left({200}\right)}{\left({2500}\right)}}}}}{{2}}{\left({200}\right)}$$
$$\displaystyle{t}=\frac{{{2500}\pm\sqrt{{{4250000}}}}}{{400}}$$
$$\displaystyle{t}=\frac{{25}}{{4}}+{\left(\frac{{5}}{{4}}\right)}\sqrt{{{17}}}=\{11.40}$$ or
$$\displaystyle{t}=\frac{{25}}{{4}}-{\left(\frac{{5}}{{4}}\right)}\sqrt{{{17}}}=\{1.09}$$
riverside walk: c = 2500
p(t) = 1000 subsitute variables into equation
$$\displaystyle{p}{\left({t}\right)}=-{200}{\left({t}^{{2}}\right)}+{2500}{t}-{c}$$
$$\displaystyle{\left({1000}\right)}=-{200}{\left({t}^{{2}}\right)}+{2500}{t}-{\left({2500}\right)}$$
$$\displaystyle{0}=-{200}{\left({t}^{{2}}\right)}+{2500}{t}-{3500}$$
$$\displaystyle{200}{t}^{{2}}-{2500}{t}+{3500}={0}$$
a = 200, b = -2500 c = 3500 quadratic equation
$$\displaystyle{t}=\frac{{{2500}\pm\sqrt{{{2500}^{{2}}-{4}{\left({200}\right)}{\left({3500}\right)}}}}}{{2}}{\left({200}\right)}$$
$$\displaystyle{t}=\frac{{{2500}\pm\sqrt{{{3450000}}}}}{{400}}$$
t = $10.89 or t =$1.61

### Relevant Questions

The table below shows the number of people for three different race groups who were shot by police that were either armed or unarmed. These values are very close to the exact numbers. They have been changed slightly for each student to get a unique problem.
Suspect was Armed:
Black - 543
White - 1176
Hispanic - 378
Total - 2097
Suspect was unarmed:
Black - 60
White - 67
Hispanic - 38
Total - 165
Total:
Black - 603
White - 1243
Hispanic - 416
Total - 2262
Give your answer as a decimal to at least three decimal places.
a) What percent are Black?
b) What percent are Unarmed?
c) In order for two variables to be Independent of each other, the P $$(A and B) = P(A) \cdot P(B) P(A and B) = P(A) \cdot P(B).$$
This just means that the percentage of times that both things happen equals the individual percentages multiplied together (Only if they are Independent of each other).
Therefore, if a person's race is independent of whether they were killed being unarmed then the percentage of black people that are killed while being unarmed should equal the percentage of blacks times the percentage of Unarmed. Let's check this. Multiply your answer to part a (percentage of blacks) by your answer to part b (percentage of unarmed).
Remember, the previous answer is only correct if the variables are Independent.
d) Now let's get the real percent that are Black and Unarmed by using the table?
If answer c is "significantly different" than answer d, then that means that there could be a different percentage of unarmed people being shot based on race. We will check this out later in the course.
Let's compare the percentage of unarmed shot for each race.
e) What percent are White and Unarmed?
f) What percent are Hispanic and Unarmed?
If you compare answers d, e and f it shows the highest percentage of unarmed people being shot is most likely white.
Why is that?
This is because there are more white people in the United States than any other race and therefore there are likely to be more white people in the table. Since there are more white people in the table, there most likely would be more white and unarmed people shot by police than any other race. This pulls the percentage of white and unarmed up. In addition, there most likely would be more white and armed shot by police. All the percentages for white people would be higher, because there are more white people. For example, the table contains very few Hispanic people, and the percentage of people in the table that were Hispanic and unarmed is the lowest percentage.
Think of it this way. If you went to a college that was 90% female and 10% male, then females would most likely have the highest percentage of A grades. They would also most likely have the highest percentage of B, C, D and F grades
The correct way to compare is "conditional probability". Conditional probability is getting the probability of something happening, given we are dealing with just the people in a particular group.
g) What percent of blacks shot and killed by police were unarmed?
h) What percent of whites shot and killed by police were unarmed?
i) What percent of Hispanics shot and killed by police were unarmed?
