A model for a company's revenue is R=-15p^2+300p+12,000, where p is the price in dollars of the company's product. What prize will maximize revenue?
Answers
[tex]\bf \textit{ vertex of a vertical parabola, using coefficients}\\\\ \begin{array}{lccclll} R = &{{ -15}}p^2&{{ +300}}p&{{ +12,000}}\\ &\uparrow &\uparrow &\uparrow \\ &a&b&c \end{array}\qquad \left(-\cfrac{{{ b}}}{2{{ a}}}\quad ,\quad {{ c}}-\cfrac{{{ b}}^2}{4{{ a}}}\right)[/tex]
so the Revenue will be the highest at [tex]\bf {{ c}}-\cfrac{{{ b}}^2}{4{{ a}}}[/tex] and that will happen at the price of [tex]\bf -\cfrac{{{ b}}}{2{{ a}}}[/tex]
[tex]R(p)=-15 p^{2}+300p+12,000[/tex]
Move the 12,000 over to the other side and set the equation equal to 0:
[tex]-15 p^{2}+300p=-12,000[/tex]
In order to complete the square, the leading coefficient on the squared term has to be a 1 and it's a -15, so we have to factor that out:
[tex]-15( p^{2}-20p)=-12,000 [/tex]
Now it's ready to complete the square on it. Do this by taking half the linear term, squaring it, and then adding it in. Our linear term is 20p. Half of 20 is 10 and 10 squared is 100, so we will add in 100 on the left. Let's do this next step in two parts:
[tex]-15( p^{2}-20p+100)=-12,000 [/tex]
That's not quite complete yet because if we add in 100 inside the parethesis on the left we have to add in that same amount on the right. But on the left, we didn't just add in 100, we added in -15 TIMES 100 cuz we can't just forget about the -15 we factored out. That looks like this then:
[tex]-15( p^{2}-20p+100)=-12,000-1,500 [/tex]
The whole reason for doing this is to create a perfect square binomial on the left which is what we have done. The perfect square binomial is this (and we are going to do the subtraction on the right at the same time):
[tex]-15(p-10)^{2}=-13,500[/tex]
If we move the right side over to the left we get this:
[tex]-15(p-10) ^{2}+13,500=0 [/tex]
which gives us a vertex of (10, 13,500). That means that at a price of $10, the max revenue will be $13,500. See how beautifully that works out!??
Related Questions
PLZ HELP NEED ANSWER ASAP! A sports club rewards teams based on overall points earned in a season. The data for points are shown in the table, where Low represents the fewest points scored and High represents the highest points scored by a single team member. Team Low High Range Mean Median IQR σ Team A 22 58 36 42.1 44 18.25 10.35 Team B 38 49 11 43.9 44.5 3.5 2.97 Team C 27 36 9 31.8 32 3.75 2.55 Part A: If the club wants to award the team that has the most consistent scoring among its team members, which team should it choose and why? Justify your answer mathematically. (5 points) Part B: If the club wants to award the team with the highest average score, which team should it choose and why? Justify your answer mathematically. (5 points)
Answers
Team Low High Range Mean Median IQR σ
Team A 22 58 36 42.1 44 18.25 10.35
Team B 38 49 11 43.9 44.5 3.5 2.97
Team C 27 36 9 31.8 32 3.75 2.55
Part A: If the club wants to award the team that has the most consistent scoring among its team members, the team that should it choose is Team B because it has only lesser points if it scored low, at 38 and it has the next to the highest score among the three team, which is 49. Also, the range of their performance is 43.9, a much better score than 42.1 in Team A and 31.8 in Team C.
Part B: If the club wants to award the team with the highest average score, Team A because it has only lesser points if it scored low, at 22 and it has the highest score among the three team, which is 58.
What is the measure of (arc) BC?
A. 55
B. 110
C. 48
D. 96
Answers
arc AB = 2*48 = 96°
Answer:
(D)[tex]96^{\circ}=(arc)BC[/tex]
Step-by-step explanation:
It is given from the figure that m∠BAC=48° and arcAC=110°.
