Given that the line passes through the points;
[tex](1,7)[/tex]And slope;
[tex]m=-\frac{5}{2}[/tex]The revenue function R in terms of the number of units sold, x, is given as R = 290x-0.52x^2where R is the total revenue in dollars. Find the number of units sold x that produces a maximum revenue?Your answer is x=What is the maximum revenue?$
Solution
Step 1:
The function reaches a maximum where the derivative is equal to 0.
Find the first derivative of the function.
Step 2:
Write the function
[tex]R(x)\text{ = 290x - 0.52x}^2[/tex]Step 3
Find the first derivative
[tex]\begin{gathered} R(x)=\text{ 290x -0.52x}^2 \\ R^{\prime}(x)\text{ = 290 - 1.04x} \end{gathered}[/tex]Step 4:
The function reaches a maximum where the derivative is equal to 0.
[tex]\begin{gathered} 290\text{ - 1.04x = 0} \\ 1.04x\text{ = 290} \\ \text{x = }\frac{290}{1.04} \\ \text{x = 278.8 }\approx\text{ 279} \end{gathered}[/tex]So the number of units which produce the maximum revenue = 279
Step 5:
Substituting this value in the original equation gives the revenue:
[tex]\begin{gathered} R\text{ = 290x - 0.52x}^2 \\ R\text{ = 290}\times279\text{ - 0.52 }\times\text{ 279}^2 \\ R\text{ = 80910 - 42034.14} \\ R\text{ = \$38875.86} \end{gathered}[/tex]Maximum revenue = $38875.86
13. You can afford a $1,800 per month mortgage payment. You've found a 30-year loan at 5.5% interest.(a) How big of a loan can you afford? $(b) How much total money will you pay the bank? $(c) How much of that money is interest? $
Step-by-step explanation:
Given
Principal = $1,800
interest rate = 5.5%
Using the below formula to calculate the mortgage
[tex]\begin{gathered} m\text{ = }\frac{p\cdot\text{ r (}1+r)^n}{(1+r)^n\text{ - 1}} \\ \text{Where P = principal, r = interest rate} \\ m\text{ = \$1800} \\ r\text{ = 5.5\% } \\ r\text{ = }\frac{5.5}{100}\text{ = 0.055} \\ \text{ since it is per month, hence the interest rate is given as} \\ r\text{ = }\frac{0.055}{12}\text{ = 0.00458} \\ n\text{ = 12 }\cdot\text{ 30} \\ n\text{ = 360} \\ 1800\text{ = }\frac{P\cdot0.00458(1+0.00458)^{360}}{(1+0.00458)^{360}\text{ - 1}} \\ 1800\text{ = }\frac{P\cdot0.00458(1.00458)^{360}}{(1.00458)^{360}\text{ - 1}} \\ 1800\text{ = }\frac{P\cdot0.00458\cdot\text{ 5.1812}}{5.1812\text{ - 1}} \\ 1800\text{ = }\frac{P\cdot\text{ 0.0237}}{4.1812} \\ \text{Cross multiply} \\ 1800\cdot\text{ 4.1812 = P }\cdot\text{ 0.0237} \\ 7526.16\text{ = P }\cdot\text{ 0.0237} \\ p\text{ = }\frac{7526.16}{0.0237} \\ P=\text{ \$317, 559. 50} \end{gathered}[/tex]Hence, the loan he can afford is $317, 559. 50
Part B
The total money he will pay to the bank is calculated as follows
Total amount = 1800 * 360
Total amount = $648, 000
Angles K and M are vertical angles. K’s measurement is 72 degrees. What is the measure of M ?
If two angles are vertical and congruent, the angle is the same:
[tex]\begin{gathered} K=M \\ M=72 \end{gathered}[/tex]The measure of M is 72°
how are the processes for converting 5/8 to decimal and to a percentage similar and how are the processes different
To convert 5 to a decimal, divide it by 8.
To convert a decimal to a percent, multiply it by 100.
What is decimal and percentage conversion?
To convert a percentage to a decimal, divide by 100. So 25% is 25/100, or 0.25. To convert a decimal to a percentage, multiply by 100 (just move the decimal point 2 places to the right) and give the % symbol.
Consider, the given fraction, 5/8
Divide 5 by 8 to change a decimal.
So, 5/8 = 0.625
Now to convert decimal 0.625 into a percent.
Multiply 0.625 by 100 to get 62.5%.
So, the conversion of fraction into decimal is 0.625 and the decimal into percent is 62.5%.
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You are given the following problem" "A car burns 0.85 gallons of gas per hour when idling. Express this rate in quarts per minute." Which of the given conversion factors will be useful to solve this problem?