You can see by the answers to part g and h, that the percentage of blacks that were unarmed and killed by police is approximately twice that of whites that were unarmed and killed by police.
j) Why do you believe this is happening?
Do a search on the internet for reasons why blacks are more likely to be killed by police. Read a few articles on the topic. Write your response using the articles as references. Give the websites used in your response. Your answer should be several sentences long with at least one website listed. This part of this problem will be graded after the due date.
Case: Dr. Jung’s Diamonds Selection
With Christmas coming, Dr. Jung became interested in buying diamonds for his wife. After perusing the Web, he learned about the “4Cs” of diamonds: cut, color, clarity, and carat. He knew his wife wanted round-cut earrings mounted in white gold settings, so he immediately narrowed his focus to evaluating color, clarity, and carat for that style earring.
After a bit of searching, Dr. Jung located a number of earring sets that he would consider purchasing. But he knew the pricing of diamonds varied considerably. To assist in his decision making, Dr. Jung decided to use regression analysis to develop a model to predict the retail price of different sets of round-cut earrings based on their color, clarity, and carat scores. He assembled the data in the file Diamonds.xls for this purpose. Use this data to answer the following questions for Dr. Jung.
1) Prepare scatter plots showing the relationship between the earring prices (Y) and each of the potential independent variables. What sort of relationship does each plot suggest?
2) Let X1, X2, and X3 represent diamond color, clarity, and carats, respectively. If Dr. Jung wanted to build a linear regression model to estimate earring prices using these variables, which variables would you recommend that he use? Why?
3) Suppose Dr. Jung decides to use clarity (X2) and carats (X3) as independent variables in a regression model to predict earring prices. What is the estimated regression equation? What is the value of the R2 and adjusted-R2 statistics?
4) Use the regression equation identified in the previous question to create estimated prices for each of the earring sets in Dr. Jung’s sample. Which sets of earrings appear to be overpriced and which appear to be bargains? Based on this analysis, which set of earrings would you suggest that Dr. Jung purchase?
5) Dr. Jung now remembers that it sometimes helps to perform a square root transformation on the dependent variable in a regression problem. Modify your spreadsheet to include a new dependent variable that is the square root on the earring prices (use Excel’s SQRT( ) function). If Dr. Jung wanted to build a linear regression model to estimate the square root of earring prices using the same independent variables as before, which variables would you recommend that he use? Why?
1
6) Suppose Dr. Jung decides to use clarity (X2) and carats (X3) as independent variables in a regression model to predict the square root of the earring prices. What is the estimated regression equation? What is the value of the R2 and adjusted-R2 statistics?
7) Use the regression equation identified in the previous question to create estimated prices for each of the earring sets in Dr. Jung’s sample. (Remember, your model estimates the square root of the earring prices. So you must actually square the model’s estimates to convert them to price estimates.) Which sets of earring appears to be overpriced and which appear to be bargains? Based on this analysis, which set of earrings would you suggest that Dr. Jung purchase?
8) Dr. Jung now also remembers that it sometimes helps to include interaction terms in a regression model—where you create a new independent variable as the product of two of the original variables. Modify your spreadsheet to include three new independent variables, X4, X5, and X6, representing interaction terms where: X4 = X1 × X2, X5 = X1 × X3, and X6 = X2 × X3. There are now six potential independent variables. If Dr. Jung wanted to build a linear regression model to estimate the square root of earring prices using the same independent variables as before, which variables would you recommend that he use? Why?
9) Suppose Dr. Jung decides to use color (X1), carats (X3) and the interaction terms X4 (color * clarity) and X5 (color * carats) as independent variables in a regression model to predict the square root of the earring prices. What is the estimated regression equation? What is the value of the R2 and adjusted-R2 statistics?
