4.8 / 4.4...............
4.8 / 14.4 = 48/144 = 1/3
After a translation, the image of P(-3,5) is P(-4,3). Find the translation rule. Write your answer in the form (x+a, y+b).
P(-3,5)
Image: (-4,3)
x coordinate:
-3+a = -4
Solve for a
a=-4+3
a= -1
y coordinate:
5+b =3
b =3-5
b= -2
So:
(x-1, y-2)
Caitlyn sells her drawings at the county fair. She wants to sell at least 85 drawings and has portraits and landscapes. She sells the portraits for $14 each and the landscapes for $10 each. She needs to sell at least $980 worth of drawings in order to earn a profit.Write a system of 2 inequalities to model this situation. Use "x" for the number of portraits and " y " for the number of landscapes.
Given:
Sell at least 85 drawing.
Sells the protraits for $14 each and landscapes for $10 each worth atlest $980
Find: Write a system inequality
Sol:.
[tex]x+y\ge85[/tex][tex]14x+10y\ge980[/tex]Graph of inequality is:
(C)
If Caitlyn sells 40 protraits and 50 landscapes is:
[tex]\begin{gathered} 14x+10y\ge980 \\ 14(40)+10(50)=1060 \end{gathered}[/tex][tex]\begin{gathered} \text{Sales } \\ x+y\ge85 \\ 40+50=90 \end{gathered}[/tex]Yes she meet minimum sale and profit is 1060.
(D)
If Caitlyn sells 80 protraits and 55 landscapes is:
[tex]\begin{gathered} \text{Sales} \\ x+y\ge85 \\ 80+55\ge85 \end{gathered}[/tex][tex]\begin{gathered} \text{Earn profit} \\ 14x+10y \\ =14(80)+10(55) \\ =1120+550 \\ =1670 \end{gathered}[/tex]Profit 1670 and yes she meet minimum sales.
What is the volume of this cone? Use 73.14. Round your answer to thenearest whole cubic centimeter, if needed.14cm10cmANSWER CHOICES ARE 1539615470140
[tex]\begin{gathered} V=3.14\cdot7^2\cdot10 \\ V=1538.6\operatorname{cm}^3 \\ V\approx1539\operatorname{cm}^3 \end{gathered}[/tex]
Select the correct answer. Consider the explicit formula of two sequences. (n) = 2n + 3 g() = 3( - 1) - Which mathematical statement is correct? g(1) > f(1) g(3) = f(3) f(1) < g(2) f(10) = g(4)
The function g is given by:
[tex]\begin{gathered} g(n)=3^{(n-1)} \\ \text{ Substitute }n=3\text{ into the equation:} \\ g(3)=3^{(3-1)}=3^2=9 \end{gathered}[/tex]The function f is given by:
[tex]\begin{gathered} f(n)=2n+3 \\ \text{ Substitute }n=3\text{ into the equation:} \\ f(3)=2(3)+3=9 \end{gathered}[/tex]Therefore,
g(3) = f(3)
given the inequality 6x - 10y ≥ 9, select all possible solutionsA. (-1, 1)B. (-3, 4)C. (2, 1)D. (4, -2)E. (2, 8)F. (5, 2)
we simplify the expression to work easier
[tex]\begin{gathered} 6x\ge9+10y \\ 6x-9\ge10y \\ \frac{6x-9}{10}\ge y \end{gathered}[/tex]now we check each point replacing the coordinates and checking the inequality
A.
[tex]\begin{gathered} \frac{6(-1)-9}{10}\ge(1) \\ \\ \frac{-6-9}{10}\ge1 \\ \\ -\frac{15}{10}\ge1 \\ \\ -1.5\ge1 \end{gathered}[/tex]A is wrong because -1.5 isnt greater than 1 , so the inequality is wrong
B.
