Kara invests $3,200 into an account with a 3.1% interest rate that is compounded quarterly. How much money will be in this account after 8 years?
Answer:
We can use the formula for compound interest to solve this problem:
A = P(1 + r/n)^(nt)
where:
A = the amount of money in the account after t years
P = the principal amount (initial investment)
r = the annual interest rate (as a decimal)
n = the number of times the interest is compounded per year
t = the number of years
Plugging in the given values:
P = $3,200
r = 0.031 (3.1% as a decimal)
n = 4 (quarterly compounding)
t = 8
A = 3200(1 + 0.031/4)^(4*8)
A = $4,100.53
Therefore, after 8 years, there will be $4,100.53 in the account.
The rectangle can be made to have rotation symmetry of order 2 by colouring one of the squares blue. Put a cross in the middle of the square which would have to be made blue.
you thought
i was feeling you
mr Li wishes to give each of his five relatives in China 500 renminbi (RMB) Find the amount he would need in Singapore dollars, if the exchange rate is S$100 to RMB510.20
Exchange rate calculation.
To find out the amount in Singapore dollars that Mr Li would need to give each of his five relatives 500 RMB each, we can follow these steps:
Calculate the total amount in RMB that Mr Li needs to give:
500 RMB/relative x 5 relatives = 2500 RMB
Convert the total amount in RMB to Singapore dollars using the given exchange rate:
2500 RMB x (S$100/RMB510.20) = S$490.64 (rounded to the nearest cent)
Therefore, Mr Li would need S$490.64 to give each of his five relatives 500 RMB each, given the exchange rate of S$100 to RMB510.20.
ChatGPT
select a random integer from -200 to 200. which of the following pairs of events re mutualyle exclusive
The pairs of events of random integers are pairs of events are,
even and odd , negative and positive integers, zero and non-zero integers.
Two events are mutually exclusive if they cannot occur at the same time.
Selecting a random integer from -200 to 200,
Any two events that involve selecting a specific integer are mutually exclusive.
For example,
The events selecting the integer -100
And selecting the integer 50 are mutually exclusive
As they cannot both occur at the same time.
Any pair of events that involve selecting a specific integer are mutually exclusive.
Here are a few examples,
Selecting an even integer and selecting an odd integer.
Selecting a negative integer and selecting a positive integer
Selecting the integer 0 and selecting an integer that is not 0.
But,
Events such as selecting an even integer and selecting an integer between -100 and 100 are not mutually exclusive.
As there are even integers between -100 and 100.
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In the Venn diagram below, event A represents the adults who drink coffee, event B represents the adults who drink tea, and event C represents the adults who drink cola.
List the region(s) which represent the adults who drink both coffee and tea.
(Stats)
Answer:
Regions 1 and 4
Step-by-step explanation:
There are 2 overlapping regions for A (coffee) and B(tea)
These are Region 4 which represents the adults who drink both coffee and tea but not cola
and
Region 1 which represents the adults who drink coffee, tea and cola
So combined these two regions we get all adults who drink both coffee and tea
The data represent the results for a test for a certain disease. Assume one individual from the group is randomly selected. Find the probability of getting someone who tested positive given that he or she had the disease.
Answer:
To find the probability of getting someone who tested positive given that he or she had the disease, we need to use the formula for conditional probability:
P(positive|disease) = P(positive and disease) / P(disease)
From the given data, we can see that there are 136 individuals who tested positive and actually had the disease. Therefore, P(positive and disease) = 136.
We can also see that there are a total of 136 + 8 = 144 individuals who actually had the disease. Therefore, P(disease) = 144.
Substituting these values into the formula, we get:
P(positive|disease) = 136 / 144
Simplifying, we get:
P(positive|disease) = 0.944
Rounding to three decimal places, we get:
P(positive|disease) ≈ 0.944
Therefore, the probability of getting someone who tested positive given that he or she had the disease is approximately 0.944.