The Inscribed Angle Theorem states that the measure of an inscribed angle is half the measure of its intercepted arc.
Now, using the above property, we have
[tex]m{\angle}BAC={\frac{1}{2}}(arc)BC[/tex]
Substituting the given values, we get
[tex]48^{\circ}=\frac{1}{2}(arc)BC[/tex]
[tex]96^{\circ}=(arc)BC[/tex]
Thus, the measure of the arc BC is 96 degrees.
Hence, option D is correct.
determine whether the sequence converges or diverges. if it converges give the limit.
48, 12, 3, 3/4, ...
Answers
The limit of the sequence is 0.
-------------------------------------------------------------
This is a geometric sequence with the first term a = 48 and r = 1/4 = 0.25 as the common ratio. Since |r| < 1, this means that the terms are steadily getting smaller and smaller. The terms will approach 0 but not actually get there.
Note: if you added up all the terms of this infinite sequence, then the sum would be S = a/(1-r) = 48/(1-0.25) = 64
Final answer:
The provided sequence is a geometric sequence that converges to 0 because its common ratio's absolute value is less than 1.
Explanation:
The sequence provided is 48, 12, 3, 3/4, suggesting a pattern where each term is divided by 4 to get the next term (this is a common ratio r = 1/4). This type of sequence is known as a geometric sequence, and to determine if it converges, we can apply the ratio test.
In a geometric sequence, successive terms approach zero if |r| < 1. Since the absolute value of the common ratio here is less than 1, the terms of the sequence will get progressively smaller and approach zero. Therefore, we can conclude that this geometric sequence converges, and the limit is 0.
How to factor:
x^3-216=0
Answers
What does exclamation point mean in math?
Answers
e.g., 5! = 1*2*3*4*5 = 120
In mathematics, an exclamation point represents a factorial, meaning the product of an integer and all the positive integers below it. For example, 5 factorial, or 5!, is 5 x 4 x 3 x 2 x 1 = 120. Unlike exponentials, factorials involve consecutive multiplication of integers.
Explanation:In mathematics, the exclamation point is used to represent a factorial. Factorial indicates the product of an integer and all the positive integers below it. For instance, 5! would be 5 x 4 x 3 x 2 x 1 = 120. This calculation method is a significant concept especially in exponential arithmetic.
Contrary to exponential notation, factorials do not deal with the concept of powers or repeated multiplication of the same number. Instead, they focus on consecutive multiplication of descending positive integers. For example, 3! is evaluated as 3 x 2 x 1 = 6, instead of 3 x 3.
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A store is having a sale on jelly beans and trail mix. For 3 pounds of jelly beans and 2 pounds of trail mix, the total cost is $14. For 5 pounds of jelly beans and 6 bounds of trail mix, the total cost is $28. Find the cost for each pound of jelly beans and each pound for trail mix.
Answers
3j=14-2t divide both sides by 3
j=(14-2t)/3
...
5j+6t=28, using j found above in this equation yields.
5(14-2t)/3+6t=28 perform indicated multiplication on left side
(70-10t+18t)/3=28 multiply both sides by 3
70-10t+18t=84 combine like terms on the left side
70+8t=84 subtract 70 from both sides
8t=14 divide both sides by 8
t=$1.75, and since j=(14-2t)/3
j=(14-2(1.75))/3
j=$3.50
So a pound of trail mix costs $1.75 and a pound of jelly beans costs $3.50.
Two angles are supplementary one angle measure 12 more than the other find the measures
Answers
Then, 12 + b + b = 180;
12 + 2b = 180;
2b = 168;
b = 84;
a = 12 + 84;
a = 96;
Shamus can paint the barn in 6 hours, and ishmael can paint the barn in 7 hours. how long will it take them to paint the barn together?
Answers
If Ishmael can paint the ban in 7 hours, the portion he can paint is 1/7
when working together the portion they can complete will be:
1/7+1/6
=13/42
time taken for them to complete the barn will be:
42/13
=3 3/13 hours
the answer is 3 3/13 hours
Jessica deposits $2000 into an account that pays simple interest at a rate of 3% per year. How much interest will she be paid in the first 2 years?