[tex]\begin{gathered} \frac{6(-3)-9}{10}\ge(4) \\ \\ \frac{-18-9}{10}\ge4 \\ \\ \frac{-27}{10}\ge4 \\ \\ -2.7\ge4 \end{gathered}[/tex]B is wrong because -2.7 isnt greater than 4
C.
[tex]\begin{gathered} \frac{6(2)-9}{10}\ge(1) \\ \\ \frac{12-9}{10}\ge1 \\ \\ \frac{3}{10}\ge1 \\ \\ 0.3\ge1 \end{gathered}[/tex]C is wrong because 0.3 isnt greater than 1
D.
[tex]\begin{gathered} \frac{6(4)-9}{10}\ge(-2) \\ \\ \frac{24-9}{10}\ge-2 \\ \\ \frac{15}{10}\ge-2 \\ \\ 1.5\ge-2 \end{gathered}[/tex]D is right because 1.5 is grreater than -2
E.
[tex]\begin{gathered} \frac{6(2)-9}{10}\ge(8) \\ \\ \frac{12-9}{10}\ge8 \\ \\ \frac{3}{10}\ge8 \\ \\ 0.3\ge8 \end{gathered}[/tex]E is wrong because 0.3 isnt greater than 8
F.
[tex]\begin{gathered} \frac{6(5)-9}{10}\ge(2) \\ \\ \frac{30-9}{10}\ge2 \\ \\ \frac{21}{10}\ge2 \\ \\ 2.1\ge2 \end{gathered}[/tex]F is Right because 2.1 is greater than 2
Find the equation of a line perpendicular to y = - 1/4 x + 9 that passes through the point (4, - 8) .
perpendicularWe were given the following information:
The equation of a line is given by: y = - 1/4 x + 9
We want to obtain the equation for a line perpendicular to this line & that passes through the point (4, -8). This is shown below:
The general equation of a straight line is given by:
[tex]\begin{gathered} y=mx+b \\ where\colon \\ m=slope \\ b=y-intercept \end{gathered}[/tex]The equation of the line given us is:
[tex]\begin{gathered} y=-\frac{1}{4}x+9 \\ \text{Comparing this with the general equation, we will deduce that:} \\ mx=-\frac{1}{4}x \\ m=-\frac{1}{4} \\ \text{Thus, the slope of this line is: }-\frac{1}{4} \end{gathered}[/tex]The relationship between the slope of a line and the slope of a line perpendicular to it is given by the statement "the product of the slopes of two lines perpendicular to one another is negative one"
This is expressed below:
[tex]\begin{gathered} m\times m_{perpendicular}=-1 \\ m_{perpendicular}=-\frac{1}{m} \\ m=-\frac{1}{4} \\ m_{perpendicular}=\frac{-1}{-(\frac{1}{4})} \\ m_{perpendicular}=4 \\ \\ \therefore m_{perpendicular}=4 \end{gathered}[/tex]Therefore, the slope of the perpendicular line is: 4
We were told that the perpendicular line passes through the point (4, -8). We will obtain the equation of the perpendicular line using the Point-Slope equation. This is shown below:
[tex]\begin{gathered} y-y_1=m(x-x_1) \\ (x_1,y_1)=(4,-8) \\ m\Rightarrow m_{perpendicular}=4_{} \\ \text{Substitute the values of the variables into the initial equation above, we have:} \\ y-\mleft(-8\mright)=4(x-4) \\ y+8=4(x-4) \\ y+8=4x-16 \\ \text{Subtract ''8'' from both sides, we have:} \\ y=4x-16-8 \\ y=4x-24 \\ \\ \therefore y=4x-24 \end{gathered}[/tex]The graphical representation of this is given below:
To approximate the length of the pond, you walk 200 ft from point A to point B, turn 80 degrees and walk 160 ft to point C. Approximate the length AC of the pond. Round your answer to 2 decimal points.
