Answer:
Step-by-step explanation:
To recover your investment in 5 years or less, you would need to earn enough to cover the cost of going to school ($30,000) as well as make up for the lost salary over the 2 years of schooling ($45,000/year * 2 years = $90,000).
Therefore, the minimum salary you would need to earn upon earning your degree is the sum of the cost of going to school and the lost salary:
Minimum salary = $30,000 + $90,000 = $120,000.
In order to recover your investment in 5 years or less, you would need to earn a minimum salary of $120,000 per year.
To recuperate the total cost of $120,000 ($30,000 tuition + $90,000 forgone salary) over 5 years, you would need to earn $24,000 more per year on top of your original $45,000 salary. This implies that the minimum salary you need to earn after graduating is $69,000 per annum.
Explanation:To determine the minimum salary, we first need to calculate your total forfeiture over the 2 years of school, which equates to real costs and opportunity costs. Firstly, the real cost is the tuition of $30,000. Secondly, the opportunity costs are the 2 years of salary you're forgoing, best understood as the wages you would've made if you hadn't gone to school. Assuming the salary of $45,000 per year, the total opportunity cost for the 2 years would be $90,000.
Therefore, the total investment is calculated as the sum costs of tuition and forgone salary i.e. $30,000 (tuition) + $90,000 (forgone salary) = $120,000. So to recover this investment in 5 years, you would need to earn an addition of $120,000 above your original salary. Meaning, you will have to recover $120,000 / 5 years = $24,000 per year on top of your initial salary to recover your total costs in the stated timeframe.
Therefore, the minimum salary you need to earn after earning your degree is equal to your original salary plus recovered investment per year: $45,000 (original salary) + $24,000 (increase) = $69,000. Hence, upon completion of your degree, you will have to earn at least $69,000 per year to recover your total investment within 5 years.
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A small grocery store selling jugs of milk, loafs of bread and pieces of cheese sold a total of 224 of those three items today. It sold
three times as many loafs of bread as jugs of milk, and it sold 4 times as many pieces of cheese as loafs of bread.
a) Write an equation that represents the information.
b) Solve the equation in (a) to find the number of jugs of milk, loafs of bread, and pieces of cheese the store sold today
Answer: the store sold 14 jugs of milk, 42 loaves of bread, and 168 pieces of cheese today.
Step-by-step explanation:
a) Let's represent the number of jugs of milk as 'm', the number of loaves of bread as 'b', and the number of pieces of cheese as 'c'. According to the given information:
The total number of items sold today is 224. Therefore, we can write the equation:
m + b + c = 224
The store sold three times as many loaves of bread as jugs of milk. We can write this as:
b = 3m
The store sold four times as many pieces of cheese as loaves of bread. We can write this as:
c = 4b
b) Now, let's solve the equation:
Substituting the value of 'b' from equation (2) into equation (3):
c = 4(3m)
c = 12m
Substituting the values of 'b' and 'c' from equations (2) and (3) into equation (1):
m + 3m + 12m = 224
16m = 224
m = 224 / 16
m = 14
Using the value of 'm' in equation (2):
b = 3m
b = 3(14)
b = 42
Using the value of 'b' in equation (3):
c = 4b
c = 4(42)
c = 168
Answer:
Step-by-step explanation:
a) Let's denote the number of jugs of milk as 'x', the number of loaves of bread as 'y', and the number of pieces of cheese as 'z'.
Based on the given information, we can form two equations:
1) The total number of items sold today is 224:
x + y + z = 224
2) The store sold three times as many loaves of bread as jugs of milk:
y = 3x
3) The store sold four times as many pieces of cheese as loaves of bread:
z = 4y
b) To solve the equations and find the number of jugs of milk, loaves of bread, and pieces of cheese the store sold today, we will substitute equation (2) and equation (3) into equation (1):
x + y + z = 224
x + 3x + 4y = 224
4x + 4y = 224
Now, substitute y = 3x into the equation:
4x + 4(3x) = 224
4x + 12x = 224
16x = 224
Divide both sides by 16:
x = 224 / 16
x = 14
Now substitute the value of x back into equation (2) to find y:
y = 3x
y = 3(14)
y = 42
Finally, substitute the value of y into equation (3) to find z:
z = 4y
z = 4(42)
z = 168
Therefore, the store sold 14 jugs of milk, 42 loaves of bread, and 168 pieces of cheese today.