Given,
A car burns 0.85 gallons of gas per hour.
We know,
0.85 gallon = 3.4 quartz.
1 hour=60 min.
Thus,
0.85 gallon per hour=3.4/60 quart per min=0.057 quart per min.
Thus the conversion factor is 0.057 quart per min.
How many pairwise comparisons are needed to learn the outcome of an election involving n=15 candidates ?
Remember that
The formula for the number of independent pairwise comparisons is k(k-1)/2, where k is the number of conditions
In this problem
k=15
substitute
15(15-1)/2=105
therefore
The answer is 105 Two different functions are represented by this graph and this table:
Answer
Option B is correct.
Function B has the greater slope.
3 is greater than 2.
Explanation
For a straight line, the slope of the line can be obtained when the coordinates of two points on the line are known. If the coordinates are (x₁, y₁) and (x₂, y₂), the slope is given as
[tex]Slope=m=\frac{Change\text{ in y}}{Change\text{ in x}}=\frac{y_2-y_1}{x_2-x_1}[/tex]For function A, we will pick two points on the line
(x₁, y₁) and (x₂, y₂) are (-2, -1) and (0, 3)
[tex]\text{Slope = }\frac{3-(-1)}{0-(-2)}=\frac{3+1}{0+2}=\frac{4}{2}=2[/tex]For function B, we will pick the two most extreme points on the table
(x₁, y₁) and (x₂, y₂) are (0, 1) and (5, 16)
[tex]\text{Slope = }\frac{16-1}{5-0}=\frac{15}{5}=3[/tex]We can easily see that function B (3) has a greater slope than function A (2).
Hope this Helps!!!
bookmarksProdigySIS Grades and Attenda...Nearpod - Classifyi...Vocational Assess..G ExploreLe1. The sliders for y= a 2 + b have been set to create the following graph. What are possible values
First we can find the value of b, which is the y-intercept of the function.
Using the point (0, 2), we have that:
[tex]\begin{gathered} y=a|x|+b \\ \\ 2=a\cdot0+b \\ b=2 \end{gathered}[/tex]Now, in order to find the value of 'a', we can use the point (2, -2):
[tex]\begin{gathered} -2=a|2|+b \\ -2=2a+2 \\ 2a=-4 \\ a=-2 \end{gathered}[/tex]So the values of 'a' and 'b' are a = -2 and b = 2.
You want to know the number of minutes that you can use on your $45.00 phone card. The card company chargesyou $0.50 for the first minute and $0.05 for cach additional minute. Solve the formula $45.00 = $0.50 + $0.05mfor m. Justify each step with an algebraic property of equality.
To solve m:
1. Subtract 0.50 in both sides of the equation (subtraction property of equality):
[tex]\begin{gathered} 45.00-0.50=0.50-0.50+0.05m \\ 44.5=0.05m \end{gathered}[/tex]2. Divide both sides of the equation into 0.05 (Division property of equality):
[tex]\begin{gathered} \frac{44.5}{0.05}=\frac{0.05}{0.05}m \\ \\ 890=m \end{gathered}[/tex]3. Rewrite the equation (Symmetric property):
[tex]m=890[/tex]Then, you can use 890 minutes on your $45.00 phone card.Drag the expressions in order from least to greatest value.
Then, from least to greatest value, the order is:
[tex]1\frac{1}{2}-\frac{5}{8}[/tex][tex]1\frac{1}{8}+\frac{1}{4}[/tex][tex]1\frac{7}{8}-\frac{1}{4}[/tex]Suppose one-seventh of the employees of a certain company work in the Southeastern region. If the company employs 247 workers in that region, what is the total number of employees working for the company?How many total employees?
Given:
One-seventh of the employees of certain companies work in the Southeastern region. If the company employs 247 workers in that region,
Required:
Find the total number of employees.
Explanation:
Let the number of employees be x.
According to the question
[tex]\begin{gathered} \frac{1}{7}x=247 \\ x=247\times7 \\ x=1729 \end{gathered}[/tex]Final Answer:
The total number of employees working is 1729.
Is there enough information given to prove that the following pairs of triangles are congruent? If so, state the postulate or theorem that supports youranswer. If not, state NONE.Word Bank:HL AA CPCTC AAS SSS None SAS
Answer: There is not enough information to conclude they are congruent, NONE.
Explanation
Postulates or theorems
• Hypotenuse Leg (HL) postulate:, when two right triangles have a congruent hypotenuse and a corresponding congruent leg, these are congruent.
,• Angle-Angle (AA) postulate:, two triangles are similar if two corresponding angles are congruent.