Answer:
276.97 ft
Explanation:
We can represent the situation with the following diagram
Now, to find the distance from A to C, we first need to find the measure of angle B, so
∠B = 180 - 80
∠B = 100
Then, using the cosine law, we can find the distance from A to C as follows
[tex]AC^2=(AB)^2+(BC)^2-2(AB)(BC)\cos (100)[/tex]Replacing the values and solving for AC, we get:
[tex]\begin{gathered} AC^2=(200)^2+(160)^2-2(200)(160)\cos (100) \\ AC^2=76713.48 \\ AC=\sqrt[]{76713.48} \\ AC=276.97\text{ ft} \end{gathered}[/tex]Therefore, the answer is 276.97 ft
The relative frequency polygon shows pulse rates or women and men.
Given:
A relative frequency polygon that shows pulse rates of women and men.
To determine the midpoints of the class with the most pulse rates for women and men, we first note that based on the given graph, the most pulse rates for women is 74.5 with more than 30% of relative frequency. While the most pulse rates for men is 64.5 with more than 40% of relative frequency.
Therefore, the midpoints of the class with the most pulse rates for women and men respectively is:
74.5, 64.5
Please help meif you can't read it it says -138=-6(6b-7)
We have the following:
[tex]\begin{gathered} -138\ge-6\cdot(6b-7) \\ \end{gathered}[/tex]solving for b:
[tex]\begin{gathered} -6\cdot(6b-7)\le-138 \\ -\frac{6}{6}\cdot(6b-7)\le-\frac{138}{6} \\ -(6b-7)\le-23 \\ (6b-7)\ge23 \\ 6b-7+7\ge23+7 \\ 6b\ge30 \\ b\ge\frac{30}{6} \\ b\ge5 \end{gathered}[/tex]Therefore, the interval is:
[tex]\lbrack5,\infty)[/tex]Fill in the blank __(x+6)+8(x+6) =4x+24
We want to fill in the blank in the equation;
let's represent the blank with a.
[tex]_{}a(x+6)+8(x+6)=4x+24[/tex]Let's factorize the right side of the equation.
[tex]\begin{gathered} a(x+6)+8(x+6)=4x+24 \\ a(x+6)+8(x+6)=4(x+6) \end{gathered}[/tex]and also the left side;
[tex]\begin{gathered} a(x+6)+8(x+6)=4(x+6) \\ (a+8)(x+6)=4(x+6) \end{gathered}[/tex]Then let's divide both sides by (x+6);
[tex]\begin{gathered} (a+8)(x+6)=4(x+6) \\ \frac{(a+8)(x+6)}{(x+6)}=\frac{4(x+6)}{(x+6)} \\ (a+8)=4 \end{gathered}[/tex]Then we can solve for a in the resulting equation.
subtract 8 from both sides.
[tex]\begin{gathered} a+8=4 \\ a+8-8=4-8 \\ a=-4 \end{gathered}[/tex]Therefore, the value to fill into the blank is -4
[tex]-4(x+6)+8(x+6)=4x+24[/tex]A recipe calls for 4 parts chocolate chips for every 6 parts of nuts. calling makes the recipe for a party and uses 18 tablespoon of nuts how many tablespoons of chocolate chips did he use
To answer this question, we need to have into account the ratio between the chocolate and the nuts. This ratio is:
[tex]\frac{4}{6}=\frac{2}{3}[/tex]That is, simplifying the ratio, we have 2 parts of chocolate and 3 parts of nuts. This ratio must be the same for the tablespoon. Then, if we have 18 tablespoons of nuts, then we can pose the situation as follows:
[tex]\frac{4}{6}=\frac{2}{3}=\frac{x}{18}\Rightarrow x=\frac{2\cdot18}{3}\Rightarrow x=\frac{36}{3}\Rightarrow x=12[/tex]Therefore, we need 12 tablespoons of chocolate to follow the recipe for the party.
x = 23 is a solution for the equation x/2 = 10 true or false
False
Here, we want to check if x = 23 is a solution for the equation
Firstly, we need to understand that what we have is a linear equation, an equation in which the highest power of the variable is 1. For this type of equations, what we expect is a single solution.