Brianna wants to purchase a vehicle. She has $15,670 saved for a down payment. Chevrolet has a 33% off sale on their Silverado. The Silverado Brianna wants costs $57,999. If she takes advantage of the sale and applies her down payment, how much will Brianna owe on her new vehicle?
Brianna will owe $23,249.33 on her new vehicle after applying the down payment and taking advantage of the 33% off sale.
To solve this problemWe need to follow these steps:
Calculate the discount on the Silverado:
Discount = 33% of $57,999
Discount = 0.33 * $57,999
Discount = $19,079.67
Subtract the discount from the original price of the Silverado:
Price after discount = $57,999 - $19,079.67
Price after discount = $38,919.33
Subtract Brianna's down payment from the price after discount:
Amount owed = Price after discount - Down payment
Amount owed = $38,919.33 - $15,670
Amount owed = $23,249.33
So, Brianna will owe $23,249.33 on her new vehicle after applying the down payment and taking advantage of the 33% off sale.
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Please help I am so lost thank you so much
The speed of the plane is equal to 120 mph.
What is speed?In Mathematics and Geometry, speed is the distance covered by a physical object per unit of time. This ultimately implies that, speed can be measured by using miles per hour (mph).
Mathematically, the speed of any a physical object can be calculated by using this formula;
Speed = distance/time
Time = distance/speed
Let the variable s represent the speed of the plane in miles per hour. Therefore, an equation that models the situation can be written as follows;
240/s = 80/s - 80
80s = 240s - 19200
19200 = 160s
s = 19200/160
s = 120 mph.
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What is the major difference between Grades 4 and 5 in terms of the teaching of probability?
Answer:
In Grade 4, students are introduced to the concept of chance and the idea that different situations have different probabilities of occurring. They learn that for many situations, there are a finite number of different possible outcomes. However, at this stage, students are not expected to calculate the probability of events occurring. In Grade 5, students continue to build on their understanding of probability and may begin to learn more advanced concepts and techniques for calculating probabilities.
Step-by-step explanation:
Find the area of the triangle below be sure to include the correct unit in your answer.
Answer:
Step-by-step explanation:
What will be the result of substituting 2 for x in both expressions below?
+4
x+6-x-2
O Both expressions equal 5 when substituting 2 for x because the expressions are equivalent.
O Both expressions equal 6 when substituting 2 for x because the expressions are equivalent.
O One expression equals 5 when substituting 2 for x, and the other equals 2 because the expressions are not
equivalent.
One expression equals 6 when substituting 2 for x, and the other equals 2 because the expressions are not
equivalent.
Both expressions equal 5 when substituting 2 for x because the expressions are equivalent.
Equivalent Algebraic expressions:Algebra is the branch of mathematics that deals with numbers and values which are represented with letters and symbols.
Sometimes, we do not want to mention a particular number, we can represent the number by a letter or a suitable symbol. This approach is algebraic.
For example, d + d = 2d
This is an example of an algebraic expressionns.
Given the algebraic expressions,
[tex]\frac{1}{2}x + 4 \\ x + 6 - \frac{1}{2}x - 2[/tex]
Substituting 2 for x in the first expression gives:
(1/2 × 2) + 4
1 + 4
5
Substituting 2 for x in the second expression gives:
2 + 6 - (1/2 ×2) - 2
8 - 1 - 2
8 - 3
5
Both expressions equal 5 when substituting 2 for x because the expressions are equivalent.
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what percent of 41.12 is 10.28 ?
What are the dimensions of the rectangle shown on the coordinate plane?
The base is 5 units and the height is 3 units.
The base is 4 units and the height is 7 units.
The base is 7 units and the height is 5 units.
The base is 7 units and the height is 3 units.
Answer:
D The base is 7 units and the height is 3 units.