Answers
answer
interest will be $120 in the first 2 years
Which of the following represents the set of possible rational roots for the polynomial shown below 2x^3+5x^2-8x-20=0
Answers
factors of constant term (-20) = +/- 1, +/-2, +/-5, +/-10, +/-20 WE call these factors p.
factors of leading coefficient (2) = +/- 1, +/-2 (call these q)
then possible roots are p/q = +/-1/1, +/- 1/2 , +/- 5/1 , +/- 5/2 and so on.
The possible rational roots are given by the ratio of the constant term to the leading coefficient. That is [tex]\pm[/tex]1/1, [tex]\pm[/tex]1/2, [tex]\pm[/tex]1/5, [tex]\pm[/tex]1/10, [tex]\pm[/tex]1/20, and so on and this can be determined by using the arithmetic operations.
Given :
Polynomial Equation -- [tex]2x^3+5x^2-8x-20=0[/tex]
The following steps can be used in order to determine the rational roots of the given polynomial equation:
Step 1 - Write the given polynomial equation.
[tex]2x^3+5x^2-8x-20=0[/tex]
Step 2 - The coefficient of [tex]x^3[/tex] is 2. So the factors of 2 are [tex]\pm[/tex]1, [tex]\pm[/tex]2.
Step 3 - The constant term of the given polynomial is -20 and the factors of the constant term are [tex]\pm[/tex]1, [tex]\pm[/tex]2, [tex]\pm[/tex]5, [tex]\pm[/tex]10, [tex]\pm[/tex]20.
Step 4 - The rational roots are given by the ratio of the constant term to the leading coefficient. That is,
[tex]\pm[/tex]1/1, [tex]\pm[/tex]1/2, [tex]\pm[/tex]1/5, [tex]\pm[/tex]1/10, [tex]\pm[/tex]1/20, and so on.
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The Hamilton Brush Company issued 2500 shares of common stock worth 100,000,000 total.What is the par value of each share
Answers
100,000,000÷2,500
=40,000 per share
The table shows the outputs y for different inputs x:
Input
(x) 2 4 6 8
Output
(y) 1 2 3 4
Part A: Do the data in this table represent a function? Justify your answer.
Part B: Compare the data in the table with the relation f(x) = 3x – 10. Which relation has a greater value when x = 8?
Part C: Using the relation in Part B, what is the value of x if f(x) = 80?
Answers
B: For the first relation, y is half of x, so f(x)=x/2. If x is 8, y is 4.
Plug in the value of x for the second function. 3(8)-10=24-10=14.
The second relation has a greater value.
C: 80=3x-10,
90=3x
x=30
LeRon draws an orange counter from one bag and a red counter from another bag. Which statement is true?
Answers
The question asks if the events are independent or dependent.
Well, let's establish what an independent and dependent event is.
An independent event is an occurrence in which two or more events do not have effect on each other, regardless what happens during each scenario.
An independent event would be (for example) "I flipped a coin and rolled a dice." Both of these are independent events, as they do not have any effect on eachother.
A dependent event is an occurrence in which two or more events have effects on eachother. For example "I pull a red card out of a deck of cards. I then pull a black card out of the same deck."
The question you have is:
LeRon draws an orange counter from one bag and a red counter from another bag. Are the events dependent or independent?
Since that LeRon draws two different counters from two separate bags, the events do not effect each other, thus making the events independent.
I hope this helps!
How to solve and why the answer is (A)
Answers
What is 5.75% of $6,000? (Express answer as dollars as cents.)
Answers
Final answer: $345.00
Answer:
The answers are:
345 dollars
34500 cents
Step-by-step explanation:
In order to determine the answers, we have to know about the percentage notation. For any "x" value that is expressed as x%, it is the same that:
x%=[tex]\frac{x}{100}[/tex]
So, first we determine the value in dollars and then we determine the value in cents, where:
1 dollar=100 cents
In dollars $:
=5.75%*$6000
=[tex]\frac{5.75}{100}[/tex]*$6000
=$345.00
In cents c:
=5.75%*6000*100
=[tex]\frac{5.75}{100}[/tex]*6000*100
=34500.00 cents
Consider the system of linear equations.