,• Corresponding Parts of Congruent Triangles are Congruent (CPCTC): ,when two triangles are congruent, their corresponding sides and angles are also congruent.
,• Angle Angle Side (AAS) Theorem: ,two angles and the non-included side of two triangles are congruent, and if the angles and the side are corresponding parts in each triangle, then the triangles are congruent.
,• Side Side Side (SSS) Postulate: i,f three sides of two triangles are congruent between each other, then the two triangles are congruent.
,• Side Angle Side (SAS) Postulate: ,two angles and the included side of two triangles are congruent, and if the angles and the side are corresponding parts in each triangle, then the triangles are congruent.
We do not know if the sides are congruent as we are not given any information about it, we just know that the three angles are congruent.
Based on the latter, we can conclude that all postulates or theorems involve the congruence of the sides with the exception of AA postulate. However, the AA postulate states that if it is true, the triangles are similar (same shape) but not necessarily congruent (same size).
Therefore, we have not enough information to conclude the triangles are congruent, we would need the to know the congruency of at least one side of both triangles.
Find the volume of a cone with a base radius of 5 yd and a height of 9 yd.
Use the value 3.14 for it, and do not do any rounding.
Be sure to include the correct unit in your answer.
yd
A
OT
5 yd
The volume of the cone is 235.5 yd.³
The dimensions of the cone are given as:
Radius of the cone = r = 5 yd.
Height of the cone = h = 9 yd.
π = 3.14
We need to calculate the volume of the cone.
Volume of a cone = 1 / 3 π r² h
Substitute the values , we get that:
V = 1 / 3 (3.14) (5)² (9) yd.³
V = 1 / 3 (3.14) (25) (9) yd.³
V = (3.14) × (25) × (3) yd.³
Simplify the expression:
V = 235.5 yd.³
Therefore, we get that, the volume of the cone is 235.5 yd.³
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Long division ( polynomial by binomial) ( x^3 - 216) / ( x-6)
Given:
[tex]\frac{x^3-216}{x-6}[/tex]We will use the long division to find the answer
The long division will be as shown in the following picture:
So, the answer will be:
[tex]x^2+6x+36[/tex]What is the equation of this graphed line?
To write this equation in slope-intercept form, y = mx + b, we need to find m and b.
m, the slope, is the distance between the points' corresponding y-value divided by the distance between the points' corresponding x-value.
[tex]m=\frac{y_2-y_1}{x_2-x_1}\\m=\frac{6+6}{-7+3}\\m=\frac{12}{-4}\\m=-\frac{12}4\\m=-\frac13[/tex]
b, the y-intercept, is shown as -5, but we can solve for it as well when it is not so easy to tell the value of b. We can solve for b by substituting known solutions of x and y after we find m.
[tex]y=mx+b\\-7=-\frac136+b\\-7=-2+b\\-7+2=-2+2+b\\b=-5[/tex]
So,
[tex]y=-\frac13x-5[/tex]
what has to be true of angles H and K in order for line AB and line CD to be parallel why
When you have two parallel lines like AB and CD and a transversal line like EF you have to know that the angles will be as follow:
All the angles with the color red will be the same between them and the ones in color green will be the same between them because the angles opposite the vertex (the union between two transversal lines) are congruent or equal:
So the answer is:
The angles H and K are equals and the why is the explanation aboveThe figure shown was created by placing the vertices of a square on the circle. Thesquare has side lengths of 7cm and the circle has a diameter of 10 cm.Which measurement is closest to the area of the shaded region of the figure insquare centimeters?
To answer this question, we need to find the area of the square, and then the area of the circle. Then, we need to subtract from the area of the circle, the area of the square.
The area of the square is given by the formula:
[tex]A_{\text{square}}=s^2[/tex]The side of the square is 7cm. Then, the area is:
[tex]A_{\text{square}}=(7\operatorname{cm})^2\Rightarrow A_{square}=49\operatorname{cm}^2[/tex]Now, the area of the circle is given by the formula:
[tex]A_{\text{circle}}=\pi\cdot r^2[/tex]The diameter of the circle is equal to 10cm. The radius of the circle is half of the measure of the diameter. Then, the radius is equal to 10/2 ---> r = 5cm. Then, we have:
[tex]A_{\text{circle}}=\pi\cdot(5\operatorname{cm})^2\Rightarrow A_{circle}=\pi\cdot25\operatorname{cm}\approx78.54\operatorname{cm}^2[/tex]Now, to find the shaded area, we need to subtract from this area, the area of the square:
[tex]A_{\text{shaded}}=A_{\text{circle}}-A_{\text{square}}=78.54\operatorname{cm}-49\operatorname{cm}=29.54\operatorname{cm}^2[/tex]Therefore, the shaded area is closest to 29.5 square centimeters (third option) (if we round our result to the nearest tenth.)