Step-by-step explanation:
The answer is d I counted the width/base then the height/length and found answer.
NEED HELP ASAP
Y
B
^^
CX
A
Previous Activity
N
Which would prove that AABC~ AXYZ? Select two
options.
OBA-BC-A
=
YX
YZ XZ
OBA = BC₁
YX
YZ
O
AC
XZ
=
=
BA
XX.
YX
AC
BC
BA = AE = 8C
YX
YZ
XZ
OBC=BA ₁ <=
XY
ZX
Next Activity
The two options that would prove that ΔABC ~ ΔXYZ include the following:
A. BA/YX = BC/YZ = AC/XZ
C. AC/XZ = BA/YX, ∠A≅∠X
What are the properties of similar triangles?In Mathematics and Geometry, two triangles are said to be similar when the ratio of their corresponding side lengths are equal and their corresponding angles are congruent.
Based on the side, side, side (SSS) similarity theorem, we can logically deduce the following congruent angles and similar triangles:
BA/YX = BC/YZ = AC/XZ (ΔABC ≅ ΔXYZ)
Based on the side, angle, side (SAS) similarity theorem, we can logically deduce the following congruent angles and similar sides:
AC/XZ = BA/YX, ∠A≅∠X
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Cecilia se compra un libro con la cuarta parte de su dinero y un cómic con la mitad del dinero que le quedaba. Al salir de la librería se da cuenta que le sobraron $8. ¿Cuál es la ecuación para conocer la cantidad de dinero que tenía en un inicio?
The equation to determine the initial amount of money Cecilia had is x = 0.
Let's denote the initial amount of money Cecilia had as "x" dollars.
According to the given information, Cecilia spent one-fourth (1/4) of her money on a book, which is (1/4)x dollars. After buying the book, she had (x - (1/4)x) dollars left.
Next, Cecilia spent half (1/2) of the remaining money on a comic, which is ((1/2)x - 8) dollars. After buying the comic, she had ((x - (1/4)x) - ((1/2)x - 8)) dollars remaining.
Since she had $8 left, we can set up the equation:
((x - (1/4)x) - ((1/2)x - 8)) = 8
To simplify the equation, we can first combine like terms:
(x - (1/4)x - (1/2)x + 8) = 8
Now, let's solve the equation step by step:
(x - (1/4)x - (1/2)x + 8) = 8
Multiplying the fractions by their common denominator, which is 4, we get:
(4x - x - 2x + 32) = 32
Simplifying further:
(x + 32) = 32
Subtracting 32 from both sides:
x = 0
Therefore, the equation to determine the initial amount of money Cecilia had is x = 0.
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I need help please!!
Answer:
25,
-7
Step-by-step explanation:
r ° q= r(q(x))= (-x+2) ^2
So, using perfect square trinomial,
[tex](-x+2)^{2} =x^{2} -4x+4[/tex]
Thus q(-3)=25
q(r(x))= (-x^2)+2=-7
An educator wants to see how the number of absences for a student in her class affects the student’s final grade. The data obtained from a sample are shown.
Test at :a 05.XCV=0.811 No. of absences -\ 10, final grade 70
By conducting a statistical test with the provided data and using a significance level of 0.05 and a critical value of 0.811, the educator can determine whether the number of absences significantly affects the final grade of the students.
To analyze the relationship between the number of absences and the final grade of students, an educator collected data from a sample. The obtained data reveals that for 10 absences, the corresponding final grade was 70. In order to determine whether the number of absences significantly impacts the final grade, a statistical test is needed.
The information provided, "Test at α = 0.05, XCV = 0.811," suggests that the educator is referring to a significance level (α) of 0.05 and the critical value (XCV) associated with this level is 0.811.
To conduct a hypothesis test, the educator needs to set up the null (H0) and alternative (Ha) hypotheses. In this case, the null hypothesis would assume that the number of absences does not have a significant effect on the final grade. The alternative hypothesis would assume that the number of absences does have a significant effect on the final grade.
Using the obtained data, the educator would perform the statistical test, such as a correlation analysis or a regression analysis, to determine the strength and significance of the relationship between absences and final grade.