7x+16y=-2
9x-4y=22
To use the linear combination method and addition to eliminate the y-terms, by which number should the second equation be multiplied?
A. -4
B. -1/4
C. 1/4
D. 4
Answers
D
To use the linear combination method and addition to eliminate the y-terms, the second equation should be multiplied by 4.
What is equation?An equation is like a relationship between two or more variables. It is expressed in equal to form. Equation of two variables looks like: ax+ by=c.
It is solved to find the value of variables.
How to solve equation?The given equations are 7x+16y= -2 and 9x-4y=22 and we have to solve using linear combination method.
First we have to change -4y to -16y because after we have to do addition so we will multiply the second equation by 4 which makes the equation as under:
36x-16y=88
Adding equation 1 and 2
7x+16y+36x-16y=-2+22
42x=20
x=20/42
x=10/21
Put the value of x in 7x+16y=-2
7(10/21)+16y=-2
10/3+16y=-2
16y=-2-10/3
16y=-16/3
y=-1/3
Hence to solve the equations we need to first multiply second equation by 4 which is option D.
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What information can be used to compare linear relationships?
Answers
Sample Response:
Linear relationships can be compared using their initial values, or y-intercepts, and their rates of change, or slopes. Initial values can tell you which relationship started with a greater value. Comparing slopes can tell you which relationship is rising or falling faster.
Determine whether the conjecture is true or false. Give a counterexample for any false conjecture. Given: ∠F ∠ F is supplementary to ∠G ∠ G and ∠G ∠ G is supplementary to ∠H ∠ H . Conjecture: ∠F ∠ F is supplementary to ∠H ∠ H .
Answers
Given: ∠F is supplementary to ∠G and ∠G is supplementary to ∠H . Conjecture: ∠F is supplementary to ∠H .
It is False.
Definition: two angles are supplementary if, and only if, they add up 180°.
=>
∠F is supplementary to ∠G => ∠F = 180° - ∠G => ∠G = 180° - ∠F
∠G is supplementary to ∠H => ∠G = 180° - ∠H
=> 180° - ∠F = 180° - ∠H
=> ∠F = ∠H,
Then, you have gotten the two angles are equal, and they will be supplementary only if ∠F = ∠ G = 90°, because 90° + 90° = 180°. In any other case, they are not supplementary.
This is a counter example:
∠G = 60°
∠F is supplementary to ∠G => ∠F = 180° - G = 180° - 120°
∠H = 30°
∠G is supplementary to ∠H => ∠G = 180° - H = ∠G = 180° - 30° = 150°
Then, ∠H + ∠F = 150° + 120° = 270° ≠ 180° => they are not supplementary.
The graph of a linear equation contains the points (4,1) and (-2,-11). Which point also lies on the graph?
A. (1,1)
B.(1,-5)
C.(1,-2)
D.(1,-7)
Answers
Answer:
Option B is correct.
(1 , -5) lies on the graph.
Step-by-step explanation:
Given the points (4,1) and (-2 , -11)
First find the linear equation for the given points.
Equation of line for two points [tex](x_1, y_1)[/tex] and [tex](x_2, y_2)[/tex]
is given by: [tex]y-y_1 = (\frac{y_2-y_1}{x_2 - x_1}) (x-x_1)[/tex]
Substitute the given points (4,1) and (-2 , -11) in above equation to find the equation of line:
[tex]y-1=(\frac{-11-1}{-2-4})(x-4)[/tex]
or
[tex]y-1=(\frac{-12}{-6})(x-4)[/tex]
or
[tex]y-1=2(x-4)[/tex]
Using distributive property on RHS ( i.e, [tex]a\cdot (b+c) = a\cdot b+ a\cdot c[/tex] )
we have;
y -1 = 2x-8
Add 1 to both sides of an equation;
y-1+1 = 2x-8+1
Simplify:
y = 2x -7
Therefore, the equation of line for the given point is: y =2x - 7 ....[1]
To find which points lies on the graph ( i.e, Line)
Substituting the given options in equation [1] we have;
A . (1,1)
Put x =1 and y =1
[tex]1 = 2\cdot 1 -7 = 2-7[/tex]
1 = -5 which is not true.