Given, y=a(x−2)(x+4)In the quadratic equation above, a is a nonzero constant. The graph of the equation in the xy-plane is a parabola with vertex (c,d). Which of the following is equal to d?A. -9aB. -8aC. -5aD. -2a
When the graph of the equation y=a(x−2)(x+4) in the xy-plane is a parabola with vertex (c,d), then the value of d is equal to option (A) -9a
To find the vertex of the parabola, we need to complete the square by factoring out the constant term a and adding and subtracting a term that will allow us to write the quadratic in the form
y = a(x - h)^2 + k,
where (h,k) are the coordinates of the vertex. We have
y = a(x - 2)(x + 4) = a(x^2 + 2x - 8x - 8) = a[(x + 1)^2 - 9]
Expanding the square and factoring out the constant term a, we get
y = a[(x + 1)^2 - 9] = a(x + 1)^2 - 9a
Comparing this to the standard form of the quadratic, we see that the vertex is at (-1,-9a). The value of d is -9a
Therefore, the correct option is (A) -9a
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Which answer choice contains all the factors of 10? • A. 1, 2, 5 О B. 1, 2, 5, 10 O C. 2, 5 O D. 1, 10
Answer:
The answer would be B (1,2,5, 10)
Step-by-step explanation:
Since one can already go into any number that greater than zero it would be a factor
2x5=10 as well so two and five would be a factor
you can also do 10x1
or 1x10 so both 10 and one would be factors of ten.
consider the following scores: 13, 18, 9, 27, 15, 15, 28, 5, 16, 21, 23, 29, 15, 15 what z-score would be earned by a person who had scored 25 points?
A person who scored 25 points in this dataset would have a z-score of 0.99.
The mean can be calculated by adding up all of the scores and dividing by the number of scores:
(13 + 18 + 9 + 27 + 15 + 15 + 28 + 5 + 16 + 21 + 23 + 29 + 15 + 15) / 14 = 18
The standard deviation can be calculated using the formula:
sqrt(sum((x - mean)^2) / (n - 1))
where x is each score in the dataset, mean is the mean of the dataset, and n is the number of scores.
Using this formula, we get:
sqrt (((13-18) ^2 + (18-18) ^2 + (9-18) ^2 + (27-18) ^2 + (15-18) ^2 + (15-18) ^2 + (28-18) ^2 + (5-18) ^2 + (16-18) ^2 + (21-18) ^2 + (23-18) ^2 + (29-18) ^2 + (15-18) ^2 + (15-18) ^2) / (14 - 1))
= 7.05
Now we can calculate the z-score of a scores of 25 using the formula:
z = (x - mean) / standard deviation
where x is the score, we are interested in, mean is the mean of the dataset, and standard deviation is the standard deviation of the dataset.
Plugging in the values, we get:
z = (25 - 18) / 7.05 = 0.99 (rounded to two decimal places)
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4 + 21 20 Step 1 of 2: Solve the inequality and express your answer in interval notation. Use decimal form for numerical values.
The solution for the given inequality "3x - 5 ≤ 7x + 1" in interval notaton is found out to be (-1.5, ∞).
First of all, we will be simplifying the inequality by subtracting 3x from both sides, we would get,
-5 ≤ 4x + 1
Next, we can subtract 1 from both the sides:
-6 ≤ 4x
Finally, we can divide both sides by 4:
-1.5 ≤ x
So the solution to the inequality is x ≥ -1.5.
Therefore, After xpressing the above inequality in interval notation, the solution is (-1.5, ∞), which means that x is greater than or equal to -1.5 and can take any value up to positive infinity.
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The complete question is :
Solve the inequality and express your answer in interval notation. Use decimal form for numerical values.
3x - 5 ≤ 7x + 1
What is the absolute
value for:
| +98|
Answer:
The absolute value of +98 is simply 98. So, | +98| = 98.
Step-by-step explanation:
Answer:
98
Step-by-step explanation:
The absolute value of a number is its POSITIVE distance from 0, so as +98 is 98 away from 0, then |+98| = 98.