If the test statistic (such as the correlation coefficient or regression coefficient) exceeds the critical value, the null hypothesis would be rejected, suggesting that the number of absences does have a significant impact on the final grade.
The educator can then interpret the results, considering factors such as the direction and strength of the relationship. Additionally, they may explore other variables or factors that could be influencing the final grade and further refine their analysis.
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An airline pricing analyst has been asked to review an airline’s flights. She has determined that 80% of all flights reach the final destination on time. Also, 30% of all flights have 100 or more passengers. When a flight has 100 or more passengers, the flight is late 60% of the time.
What is the probability the flight has 100 or more passengers and does arrive on time?
The probability that a flight has 100 or more passengers and arrives on time is approximately 0.12 or 12%.
Let A be the event that a flight has 100 or more passengers, and B be the event that the flight arrives on time. We want to find the probability of the intersection of these two events, P(A ∩ B).
We are given:
- P(B) = 0.80, the probability that any given flight arrives on time
- P(A) = 0.30, the probability that any given flight has 100 or more passengers
- P(B | A) = 0.40, the conditional probability that a flight arrives on time given that it has 100 or more passengers (since 60% of such flights are late)
We can use the formula for conditional probability to find P(A ∩ B):
P(A ∩ B) = P(B | A) * P(A)
= 0.40 * 0.30
= 0.12
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can someone please help me, I don't know how to do this
Answer:
x = 82
Step-by-step explanation:
x and 98 are same- side exterior angles. They are on the same side of the transversal and are outside the parallel lines.
same- side exterior angles sum to 180° , so
x + 98 = 180 ( subtract 98 from both sides )
x = 82
[tex]x[/tex] and [tex]98^{\circ}[/tex] are same side exterior angles which add up to [tex]180^{\circ}[/tex].
Therefore
[tex]x+98^{\circ}=180^{\circ}\\x=82^{\circ}[/tex]
it is my first time taking my baby to the cinemas in Junes 2023, and the cinemas have sales because there are tons of kids' movies to be seen. For adults the ticket costs 70$ and for children it costs 30$, which tickets sell like 1000$ a day leading to 31000 a month. Calculate the number of tickets that were sold for adults and children in a day. A+C=1000 70+30=31000.
A+C=1000
70+30=31000
if we wanted to extend this discussion beypnd what has been shared so far, what additional question could we ask?
Step-by-step explanation:
If we wanted to extend the discussion beyond what has been shared so far, an additional question we could ask is:
"What is the ratio of adult tickets to children's tickets sold in a day?"
This question would provide insight into the distribution of ticket sales between adults and children and help us understand the demand for different movie genres or screenings among the audience.
Find the distance between the points A and B given below.
(That is, find the length of the segment connecting A and B.)
Round your answer to the nearest hundredth.
1 unit
A
B
Answer:
I wish you good luck in finding your answer
help me on this question ive been stuck on this
Which scatter diagram fits the given paired data
Systolic Blood Pressure
Diastolic Blood Pressure
119 125
76
84
131 130 123 118 142
91
84
89
76
91
138 125 131 128 140 123
93 78 88 75 94 76
the correct scatter diagram that fits the given paired data is option A, as shown in the attached image.
The scatter plot that fits the given paired data is as follows:
Firstly, the data needs to be paired correctly: Systolic Blood Pressure | Diastolic Blood Pressure119 125131 130123 118118 142138 125131 128140 123Then the data pairs are plotted on a graph to create a scatter plot. It is a graph that shows the relationship between two sets of data. In this case, the Systolic blood pressure is on the horizontal axis (X-axis) and the Diastolic blood pressure is on the vertical axis (Y-axis).
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Verify that the given point is on the curve. Find the lines that are a. tangent and b. normal to the curve at the given point.
The point (2, 7π / 2) does not belong to the implicit curve 8 · x · y + π · sin y = 55π and tangent and normal lines cannot be determined.
Does a point given belong to an implicit curve?