Thus;
[tex]\begin{gathered} \frac{x}{2}\text{ = 10} \\ \\ x\text{ = 2 }\times\text{ 10} \\ \\ x\text{ = 20} \end{gathered}[/tex]Since x = 20 is the only possible solution, then x = 23 as a solution must be incorrect and false
4Which expression is equivalent to 3*25?--45+34 1 1+3x+4³03x-y-4 5+3+3X-3x - 1 - 1 - 3 x -1 13X-43 152+3544YyIN1|2
Explanation
If we look at the given options we can see that
[tex]\begin{gathered} \frac{1}{3}x-\frac{1}{4}y-\frac{4}{5}+\frac{1}{3}x-\frac{1}{4}y \\ rearrange\text{ terms} \\ =\frac{1}{3}x+\frac{1}{3}x-\frac{1}{4}y-\frac{1}{4}y-\frac{4}{5} \\ =\frac{x+x}{3}-\frac{y+y}{4}-\frac{4}{5} \\ =\frac{2x}{3}-\frac{2y}{4}-\frac{4}{5} \\ =\frac{2x}{3}-\frac{1}{2}y-\frac{4}{5} \end{gathered}[/tex]Answer: Option 2
The quotient form states that:a) The polynomial P(x) is equal to the quotient Q(x) plus the reminder Rb) The reminder R will always be a non-zero numberc) The polynomial P(x) is equal to the divisor d(x) times the quotient Q(x) plus the reminder Rd) The divisor d(x) is always a factor of P(x)
If we have a polynomial P(x) and divide it by d(x) we get
[tex]P(x)=d(x)\cdot Q(x)+R(x)[/tex]It's similar to what happens in normal divisions.
Therefore, the correct statement is:
c) The polynomial P(x) is equal to the divisor d(x) times the quotient Q(x) plus the reminder R
Rewrite the rational expression as an equivalent rational expression with the given denominator
Let z be the required expression.
The given is
[tex]\frac{3}{14x+98}=\frac{z}{14y(x+7)}[/tex]Using the cross product method, we get
[tex]3\times14y(x+7)=z(14x+98)[/tex][tex]42y(x+7)=z(14x+98)[/tex]Dividing both sides by 14x+98, we get
[tex]\frac{42y\mleft(x+7\mright)}{14x+98}=\frac{z\mleft(14x+98\mright)}{14x+98}[/tex][tex]\frac{42y\mleft(x+7\mright)}{14(x+7)}=z[/tex][tex]\frac{42y}{14}=z[/tex][tex]3y=z[/tex]Hence the answer is
[tex]\frac{3}{14x+98}=\frac{3y}{14y(x+7)}[/tex]what is the distance from the origin to point P graphed on the complex plane below?√7√29729
SOLUTION:
Step 1:
In this question, we are given the following:
Step 2:
From the graph, we can see that:
Using the Pythagoras' theorem,
[tex]\begin{gathered} p^2=2^{2\text{ }}+5^2 \\ p^2=\text{ 4+ 25} \\ p^2=\text{ 29} \\ \text{square}-\text{root both sides, we have that:} \\ p\text{ =}\sqrt[]{29} \end{gathered}[/tex]CONCLUSION:
The final answer is:
[tex]\sqrt[]{29}[/tex]If AB=18cm,AC=14cm,BC=8cm and XY=4cm how do i find AX and AY?