(4+4i)(5-i) perform the indicated operation and express the result as a simplified complex number
Answer:
24 + 16i
Step-by-step explanation:
[tex](4+4i)(5-i)\\=(4)(5)+(4)(-i)+(4i)(5)+(4i)(-i)\\=20-4i+20i-4i^2\\=20+16i-4(-1)\\=20+16i+4\\=24+16i[/tex]
HELP ASAP WILL GIVE BRAINLYEST AND 100 POINTS
IF YOU DON"T TRY TO ANSWER THE QUESTION RIGHT I WILL REPORT YOU
Answer:
The order from least to greatest is:
8.2 x 10^-7 < 5.8 x 10^-5 < 1.2 x 10^3 < 9.7 x 10^3 < 3.4 x 10^6
Answer:
it is already in the correct order
Step-by-step explanation:
Use Mathematical Induction to prove the sum of Arithmetic Sequences:
n
∑
j
=
1
(
a
+
(
j
−
1
)
d
)
=
n
2
(
2
a
+
(
n
−
1
)
d
)
Answer:
We will use mathematical induction to prove the formula for the sum of arithmetic sequences:
For n=1, we have:
∑j=1^1(a + (j-1)d) = a
On the other hand, we have:
n/2(2a + (n-1)d) = 1/2(2a) = a
Thus, the formula holds for n=1.
Assuming the formula holds for n=k, we will prove that it holds for n=k+1.
We have:
∑j=1^(k+1)(a + (j-1)d) = (a + kd) + ∑j=1^k(a + (j-1)d)
Using the formula for n=k, we can write:
∑j=1^k(a + (j-1)d) = k/2(2a + (k-1)d)
Substituting this back into the first equation, we have:
∑j=1^(k+1)(a + (j-1)d) = (a + kd) + k/2(2a + (k-1)d)
Simplifying the right-hand side, we get:
∑j=1^(k+1)(a + (j-1)d) = 1/2(2a + (2k+1)d)
But (k+1)/2(2a + kd + d) = 1/2(2a + (2k+1)d), so the formula holds for n=k+1.
Therefore, by mathematical induction, the formula for the sum of arithmetic sequences is proved.
A normal distribution is informally described as a probability distribution that is "bell-shaped" when graphed. Draw a rough sketch of a curve having the bell shape that is characteristic of a normal distribution.
Choose the correct answer below.
A.
A symmetric curve is plotted over a horizontal scale. From left to right, the curve starts on the horizontal scale and rises at a decreasing rate to a central peak before falling at an increasing rate to the horizontal scale.
B.
A symmetric curve is plotted over a horizontal scale. From left to right, the curve starts above the horizontal scale, falls from the horizontal at an increasing rate, then falls at a decreasing rate to a central minimum before rising at an increasing rate, then rising at a decreasing rate, and finally becoming nearly horizontal.
C.
A symmetric curve is plotted over a horizontal scale. From left to right, the curve starts on the horizontal scale, rises from horizontal at an increasing rate, then rises at a decreasing rate to a central peak before falling at an increasing rate, then falling at a decreasing rate, and finally approaches the horizontal scale.
The correct answer is C. A normal distribution is a symmetric probability distribution that is bell-shaped when graphed. When plotted on a horizontal scale, the curve starts on the horizontal axis, rises to a central peak, and then falls back to the horizontal axis.
The curve is symmetric, meaning that the left and right halves of the curve are mirror images of each other. The curve approaches the horizontal axis but never touches it, which indicates that there is a non-zero probability of observing values at any distance from the mean, although the probability decreases as the distance from the mean increases.
Normal distribution is a type of probability distribution that is commonly found in natural and social phenomena, where the majority of the observations tend to cluster around the mean, with fewer observations further away from the mean.
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Find equations of the tangent line and normal line to the curve y
=
x
4
+
2
e
x
at the point (0,2)
The equation of the tangent line is y = 2x + 2., and the equation of the normal line is y = -1/2 x + 2.
To find the equations of the tangent and normal lines to the curve y = x⁴ + [tex]2e^{X}[/tex] at the point (0,2), we will need to find the slope of the curve at that point, and then use point-slope form to write the equations of the tangent and normal lines.
First, we can find the slope of the curve at the point (0,2) by taking the derivative of the function and evaluating it at x = 0:
y = x⁴ + [tex]2e^{X}[/tex]
y' = 4x³ + [tex]2e^{X}[/tex]
y'(0) = 4(0)³ + 2e⁰ = 2
So the slope of the curve at the point (0,2) is 2.
Next, we can use point-slope form to write the equation of the tangent line. The point-slope form of a line will be given by:
y - y₁ = m(x - x₁)
where m will be the slope of the line and (x₁, y₁) is a point on the line.
For the tangent line at (0,2), we have:
y - 2 = 2(x - 0)
Simplifying, we get:
y = 2x + 2
So the equation of tangent line is y = 2x + 2.