You need a 50% alcohol solution. On hand, you have a 300 mL of a 40% alcohol mixture. You also
have 80% alcohol mixture. How much of the 80% mixture will you need to add to obtain the desired
solution?
You will need
—————————mL of the 80% solution
You will need 100 mL of the 80% solution and solve the question by using percentage concept.
What is the percentage?
A percentage is a number or ratio that can be expressed as a fraction of 100.
Assume that x mL of 80% solution is needed.
The amount of alcohol in 300 mL of a 40% alcohol mixture is
300 mL × 40% = 300 × (40/100) = 120 ml.
The amount of alcohol in 300 mL of a 40% alcohol mixture is
300 mL - 120 mL = 180 mL.
The amount of alcohol in x mL of a 80% alcohol mixture is
x mL × 80% = x × (80/100) = (8x)/10 ml.
The amount of alcohol in x mL of a 80% alcohol mixture is
x - (8x)/10 ml = (2x)/10 mL.
Total amount of alcohol is 120 + (8x)/10 mL
Total amount of water is 180 + (2x)/10 mL
The meaning of 50% alcohol solution is the amount of alcohol and water is equal.
120 + (8x)/10 = 180 + (2x)/10
(8x)/(10)- (2x)/10 = 180 -120
(6x)/(10) = 60
x = 100
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For the equation, find three ordered pairs solutions by completing the table. Then use any of the ordered pairs to graph the equation X-y=3
for y=0
[tex]\begin{gathered} x-0=3 \\ x=3\to y=0 \end{gathered}[/tex]To graph this function we solve for and give x values:
[tex]y=x-3[/tex][tex]\begin{gathered} x=0\to y=-3 \\ \\ x=1\to y=-2 \\ \\ x=2\to y=-1 \\ \\ x=3\to y=0 \end{gathered}[/tex]Equivalent equations have exactly the same solution set. Select Yes or No in thedropdowns to indicate whether each equation is equivalent to this equation.
The given equation is:
[tex]4x+3=\frac{5}{2}x-7[/tex]Solve the equation for x:
[tex]\begin{gathered} 4x+3=\frac{5}{2}x-7 \\ \text{Collect like terms:} \\ \Rightarrow4x-\frac{5}{2}x=-7-3 \\ \Rightarrow\frac{3}{2}x=-10 \\ \Rightarrow3x=-20 \\ \Rightarrow x=-\frac{20}{3} \end{gathered}[/tex]Next substitute the solution into each equation.
The equation that the solution satisfies is equivalent to the original equation.
Check for the first equation:
[tex]\begin{gathered} 4x=\frac{5}{2}x-4;x=-\frac{20}{3} \\ \Rightarrow4(-\frac{20}{3})=\frac{5}{2}(-\frac{20}{3})-4 \\ \Rightarrow-\frac{80}{3}=-\frac{50}{3}-4 \\ \Rightarrow-\frac{80}{3}\ne-\frac{62}{4} \end{gathered}[/tex]Since the solution does not satisfy the equation, it follows that the equation is not equivalent to the original equation.
Hence, select NO for the first equation.
Use the same procedure to check for the other equations.
Only the third and fourth equations are equivalent to the original equation, so feel yes for them, but no for the first and second.
Express the function graphed on the axes below as a piecewise function.10802-108-6-422246810-10
From the graph it is clear that the function is a horizontal line from -6 to -3 and an oblique line from -3 to 3 so the equation from -6 to -3 is:
[tex]f(x)=5[/tex]The equation of line using two point form from -3 to 3 is found by knowing the points.The line passes through (-2,6) and (0,4) so it follows:
[tex]\begin{gathered} \frac{y-6}{x+2}=\frac{4-6}{0+2} \\ y-6=-(x+2) \\ y=-x-2+6=-x+4 \\ y=f(x)=-x+4 \end{gathered}[/tex]Hence the piecewise function is given by:
[tex]f(x)=\begin{cases}5,-6\leq x<-3 \\ -x+4,-3Hence the function shown above is the required piecewise function.The order in which you perform operations matters in a numerical expression. O A. True B. False
Answer:
A. True
Explanation:
Consider the numerical expression below:
[tex]9+2\times3[/tex]If we solve from the left to the right, we have:
[tex]9+2\times3=11\times3=33\text{ (Wrong Solution)}[/tex]This result is incorrect.