To answer this question, we can see that:
Then, we have that both triangles are similar triangles since the side CB and XY are parallel and then the interior angles are congruent. Then, we can write the next proportions:
[tex]\frac{CB}{XY}=\frac{AC}{AX}=\frac{AB}{AY}[/tex]We have that the ratio between CB and XY is:
[tex]\frac{CB}{XY}=\frac{8}{4}\Rightarrow\frac{CB}{XY}=2[/tex]In other words, we can say that the side CB is twice the measure of side XY or:
[tex]\frac{CB}{XY}=\frac{AC}{AX}=\frac{AB}{AY}=2[/tex]And we can also say that:
[tex]\frac{XY}{CB}=\frac{AX}{AC}=\frac{AY}{AB}=\frac{1}{2}[/tex]Using these proportions is easy to find the values of AX and AY as follows:
Finding the value of AXWe have that AC = 14cm, then:
[tex]\frac{AX}{AC}=\frac{1}{2}\Rightarrow AX=\frac{1}{2}\cdot AC\Rightarrow AX=\frac{1}{2}\cdot14\operatorname{cm}\Rightarrow AX=7\operatorname{cm}[/tex]Finding the value of AYWe have that AB = 18cm, then we have:
[tex]\frac{AY}{AB}=\frac{1}{2}\Rightarrow AY=\frac{1}{2}\cdot AB\Rightarrow AY=\frac{1}{2}\cdot18\operatorname{cm}\Rightarrow AY=9\operatorname{cm}[/tex]In summary, therefore, the value for AX = 7cm, and the value for AY = 9cm.
what is 36 muiltiplyed by 100 equal
A group of 7 students was asked, "How many hours did you watch television last week?" Here are their responses.17, 20, 19, 17, 18, 4, 6Find the mean number of hours for these students.If necessary, round your answer to the nearest tenth.hours
SOLUTION
The mean of a set of data is the sum of the datas divided by the number of data given.
Hence
[tex]\begin{gathered} \text{Mean}=\frac{sum\text{ of data}}{n} \\ \text{Where n=number of data given } \end{gathered}[/tex]Given the set of data below:
[tex]\begin{gathered} 17,\: 20,\: 19,\: 17,\: 18,\: 4,\: 6 \\ \text{Hence} \\ n=7 \end{gathered}[/tex]The sum is
[tex]17+20+19+17+18+4+6=101[/tex]Hence
The mean number of hour is
[tex]\text{Mean}=\frac{101}{7}=14.4286[/tex]Therefore
The mean number of hours for these students is 14. 4
1.1. Which statement explains why the two systems of equations below have thesame solution?A6x + 8y = -102x - 5y = 12B8x + 3y = 212x + 16y = -20
Let:
[tex]\begin{gathered} 6x+8y=-10_{\text{ }}(1) \\ 2x-5y=12_{\text{ }}(2) \\ 8x+3y=2_{\text{ }}(3) \\ 12x+16y=-20_{\text{ }}(4) \end{gathered}[/tex][tex]\begin{gathered} (4)=2(1) \\ so\colon \\ 2(6x+8y)=2(-10)\equiv12x+16y=-20 \\ 12x+16y=-20\equiv12x+16y=-20 \end{gathered}[/tex]Therefore, (4) is a Scalar Multiple of (1).
[tex]\begin{gathered} (1)+(2) \\ 6x+2x+8y-5y=-10+12 \\ 8x+3y=2\equiv(3) \\ so\colon_{} \\ (1)+(2)\equiv(3) \end{gathered}[/tex]Therefore, (3) is a linear combination of (1) and (2)
how would you find the inverse of [tex]g(x) = \sqrt[3]{x + 1} [/tex]
We have the function
[tex]g(x)=\sqrt[3]{x+1}[/tex]and we would like to find the inverse of it.
To do this, first we change the name of the function to y (this way is more easy to manipulate the expression), then
[tex]y=\sqrt[3]{x+1}[/tex]Now we need to solve this equation for x, let's do this.
[tex]\begin{gathered} y=\sqrt[3]{x+1} \\ y^3=(\sqrt[3]{x+1})^3 \\ y^3=x+1 \\ x=y^3-1 \end{gathered}[/tex]Once we do this the x is the inverse function. Now we relabeled the variables, x would be g^(-1) and y will be x.