To find the equation of the normal line, we need to find the negative reciprocal of the slope of the tangent line (since the slopes of perpendicular lines are negative reciprocals of each other). So the slope of the normal line will be:
m = -1/2
Using point-slope form again, the equation of the normal line is:
y - 2 = (-1/2)(x - 0)
Simplifying, we get:
y = -1/2 x + 2
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Paul borrowed
$
6
,
000
from a credit union for
5
years and was charged simple interest at a rate of
5.45
%
. What is the amount of interest he paid at the end of the loan?
Paul paid $1,635 in interest at the end of the loan.
What is simple interest?Simple Interest (S.I.) is the method of calculating the interest amount for a particular principal amount of money at some rate of interest.
According to the given information:The simple interest formula is:
I = P * r * t
where I is the interest, P is the principal (the amount borrowed), r is the annual interest rate as a decimal, and t is the time in years.
In this problem, P = $6,000, r = 0.0545 (since the interest rate is given as 5.45%), and t = 5 years. Plugging in these values, we get:
I = 6,000 * 0.0545 * 5 = $1,635
Therefore, Paul paid $1,635 in interest at the end of the loan.
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What is the value of the underlined digit?
5(3)
Enter the correct answer in the box.
Answer: tens
Step-by-step explanation:
Please help me solve this my head hurts
a. Nοne οf these measure οf central tendency dοn't exist fοr this data set. Optiοn d) is cοrrect
b. As the sum οf all οbservatiοns wοuld chance, the mean wοuld be affected by the change. Optiοn a) is cοrrect
What is central tendency?In statistics, the central tendency is the descriptive summary οf a data set. Thrοugh the single value frοm the dataset, it reflects the centre οf the data distributiοn. Mοreοver, it dοes nοt prοvide infοrmatiοn regarding individual data frοm the dataset, where it gives a summary οf the dataset. Generally, the central tendency οf a dataset can be defined using sοme οf the measures in statistic.
c.
Suppοse that, the largest measurement 97 is remοved.
The number οf οbservatiοns as well as the sum οf all οbservatiοns wοuld change.
Therefοre, the median and the mean wοuld be changed and οptiοn a) and b) are cοrrect.
d.
Since there are three mοdes, the mean, median and mοde must nοt be cοmpared tο each οther fοr skewness.
Instead, it is required tο grοup data in intervals and οbserve the pattern οf classes versus frequencies, as displayed in histοgram.
Therefοre, the distributiοn appears rοughly symmetric and οptiοn c) is cοrrect.
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locate the absolute extrema of the function
on the closed interval
Answer:
To find the integral of f(x) = 2x + 5/3 over the interval [0, 5], we can use the definite integral formula:
∫[a,b] f(x) dx = F(b) - F(a)
where F(x) is the antiderivative of f(x).
First, we find the antiderivative of f(x):
F(x) = x^2 + (5/3)x + C
where C is the constant of integration.
Next, we evaluate F(5) and F(0):
F(5) = 5^2 + (5/3)(5) + C = 25 + (25/3) + C
F(0) = 0^2 + (5/3)(0) + C = 0 + 0 + C
Subtracting F(0) from F(5), we get:
∫[0,5] f(x) dx = F(5) - F(0)
= 25 + (25/3) + C - C
= 25 + (25/3)
= 100/3
Therefore, the definite integral of f(x) = 2x + 5/3 over the interval [0, 5] is 100/3.
£4100 is deposited into a bank paying 13.55% interest per annum , how much money will be in the bank after4 years
Answer:
To solve this problem, we can use the formula for compound interest:
A = P(1 + r/n)^(n*t)
Where:
A = the amount of money in the account after the specified time period
P = the initial principal amount (the amount deposited)
r = the annual interest rate (as a decimal)
n = the number of times the interest is compounded per year
t = the time period in years
In this case:
P = £4100
r = 13.55% = 0.1355
n = 1 (interest is compounded once per year)
t = 4 years
Plugging these values into the formula, we get:
A = £4100(1 + 0.1355/1)^(1*4)
A = £4100(1.1355)^4
A = £4100(1.6398)
A = £6717.58
Therefore, the amount of money in the account after 4 years will be £6717.58.