In this question we find the definition of an implicit curve, in which we must determine if point (2, 7π / 2) belongs to the curve. First, we check that point:
8 · x · y + π · sin y = 55π
8 · 2 · (7π / 2) + π · sin (7π / 2) = 55π
56π + 0.191π = 55π
56.191π = 55π
56.191 = 55 (CRASH!)
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Ki Tae uses 54 meters of fencing to make a 6-sided outdoor dog pen. Two of the sides of the dog pen are each 15 meters long. The remaining 4 sides each have the same length.
Ki Tae used 54 meters of fencing to construct a 6-sided outdoor dog pen. Two of the sides are each 15 meters long, while the remaining four sides are each 6 meters long.
Let's solve the problem step by step. We know that Ki Tae used a total of 54 meters of fencing to construct a 6-sided outdoor dog pen. Two of the sides have a length of 15 meters each.
Let's denote the length of the remaining four sides as "x."
Since the dog pen has six sides, we can set up an equation based on the total length of the fencing:
15 + 15 + x + x + x + x = 54
Simplifying the equation, we have:
30 + 4x = 54
Subtracting 30 from both sides, we get:
4x = 24
Dividing both sides by 4, we find:
x = 6
Therefore, each of the remaining four sides has a length of 6 meters.
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Which expression is always equivalent to sin x when 0° < x < 90°?
(1) cos (90°- x)
(3) cos (2x)
(2) cos (45° - x)
(4) cos x
The expression that is always equivalent to sin x when 0° < x < 90° is (1) cos (90° - x). Option 1
To understand why, let's analyze the trigonometric functions involved. In a right triangle, the sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. Since we are considering angles between 0° and 90°, we can guarantee that the side opposite the angle will always be the shortest side of the triangle, and the hypotenuse will be the longest side.
Now let's examine the expression cos (90° - x). The cosine of an angle is defined as the ratio of the length of the side adjacent to the angle to the length of the hypotenuse. In a right triangle, when we subtract an angle x from 90°, we are left with the complementary angle to x. This means that the remaining angle in the triangle is 90° - x.
Since the side adjacent to the angle 90° - x is the same as the side opposite the angle x, and the hypotenuse is the same, the ratio of the adjacent side to the hypotenuse remains the same. Therefore, cos (90° - x) is equivalent to sin x for angles between 0° and 90°.
On the other hand, options (2) cos (45° - x) and (3) cos (2x) do not always yield the same value as sin x for all angles between 0° and 90°. The expression cos x (option 4) is equivalent to sin (90° - x), not sin x.
Option 1 is correct.
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A piece of wood is in the shape of a rectangular prism with a length of 10 inches, a width of 4 inches, and a height of 5 inches. You cut the wood in half to form two pieces of wood, each with a length of 5 inches. What is the percent increase in the total surface area? Round your answer to the nearest hundredth, if necessary. %
Answer: 18.18%
Step-by-step explanation:
First, let's calculate the surface area of the original piece of wood. The surface area (SA) of a rectangular prism is given by the formula:
[tex]$$SA = 2lw + 2lh + 2wh$$[/tex]
where [tex]\(l\)[/tex] is the length, [tex]\(l\)[/tex] is the width, and [tex]\(h\)[/tex] is the height. For the original piece of wood, [tex]\(l = 10\) inches[/tex], [tex]\(w = 4\) inches[/tex], and [tex]\(h = 5\) inches[/tex].
After the piece of wood is cut in half, the length becomes 5 inches, but the width and height remain the same. So, for each of the two new pieces of wood, [tex]\(l = 5\) inches[/tex], [tex]\(w = 4\) inches[/tex], and [tex]\(h = 5\) inches[/tex]. The total surface area of the two new pieces of wood is twice the surface area of one of the new pieces.
The percent increase in the total surface area is given by the formula:
[tex]$$\text{Percent Increase} = \frac{\text{New Total SA} - \text{Original SA}}{\text{Original SA}} \times 100\%$$[/tex]
Let's calculate these values.
The percent increase in the total surface area when the piece of wood is cut in half is approximately 18.18%.
Tenía unas matas en el vivero.
Sembré 23 el lunes, 28 el
martes, 29 el miércoles. Si el
jueves tenía 90, ¿con cuántas
matas empecé?