Therefore
[tex]g^{-1}(x)=x^3-1[/tex]To make sure we are right let's do the composition between the original function and its inverse (we know that the result should be the identity function).
[tex]\begin{gathered} (g\circ g^{-1})(x)=g(g^{-1}(x)) \\ =g(x^3-1) \\ =\sqrt[3]{x^3-1+1} \\ =\sqrt[3]{x^3} \\ =x \end{gathered}[/tex]since this holds, our solution is correct.
Which graph shows a polynomial function that does not have a negative interval?
In this case, we'll have to carry out several steps to find the solution.
Step 01:
Data:
Graph
Step 02:
We must analyze the graphs to find the solution.
Positive Interval (above x-axis)
That is the full solution.
WILL GIVE BRAINLYDirections: Click on a box to choose each answer you want to select. YoIdentify each classification of the realnumber system that can describe the setof numbers shown.{-V36, 35%, 2} Whole irrational natural integer real or rational
You have the following set of numbers:
{√(36), 35%, -2}
- The first number is √(36) = 6, that is, it is a natural number
- The second number is a while number
- The third number is an integer (because is a whole number wit a minus sign)
Hence, the correct option is the fourth one
- Natural numbers, whole numbers, integers
how to solve? |b+1|=5
Answer:
b = 4 and -6
Explanation:
Given the equation:
|b + 1| = 5
|b + 1| represents the absolute value of (b + 1)
This equation gives rise to two equations
b + 1 = 5 ......................................(1)
b + 1 = -5 .......................................(2)
Solve equation (1)
b + 1 = 5
Subtract 1 from both sides of the equation
b + 1 - 1 = 5 - 1
b = 4
Solve equation (2)
b + 1 = -5
Subtract 1 from both sides of the equation
b + 1 - 1 = -5 -1
b = -6
Therefore, the values for b are 4 and -6
Antoine is picking out some movies to rent, and he has now the down his selection to 4 Mysteries, 3comedies, 3 horror films, and 6documentaries. How many different combinations of 9movies can he rent if he wants all 4 Mysteries?
Given:
Movies to rent:
4 = mysteries
3 = comedies
3 = horror
6 = documentaries
Asked: How many different combinations of 9 movies can he rent if he wants all 4 Mysteries?
Solution:
The total choices Antoine has is 4+3+3+6 = 16 movies to rent.
Out of the 16 movies, he wants all 4 mystery movies so, we will subtract 4 to the total number of movies to rent.
We will have a new total number of movies to rent of 16 - 4 = 12
Antoine will rent 9 movies in total.
Out of 9 movies, 4 movies are mystery movies. 9 - 4 = 5
The combination of the remaining 5 slots out of 9 with the 12 other categories of movies would be:
[tex]\begin{gathered} nCr\text{ = }\frac{n!}{r!(n-r)!} \\ 12C5\text{ = }\frac{12!}{5!(5-1)!} \\ 12C5\text{ = 792} \end{gathered}[/tex]ANSWER: There are 792 different combinations Antoine can get.
Sara runs along a circular track. The diameter of the track is 75 yards. She will jog 2 laps around the track. Which is the closest to the amount of yards Sara will run?
Sara will run 8836 square yards
Explanation:Given that the diameter of the track Sara runs is 75 yards.
The radius is half the diameter, and so, 37.5
Sara will jog 2 laps around the track, therefore, the amount of yards Sara will run is obtained by finding twice the area of the circle she is going to cover.
Area of a circle is:
[tex]A=\pi r^2[/tex]Where r is the radius
Using r = 37.5
[tex]\begin{gathered} A=\pi(37.5)^2 \\ =1406.25\pi \\ =4417.9 \end{gathered}[/tex]Since she is jogging 2 laps, we have the yards she will cover to be:
2(4417.9)
= 8835.8
Approximately 8836 square yards
In the United States, 1000 residents aged 15 or older were surveyed and 870 replied that they were satisfied with the water quality. The 90% confidence interval estimate of all U.S residents satisfied with their water quality is _________.