10) Use the regression equation identified in the previous question to create estimated prices for each of the earring sets in Dr. Jung’s sample. (Remember, your model estimates the square root of the earring prices. So you must square the model’s estimates to convert them to actual price estimates.) Which sets of earrings appear to be overpriced and which appear to be bargains? Based on this analysis, which set of earrings would you suggest that Dr. Jung purchase?
class council determined that its profit from the upcoming homecoming dance is directly related to the ticket price for the dance. Looking at past dances, the council determined that the profit pp can be modeled by the function p(t)=−12t^2+480t+30, where tt represents the price of each ticket. What should be the price of a ticket to the homecoming dance to maximize the council's profit? Price
The mill Mountain Coffee shop blends coffee on the premises for its customers. it sells three basic blends in 1- pound bags, Special , Mountain dark, and Mill regular. It uses four different types of coffee to produce the blends- Brazilian, mocha,Columbian, and mild. The shop used the following blend recipe requirements :
$$\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{l}\right|}{l}{\left|{l}\right|}\right\rbrace}{h}{l}\in{e}\text{Blend}&\text{Mix requirement}&\text{Selling price/lb(\\)}\backslash{h}{l}\in{e}\text{special}&\text{at least 40% columbian,}&{6.50}\backslash&\text{at least 30% mocha}\backslash{h}{l}\in{e}\text{Dartk}&\text{at least 60% Brazillian}&{5.25}\backslash&\text{no more than 10% mid}\backslash{h}{l}\in{e}\text{Regular}&\text{no more than 60% mid}&{3.75}\backslash&\text{at least 30% Brazillian}\backslash{h}{l}\in{e}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}$$
The cost of Brazilian coffee is 2.00 per pound, the cost of mocha is $2.75 per pound, the cost of Columbian is$2.90 per pound,and the cost of mild is $1.70 per pound. The shop has 110 pounds of Brazilan coffee. 70 pounds of mocha, 80 pounds of Columbian, and 150 pounds of mild coffee available per week. The shop wants to know the amount of each blend it should prepare each week to maximize profit. a. Formulate a linear programming model b. Solve this model asked 2021-03-11 Let's say the widget maker has developed the following table that shows the highest dollar price p. widget where you can sell N widgets. Number N Price p $$200 53.00$$ $$250 52.50$$ $$300 52.00$$ $$35051.50$$ (a) Find a formula for pin terms of N modeling the data in the table. (b) Use a formula to express the total monthly revenue R, in dollars, of this manufacturer in month as a function of the number N of widgets produced in a month. $$R=$$ Is Ra linear function of N? (c) On the basis of the tables in this exercise and using cost, $$C= 35N + 900$$, use a formula to express the monthly profit P, in dollars, of this manufacturer asa function of the number of widgets produced in a month $$p=$$ (d) Is Pa linear function of N2 e. Explain how you would find breakeven. What does breakeven represent? asked 2020-10-23 The close connection between logarithm and exponential functions is used often by statisticians as they analyze patterns in data where the numbers range from very small to very large values. For example, the following table shows values that might occur as a bacteria population grows according to the exponential function P(t)=50(2t): Time t (in hours)012345678 Population P(t)501002004008001,6003,2006,40012,800 a. Complete another row of the table with values log (population) and identify the familiar function pattern illustrated by values in that row. b. Use your calculator to find log 2 and see how that value relates to the pattern you found in the log P(t) row of the data table. c. Suppose that you had a different set of experimental data that you suspected was an example of exponential growth or decay, and you produced a similar “third row” with values equal to the logarithms of the population data. How could you use the pattern in that “third row” to figure out the actual rule for the exponential growth or decay model? asked 2020-11-12 Finance bonds/dividends/loans exercises, need help or formulas Some of the exercises, calculating the Ri is clear, but then i got stuck: A security pays a yearly dividend of 7€ during 5 years, and on the 5th year we could sell it at a price of 75€, market rate is 19%, risk free rate 2%, beta 1,8. What would be its price today? 