Shallow Drilling, Inc. has 76,650 shares of common stock outstanding with a beta of 1.47 and a market price of $50.00 per share. There are 14,250 shares of 6.40% preferred stock outstanding with a stated value of $100 per share and a market value of $80.00 per share. The company has 6,380 bonds outstanding that mature in 14 years. Each bond has a face value of $1,000, an 8.00% semiannual coupon rate, and is selling for 99.10% of par. The market risk premium is 9.79%, T-Bills are yielding 3.21%, and the tax rate is 26%. What discount rate should the firm apply to a new project's cash flows if the project has the same risk as the company's typical project?
Group of answer choices
The discount rate that should be applied to a new project's cash flows is the Weighted Average Cost of Capital (WACC). To calculate WACC, you need to first calculate the cost of debt. This is done by taking the face value of the bonds ($1000) multiplied by the coupon rate (8%) multiplied by (1 - the tax rate (26%)), which equals 5.92%. The cost of debt is then calculated by taking the market value of the debt (6,380 x $1,000 x 99.1%) and dividing this by the total market value of the debt plus the market value of the equity (6,380 x $1,000 x 99.1% + 76,650 x $50 + 14,250 x $80), which equals 5.22%.
Next, you need to calculate the cost of equity using the Capital Asset Pricing Model (CAPM). This is done by taking the risk-free rate (3.21%) plus the market risk premium (9.79%) multiplied by the firm's beta (1.47), which equals 17.18%.
The WACC is then calculated by taking the cost of equity multiplied by the proportion of equity (76,650 x $50 + 14,250 x $80 divided by the total market value of the debt plus the market value of the equity) plus the cost of debt multiplied by the proportion of debt (6,380 x $1,000
I will mark you brainiest!
What is the length of LJ?
A) 23.0
B) 17.0
C) 4.7
D) 3.5
Answer:
The right answer is below
Step-by-step explanation:
4.7 is the length of LJ
no algebra pls thanks uv
Answer: z = 210
Step-by-step explanation: Let's assume the original price of the present was x dollars.
Amanda agreed to pay 30% of the original price, so her contribution was 0.3x dollars.
The remaining amount to be paid is (1 - 0.3)x = 0.7x dollars.
Gabriel agreed to pay 2/5 of the remaining amount, so he paid (2/5)(0.7x) = 0.28x dollars.
The balance amount to be paid by Daniel is (1 - 0.3 - 2/5)(x) = 0.42x dollars.
When the price increased by 25%, the new price became 1.25x dollars.
We can set up an equation based on Amanda's contribution and solve for x:
0.3x = 63
x = 210
Therefore, the original price of the present was $210.
Calculate the derivative of the following function and simplify.
y = [tex]e^{x} csc x[/tex]
Answer:
To find the derivative of this function, we'll use the product rule and the chain rule. Let's begin by writing the function in a more readable form using parentheses:
y = e^x * csc(x) * (1 / x) * csc(x)
Now we can apply the product rule, letting u = e^x and v = csc(x) * (1 / x) * csc(x):
y' = u'v + uv'
To find u' and v', we'll need to use the chain rule.
u' = (e^x)' = e^x
v' = (csc(x) * (1 / x) * csc(x))'
= (csc(x))' * (1 / x) * csc(x) + csc(x) * (-1 / x^2) * csc(x) + csc(x) * (1 / x) * (csc(x))'
= -csc(x) * cot(x) * (1 / x) * csc(x) - csc(x) * (1 / x^2) * csc(x) - csc(x) * (1 / x) * csc(x) * cot(x)
= -csc(x) * [cot(x) * (1 / x) + (1 / x^2) + (cot(x) / x)]
Now we can substitute these into the product rule formula:
y' = e^x * csc(x) * (1 / x) * csc(x) * [-cot(x) * (1 / x) - (1 / x^2) - (cot(x) / x)] + e^x * (-csc(x) * cot(x) * (1 / x) * csc(x) - csc(x) * (1 / x^2) * csc(x) - csc(x) * (1 / x) * csc(x) * cot(x))
Next, we can simplify this expression. One way to do this is to factor out common terms:
y' = e^x * csc(x) * (1 / x) * csc(x) * [-cot(x) * (1 / x) - (1 / x^2) - (cot(x) / x)] - e^x * csc(x) * cot(x) * (1 / x) * csc(x) * [1 + (cot(x) / x)]
Now we can simplify further by combining like terms:
y' = e^x * csc(x) * (1 / x) * csc(x) * [-cot(x) * (2 / x) - (1 / x^2)] - e^x * csc(x) * cot(x) * (1 / x) * csc(x) * [1 + (cot(x) / x)]
= e^x * csc(x) * (1 / x) * csc(x) * [-2cot(x) / x - 1 / x^2 - cot(x) / x^2] - e^x * csc(x) * cot(x) * (1 / x) * csc(x) * [1 + cot(x) / x]
At this point, the derivative is simplified as much as possible.