However, by the order of operations, multiplication comes before addition, so we rightly have:
[tex]9+2\times3=9+6=15[/tex]We see that if we do not follow the order of operations, our results will not be accurate.
Therefore, the order in which you perform operations matters in a numerical expression.
The answer is True.
8. Irene's score on Test 1 was 120. Her score on test 2 was 108. What is the percent decrease from Test 1 to Test 2? A. 10% B. 11% C. 12% D. 13%
Test 1 : 120
Test 2 : 108
120 x = 108
x= 108/120
x= 0.9
0.9 x 100 = 90%
100%-90% = 10%
A.10%
Sketch the graph of the equation. y= 1/2x− 3/2.
Use the graphing tool to graph the line.
The graph of the line is given below:
What is graph?
The collection of ordered pairings where f(x)=y exists is the graph of a function f. These pairs are Cartesian coordinates of points in two-dimensional space and so form a subset of this plane in the typical situation when x and f(x) are real integers.
The graph of the given equation is,
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what is this 3+12c-4c
what is this 3+12c-4c
step 1
combine like terms
3+(12c-4c)
answer is
3+8cFour students graphed the system of equations shown below. Which graph is correct?
Explanation
Step 1
Let
[tex]\begin{gathered} y_1=-\frac{3}{4}x+4 \\ y_2=\frac{1}{2}x-1 \end{gathered}[/tex]a) graph y1, to draw the line, we need 2 points ( coordinates)so
i)when x=0
[tex]undefined[/tex]The function C(x) = 300x + 180 gives the cost for a college to offer x sections of an introductory class in CPR (cardiopulmonary resuscitation). The function R(x) = 390x gives the amount of revenue the college brings in when offering x sections of CPR. Find the point where the cost equals the revenue by graphing each function on the same coordinate system. (x, y) =
Cost = Revenue
Then
300x + 180 = 390x
x= (390-300)/180= 180/90 = 2
Now graph both functions
Vijay inherited some money from his grandfather and put it in a bank account that earns 6% interest compounded annually. After 3 years, Vijay had $4,000.00 in the bank account. How much interest did he earn? Round your answer to the nearest cent.
We have in this question a case of Annual Compound Interest. The formula for this case is as follows:
[tex]A=P(1+r)^n_{}_{}[/tex]Where:
A = accrued amount.
P = Principal (starting amount).
r = interest rate.
n = number of years.
We have from the question:
r = 6%
n = 3 years.
P = $4,000.00
A = it is the unknown amount.
We can determine A to then find the interest he earned. Thus, we can proceed as follows:
[tex]A=4000(1+0.06)^3\Rightarrow A=4000(1.06)^3\Rightarrow A=4764.064[/tex]This previous amount is the accrued amount (the starting amount plus the interest after 3 years annually compounded).
Therefore, the amount Vijay earned in interests (after 3 years) is:
[tex]4764.064-4000=764.064[/tex]And, rounding this amount to the nearest cent, we have that, finally, the earned interest (after 3 years) is $764.06.
In a horse race with 6 horses, in a horse race you have 6 horses, you make a bet by predicting the ranking of all 6 horses. If you place your bet at random, whatis the probability that you will get the first and second horse correct and in the correct order?Give your answer as a fraction.
Answer
Probability that one will get the first and second horse correct and in the correct order = ½
Explanation
The probability of an event is given as
[tex]\text{Probability of an event = }\frac{Number\text{ of elements in that event}}{Total\text{ number of elements in the sample space}}[/tex]For this question, we need to calculate the probability of getting the first and second horse correctly.
Number of elements in the event = Number of predictions with the first and second horse correct in that order = 1 × 1 × 6 × 5 × 4 × 3 = 360
Total number of elements in the sample space = Total number of possible arrangements for the horses = 6 × 5 × 4 × 3 × 2 × 1 = 720
Probability that one will get the first and second horse correct and in the correct order = (360/720) = ½
Hope this Helps!!!
Answer the question below. Be sure to show your work
ANSWER:
We have to find new side-lengths of PQR triangle.
Original sides are.
[tex]\begin{gathered} PQ\text{ = 8cm} \\ QR=17\operatorname{cm} \\ RP=15\operatorname{cm} \end{gathered}[/tex]After multiplying by 2.5 we get
[tex]\begin{gathered} PQ^{\prime}=2.5\times PQ=(8\times2.5)cm=20\operatorname{cm} \\ QR^{\prime}=(17\times2.5)cm=42.5\operatorname{cm} \\ RP^{\prime}=(15\times2.5)cm=37.5\operatorname{cm} \end{gathered}[/tex]These are the new sides of the P'Q'R' triangle.