2.1 And if its dividend growths 1,7% each year along these 5 years-what would be its price? A security pays a constant dividend of 0,90€ during 5 years and thereafter will be sold at 10 €, market rate 18%, risk free rate 2,5%, beta 1,55, what would be its price today? At what price have i purchased a security if i already made a 5€ profit, and this security pays dividends as follows: first year 1,50 €, second year 2,25€, third year 3,10€ and on the 3d year i will sell it for 18€. Market rate is 8%, risk free rate 0,90%, beta=2,3. What is the original maturity (in months) for a ZCB, face value 2500€, required rate of return 16% EAR if we paid 700€ and we bought it 6 month after the issuance, and actually we made an instant profit of 58,97€ You'll need 10 Vespas for your Parcel Delivery Business. Each Vespa has a price of 2850€ fully equipped. Your bank is going to fund this operation with a 5 year loan, 12% nominal rate at the beginning, and after increasing 1% every year. You'll have 5 years to fully amortize this loan. You want tot make monthly installments. At what price should you sell it after 3 1/2 years to lose only 10% of the remaining debt. asked 2021-02-06 A.Which figure shows the loop that the must beused as the Ampèrean loop for finding for inside the solenoid? B.Find , the z component of the magnetic field insidethe solenoid where Ampère's law applies. Express your answer in terms of,,,,and physical constants such as . C.The magnetic field inside a solenoidcan be found exactly using Ampère's law only if thesolenoid is infinitely long. Otherwise, the Biot-Savart law must beused to find an exact answer. In practice, the field can bedetermined with very little error by using Ampère's law, aslong as certain conditions hold that make the field similar to thatin an infinitely long solenoid. Which of the following conditions musthold to allow you to use Ampère's law to find a goodapproximation? Consider only locations where the distance from the ends ismany times . Consider any location inside the solenoid, as long asis much larger than for the solenoid. Consider only locations along the axis of the solenoid. Answer? Show transcribed image text A. Which figure shows the loop that the must beused as the Amp?rean loop for finding B_z(r) for r inside the solenoid? B. Find B_z(r), the z component of the magnetic field insidethe solenoid where Amp?re's law applies. Express your answer in terms ofL,D,n,I,and physical constants such as mu_0. C. The magnetic field inside a solenoidcan be found exactly using Amp?re's law only if thesolenoid is infinitely long. Otherwise, the Biot - Savart law must beused to find an exact answer. In practice, the field can bedetermined with very little error by using Amp?re's law, aslong as certain conditions hold that make the field similar to thatin an infinitely long solenoid. Which of the following conditions musthold to allow you to use Amp?re's law to find a goodapproximation? Consider only locations where the distance from the ends ismany times D. Consider any location inside the solenoid, as long asLis much larger than Dfor the solenoid. Consider only locations along the axis of the solenoid. asked 2020-10-19 The purchase price of a home y (in$1000) can be approximated based on the annual income of the buyer $$x_1$$ (in $1000) and on the square footage of the home $$x_2 (\text{ in } 100ft^2)$$ according to $$y=ax_1+bx_2+c$$ The table gives the incomes of three buyers, the square footages of the home purchased, and the corresponding purchase prices of the home. a) Use the data to write a system of linear equations to solve for a, b, and c. b) Use a graphing utility to find the reduced row-echelon form of the augmented matrix. c) Write the model $$y=ax_1+bx_2+c$$ d) Predict the purchase price for a buyer who makes$100000 per year and wants a $$2500ft^2$$ home.
$$\begin{array}{|c|c|} \hline Number\ N & Price\ p\\ \hline 200 & 53.00\\ \hline 250 & 52.50\\\hline 300 & 52.00\\ \hline 350 & 51.50\\ \hline \end{array}$$
$$\displaystyle{p}=$$
$$\displaystyle{R}=$$
(c) On the basis of the tables in this exercise and using cost, $$\displaystyle{C}={35}{N}+{900}$$, use a formula to express the monthly profit P, in dollars, of this manufacturer as a function of the number of widgets produced in a month.
$$\displaystyle{P}=$$