If 23 plants were planted on Monday, 28 plants on Tuesday and 29 plants on Wednesday, and 90 plants were planted on Thursday, we can determine that we started with 10 plants initially.
To determine how many plants to start with initially, we need to perform a series of calculations based on the information provided.
On Monday 23 plants were planted, on Tuesday 28 plants were planted and on Wednesday 29 plants were planted. If on Thursday there were a total of 90 plants, we can add all the plants planted until Thursday and then subtract them from the total to obtain the initial amount.
Adding the plants planted:
23 + 28 + 29 = 80
Then, we subtract this amount from Thursday's total:
90 - 80 = 10
Therefore, we started with 10 plants initially.
In summary, if 23 plants were planted on Monday, 28 plants on Tuesday and 29 plants on Wednesday, and 90 plants were planted on Thursday, we can determine that we started with 10 plants initially. This is obtained by adding the plants planted until Thursday and then subtracting that amount from the total for Thursday.
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in this chart, × is the length of a persons forearm in centimeters and y is the persons height in centimeters. the question is if someones forearm (x) is 24.5 cm, how tall would they be? how do i find this? and how would i make a linear regression graph? thanks
The height of a person whose length of forearm is 24.5 cm is equal to 163.38 centimeters.
How to construct and plot the data in a scatter plot?In this exercise, we would plot the length of forearm on the x-axis of a scatter plot while height would be plotted on the y-axis of the scatter plot through the use of Microsoft Excel.
On the Microsoft Excel worksheet, you should right click on any data point on the scatter plot, select format trend line, and then tick the box to display a linear equation for the line of best fit on the scatter plot;
y = 3.01x + 89.63
Based on the equation of the line of best fit above, the height of a person whose length of forearm is 24.5 cm can be determined as follows;
y = 3.01x + 89.63
y = 3.01(24.5) + 89.63
y = 163.375 ≈ 163.38 centimeters.
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Solve the inequality.
x
_ -3 >2
4
The solution to the inequality[tex](x/4) - 3 > 2 is x > 20.[/tex]
To solve the inequality [tex](x/4) - 3 > 2,[/tex]we'll follow these steps:
Step 1: Eliminate the fraction by multiplying both sides of the inequality by the denominator, which is 4 in this case. This step allows us to get rid of the fraction and simplify the inequality.
[tex](x/4) - 3 > 2[/tex]
Multiply both sides by 4:
[tex]4 * [(x/4) - 3] > 4 * 2[/tex]
This simplifies to:
x - 12 > 8
Step 2: Isolate the variable on one side of the inequality by adding 12 to both sides:
x - 12 + 12 > 8 + 12
This simplifies to:
x > 20
So, the solution to the inequality is x > 20. This means that any value of x greater than 20 will satisfy the inequality.
To represent this solution graphically, we can plot the number line and shade the region to the right of 20, indicating that any value greater than 20 is a valid solution.
---------------------------------
-∞ 20 +∞
--------------------------
●=================
---------------------------------
In the number line above, the shaded region represents the solution x > 20. Any value to the right of 20, including 20 itself, will satisfy the original inequality.
In summary, the solution to the inequality [tex](x/4) - 3 > 2 is x > 20.[/tex]
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Because f(x) ___ its inverse is a function.
○ is one to one
X is a function
○ Passes the vertical line test
The inverse of the function f(x) = 2·x - 4, is the option;
g(x) = (1/2)·x + 2
The completed statement is; Because f(x) is one to one, its inverse is a function
What is the inverse of a function?The inverse of a function is one that takes the output of a specified function to produce the input of the function.