Given:
Here In the United States, 1000 residents aged 15 or older were surveyed and 870 replied that they were satisfied with the water quality is given.
Required:
Interval of 90% confidence level.
Explanation:
The formula to find the interval of confidence level is as below
[tex]=(p^{\prime}-z*\sqrt[]{\frac{p^{\prime}(1-p^{\prime})}{n}},\text{p' }+z*\sqrt[]{\frac{p^{\prime}(1-p^{\prime})}{n}})[/tex]Now we have to find the value of all
z=1.64485
p'=870/1000=0.87
n=1000
Now put the all values
[tex](0.87-1.64485*\sqrt[]{\frac{0.87(1-0.87)}{1000}},0.87+1.64485*\sqrt[]{\frac{0.87(1-0.87)}{1000}})[/tex][tex](0.87-0.0175,0.87+0.0175)[/tex][tex](0.8525,0.8875)[/tex]Final answer:
Confidence interval is (0.8525,0.8875)
If ai6 and an=3an-1 + n then find the value of az.Answer:
Answer
a₃ = 63
Explanation
We are told to find the 3rd term of this sequence.
Then, we are given that
first term = a₁ = 6
nth term = aₙ = 3aₙ₋₁ + n
To find the third term, we need to find the second term first
second term = a₂ = 3a₁ + 2 = 3 (6) + 2 = 18 + 2 = 20
Third term = a₃ = 3a₂ + 3 = 3 (20) + 3 = 60 + 3 = 63
Hope this Helps!!!
1. Lines m and n are perpendicular. If the slope of line m is zero, then what is the slope of line n? If you were to sketchlines m and n, what type of lines would you sketch?2. On a graph, create your own set of perpendicular lines similar to lines m and n. Choose two points on each line andprove through the slope formula, that the lines are perpendicular to each other.
when lines are perpendicular to each other, they make 90 degrees to each other
whenever you have a slope of 0, it means the y-axis remains constant while the x-axis increases
since m has a slope of zero, that's why it's in that position while n increases
slope = y2 - y1 / x2 - x1
A machine can produce either nuts or bolts but not both at the same time. the machine can be used at most eight hours per day. furthermore at most six hours a day can be used for making nuts and at most five hours a day can be used for making bolts. there is a two dollar profit for each hour the machine makes nuts and a three dollar profit for every hour the machine makes bolts. how many hours a day should be spent on making each item in order to maximize profit? What is the maximum profit?
The number of hours spend in a day by machine on preparing nuts and bolts are - 3 hours for nuts and 5 hours for bolts to get the maximum profit of 21.
What is meant by the term maximize?When we talk about maximizing or minimizing a function, we are referring to what the maximum and minimum possible values of that function are.This can be expressed in the form of global or local range.When we say global range, we mean the maximum value of a function over the entire range of input that the function can be defined over (the domain of a function).For the given question
Let 'x' be the number of hours spend by machine for making nuts.
Let 'y be the number of hours spend on bolts.
Thus,
x + y ≤ 8
x ≤ 6 and x ≥0
y ≤ 5 and y ≥ 0.
The maximum profit is given by equation;
f(x,y) = 2x + 3y
Plot the graph for the given conditions using lines.
Consider the points from the graph and check for maximum profit.
f(0,0) = 2(0) + 3(0) = 0
f(0,5) = 2(0) + 3(5) = 15
f(6,0) = 2(6) + 3(0) = 12
f(3,5) = 2(3) + 3(5) = 21 (maximum value)
f(6,2) = 2(6) + 3(2) = 18
Thus, the number of hours spend in a day on preparing nuts and bolts are - 3 hours for nuts and 5 hours for bolts to get the maximum profit of 21.
To know more about the maximize, here
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