Similarly
B. for (1, -5)
[tex]-5= 2\cdot 1 -7 = 2-7[/tex]
-5 = -5 which is true.
C. for (1, 2)
[tex]2= 2\cdot 1 -7 = 2-7[/tex]
2 = -5 which is not true.
And
D. For (1 , -7)
[tex]-7= 2\cdot 1 -7 = 2-7[/tex]
-7 = -5 which is also not true.
Therefore, the only point which lies on the line graph [1] is; (1 ,-5)
Suppose that some country had an adult population of about 50 million, a labor-force participation rate of 60 percent, and an unemployment rate of 5 percent. how many people were unemployed?
Answers
unemployment rate of 5%....0.05(30 mill) = 1.5 mill
1.5 million were unemployed <==
** if the unemployment rate is 5%, that means 5% of the labor force, not the entire adult population because some of the adult population might be too old or disabled and cant participate in the labor force. The labor force is people that are able to work.
Based on the labor force participation rate, the number of unemployed people is 1.5 million people.
The unemployed rate is based on the labor force participation rate and not the adult population.
The labor force is 60% and in absolute value this is:
= 60% x 50 million people
= 30 million people
The unemployment rate is 5% of this:
= 5% x 30 million
= 1.5 million people
In conclusion, there are 1.5 million unemployed people
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Collinear points are not necessarily coplanar.. true or false
Answers
The given statement is a false statement.
Step-by-step explanation:Collinear Points--
Two or more points are said to be collinear if they lie on a straight line.
Coplanar--
Set of points are said to be coplanar if they lie on the same plane.
If the set of points are collinear then we may get a line such that it will contain all these points and as we know that line always lie in a plane also there may be infinite number of planes which may contain the same line.
So, all the points on the line will also lie in the same plane.
Yes, collinear points are necessarily coplanar.
Use the functions f(x) = 2x + 7 and g(x) = 6x − 5 to complete the function operations listed below.
Part A: Find (f + g)(x). Show your work. (3 points)
Part B: Find (f ⋅ g)(x). Show your work. (3 points)
Part C: Find f[g(x)]. Show your work. (4 points)
Answers
We are given the two functions:
f(x) = 2x + 7
g(x) = 6x – 5
Part A. Find (f + g)(x)
(f + g)(x) = f(x) + g(x)
(f + g)(x) = 2 x + 7 + 6 x – 5
(f + g)(x) = 8 x + 2
Part B. Find (f ⋅ g)(x)
(f ⋅ g)(x) = f(x) ⋅ g(x)
(f ⋅ g)(x) = (2 x + 7) (6 x – 5)
(f ⋅ g)(x) = 12 x^2 – 10 x + 42 x – 35
(f ⋅ g)(x) = 12 x^2 + 32 x – 35
Part C. Find f[g(x)]
f[g(x)] = 2 (6 x – 5) + 7
f[g(x)] = 12 x – 10 + 7
f[g(x)] = 12 x - 3
How do I simplify this?
Answers
[sqrt(x-2)+5/sqrt(x-2)]/(x-2)
simplifying the numerator we get:
sqrt(x-2)+5/sqrt(x-2)
=[x-2+5]/sqrt(x-2)
=(x+3)/sqrt(x-2)
therefore our entire expression will be:
[(x+3)/sqrt(x-2)]/(x-2)
[(x+3)/(x-2)^(1/2)]*1/(x-2)
applying the law if indices we get
=(x+3)/(x-2)^(3/2)
if y varies inversely as x, and y=20 as x=3, find y for the x value 4
Answers
Please help! Thanks (:
Answers
So the answer is choice B
Note: there are four zeros between the decimal point and the digit '4' in the value 0.0000416
Thirty pounds of a salt-water solution contains fifteen percent salt. How much salt is in the water?