The inverse of the function f(x) = 2·x - 4, can be found by making x the subject of the function equation as follows;
f(x) = 2·x - 4
f(x) + 4 = 2·x
2·x = f(x) + 4
x = (f(x) + 4)/2 = f(x)/2 + 2
x = f(x)/2 + 2
Substituting f(x) = x and x = g(x) in the above equation, we get;
g(x) = x/2 + 2
The inverse of the function is therefore, g(x) = (1/2)·x + 2The function f(x) = 2·x - 4 is a one to one function, and the condition of a one to one function guarantees that the inverse of the function is also a function
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Answer:
A
Step-by-step explanation:
is one to one
21. Find the surface area of the figure below.
20 mm
17.mm
A. 969 mm²
B. 984 mm²
17 mm
C. 1,040 mm²
D. 1,105 mm²
Answer:
Step-by-step explanation:
Last year, Ali biked b miles. This year, he biked 358 miles. Using b, write an expression for the total number of miles he biked
Let f(x) = 4x² - 2x +11
The slope of the tangent line to the graph of f(x) at the point (3, 41)
Slope =
M=
B=
Answer:
f(x) = 4x² - 2x + 11
f'(x) = 8x - 2
m = f'(3) = 8(3) - 2 = 24 - 2 = 22
41 = 22(3) + b
41 = 66 + b
b = -25
y = 22x - 25
John Doe produces two kinds of men’s shirts: polo and t-shirts. Polo shirts require 2 hours in the pattern and cutting section and 1 hour in the sewing section. T-shirts require 1 hours in the pattern and cutting section and 2 hours in the sewing section. The pattern and cutting section has 84 hours available weekly. The sewing section has 106 hours available weekly. Past sales indicate that at most 36 polo shirts can be sold. The profit on each polo shirt is $30 and on each t-shirt is $22. How many of each kind should the company produce in order to maximize its profit?
a) Define your variables (2):
b) Constraints (5):
c) Objective function (1):
d) Graph (label the axes) and Work
The evaluation of the constraints with regards to the production of the polo and t-shirts, and to maximize the profit, using linear programming indicates that we get;
a. x = The number of polo shirts produced, y = The number of t-shirts produced
b. The inequalities representing the constraints are;
2·x + y ≤ 84
x + 2·y ≤ 106
x ≤ 36
x ≥ 0, y ≥ 0
c. P = 30·x + 22·y
d. Please find attached the graph of the feasible region
To maximize profit, the company should produce;
21 polo shirts and 43 t-shirts
What is linear programming?Linear programming is a method used for optimizing (maximizing or minimizing a value) operations with some specified constraints.
a. The details indicates that the question is related to linear programming. Let x represent the number of polo shirt produced, and let y represent the number of t-shirts produced.
x = The number of polo shirt produced
y = The number of t-shirts produced
b) The constraints are;
Pattern and cutting section; 2·x + y ≤ 84
Sewing section; x + 2·y ≤ 106
Sales constraints; x ≤ 36
The values of x and y are non negative, numbers, therefore;
x ≥ 0, y ≥ 0
c) The objective of the company is to maximize profit, P, therefore, the objective function is; P = 30·x + 22·y
d) The graph can be plotted from the constraint inequalities, by making y the subject in the inequalities that includes both x and y as follows;
2·x + y ≤ 84, therefore; y ≤ 84 - 2·x
x + 2·y ≤ 106, therefore; y ≤ 53 - x/2
x ≤ 36
Please find attached the graph of the inequalities, showing the feasible region which is the polygon with boundaries which are the lines representing the constraints.
The objective function evaluated at the vertices of the feasible region indicates that we get;
[tex]\begin{tabular}{ | l | l | c | }\cline{1-3}(x, y)& 30\cdot x + 22\cdot y & P(\$) \\ \cline{1-3}(0, 53 & 30\times 0 + 22\times 53 & 1166 \\\cline{1-3}(21, 42.5 & 30\times 21 + 22\times 42.5 & 1565 \\\cline{1-3}(32, 12) & 30\times 36 + 22\times 12 & 1344 \\\cline{1-3}(36, 0) & 30\times 36 + 22\times 0 & 1080 \\\cline{1-3}(0, 0) & 30\times 0 + 22\times 0& 0 \\\cline{1-3}\end{tabular}[/tex]
The feasible region and the objective function indicates that the values of x and y that maximizes the profit is; (x, y) = (21, 42.5)
Therefore, to maximize profit, the number of polo and t-shirts the company should produce are 21, and 42.5 ≈ 43 respectively.
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