1.5 lb
3 lb
4.5 lb
(I came up with 2 answers and I need someone with the same answer, the two options that I came up with is 1.5 lb. or 4.5lb.
Answers
answer
4.5 lb
A fantasy board game makes use of four dice. The first is a standard 6-faced die, the second die has 4 faces, and the other two have 10 faces each. If all the dice are thrown together and their combination dictates a certain outcome of the game, how many possible outcomes exist?
Answers
Answer:
2400
Step-by-step explanation:
Given : The first is a standard 6-faced die, the second die has 4 faces, and the other two have 10 faces each.
To Find: If all the dice are thrown together and their combination dictates a certain outcome of the game, how many possible outcomes exist?
Solution:
No. of faces of first die = 6
No. of faces of second die = 4
No. of faces of third die = 10
No. of faces of fourth die = 10
Now we are supposed to find the the number of possible outcomes exist.
So, Number of possible outcomes if all the dice are thrown together :
= [tex]6 \times 4 \times 10 \times 10[/tex]
=[tex]2400[/tex]
Hence there are 2400 possible outcomes if all the dice are thrown together and their combination dictates a certain outcome of the game.
George plans to cover his circular pool for the upcoming winter season. The pool has a diameter of 20 feet and extends 12 inches beyond the edge of the pool. A rope runs along the edge of the cover to secure it in place.
a.
What is the area of the pool cover?
b.
What is the length of the rope?
Answers
The pool cover extents 12 inches beyond the edge of the pool.
12 inches = 1 foot
Therefore, the radius of the pool cover is : r = 10 + 1 = 11 ft.
a. The area of the pool cover:
A = r² π = 11² π = 121 π ft²
b. The length of the rope:
l = 2 r π = 2 · 11 π = 22 π ft.
Answer:
Given the data, the pool has a diameter of 20 feet, so, R=10ft.
The pool cover expands 12 inches beyond the pool, so, 12in.= 1 foot
Therefore, the radius of the pool cover will then be: 10+1= 11ft.
ANSWER A (pool cover)) r² π = 11² π = 121 π ft²
A.)= 121 π ft²
ANSWER B (rope)) l = 2 r π = 2 · 11 π = 22 π ft.
B.)= 22 π ft
Step-by-step explanation:
I got a 100% on it on founders edtell
The figure shows the location of 3 points around a lake. The length of the lake, BC, is also shown. (Figure is not drawn to scale.) A picture of a right triangle ABC with right angle at B is shown. The length of the side AC is labeled as 9 miles. The length BC of the triangle is labeled as 6 miles. This length BC is also the length of an irregular gray shaded shape. Which of the following options is closest to the distance (in miles) between points A and B? 6.71 miles 7.35 miles 8.66 miles 10.82 miles.
Answers
use the Pythagorean theorem
6^2 + x^2 = 9^2
36 + x^2 = 81
81-36= 45
square root of 45 is approximately 6.71 miles
Answer:
6^2 + x^2 = 9^2
36 + x^2 = 81
81-36= 45
square root of 45 is approximately 6.71 miles
Step-by-step explanation:
Using the theram theroy is very important to know the fulma and steps
Kwamee created a coffee blend for his cafe by mixing Kona beans and Fuji beans.Kona beans cost $11 per pound and Fuji beans cost $7.50 per pound.He bought a total of 23 pounds of coffee beans and it cost $197. Write a system of equations that can be used to determine how many pounds of Kona and Fuji beans Kwamee bought.
Answers
K + F = 23
Now let's see what we can do with the dollars. Again, we don't know how much he bought of each kind of coffee, but we do know that Kona beans cost $11 per pound and that Fuji beans cost $7.50 per pound, and we know that he spent a total of $197. So let's set that up:
11K + 7.50F = 197
Those are your 2 equations. It doesn't say you need to solve them, so you're done.