(please could you kindly mark my answer as brainliest)
what is the distance between-48 and -12
Answer:36
Step-by-step explanation:-48+12
A local hamburger shop sold a combined total of 688 hamburgers and cheeseburgers on Monday. There were 62 fewer cheeseburgers sold than hamburgers. How many hamburgers were sold on Monday? I hamburgers
Answer: 375 hamburgers
Step-by-step explanation:
Let's assume that x is the number of hamburgers sold on Monday.
According to the problem, the number of cheeseburgers sold is 62 fewer than the number of hamburgers sold. So the number of cheeseburgers sold is x-62.
The total number of hamburgers and cheeseburgers sold is 688. So we can set up an equation:
x + (x-62) = 688
Simplifying this equation, we get:
2x - 62 = 688
Adding 62 to both sides, we get:
2x = 750
Dividing both sides by 2, we get:
x = 375
Therefore, 375 hamburgers were sold on Monday.
Please figure out #3. I’ll mark brainliest for right answer.
Answer:
We are given the cost equations for Emma's and Madison's text message plans as:
Emma's Plan: y = 0.10x + 10
Madison's Plan: y = 0.15x
where y is the cost in dollars and x is the number of texts sent. We are also told that Emma and Madison paid the same amount in one month. Let's set the two equations equal to each other and solve for x:
0.10x + 10 = 0.15x
Subtracting 0.10x from both sides, we get:
10 = 0.05x
Dividing both sides by 0.05, we get:
x = 200
Therefore, Emma and Madison sent 200 text messages in one month to pay the same amount
Question 2
State the probability that a randomly selected, normally
distributed value lies between
a) o below the mean and o above the mean (round to
the nearest hundredths)
b) 20 below the mean and 20 above the mean (round
to the nearest hundredths)
The probability of a randomly selected, normally distributed value lying between 20 below the mean and 20 above the mean is 0.9545 (rounded to the nearest hundredth).
What is the fundamental concept of probability?A number between zero and one represents the probability that an occurrence will take place. An event is a predefined set of random variable outcomes. Only one mutually exclusive event can occur at a time. Exhaustive events encompass or include all possible outcomes.
We can calculate the probabilities of a randomly chosen value falling between different z-scores using the provided standard normal distribution table.
a) The probability of a value being 0 below or above the mean is the same as the probability of a value being -1 to 1 standard deviations from the mean. According to the standard normal distribution table, the probability of a z-score between -1 and 1 is 0.6827. As a result, the probability of a randomly chosen, normally distributed value falling between 0 below and 0 above the mean is 0.6827. (rounded to the nearest hundredth).
b) The probability of a value falling between 20 and 20 standard deviations from the mean is the same as the probability of a value falling between -20/10 and 20/10 standard deviations from the mean (since the standard deviation is 10). According to the standard normal distribution table, the probability of a z-score between -2 and 2 is 0.9545. As a result, the probability of a randomly chosen, normally distributed value falling between 20 below and 20 above the mean is 0.9545. (rounded to the nearest hundredth).
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This is for a test, teacher said we could use any source available so help is appreciated
a. P(x) = 50x - 4000.1.
b. The company must sell at least 81 units of their product to have a positive profit.
c. The company should try to sell as many units as possible within the given range to maximize their profit.
What is revenue function?A revenue function is a mathematical equation or formula that represents the amount of money a company or organization generates from the sale of its products or services. It is a function that relates the price of a product or service to the quantity sold and represents the total revenue earned by a company at a given price and level of production.
What is profit function?A profit function is a mathematical equation or formula that represents the amount of profit a company or organization generates from the sale of its products or services. It is a function that relates the price of a product or service, the cost of production, and the quantity sold and represents the total profit earned by a company at a given price and level of production.
In the given question,
(a) The profit function can be obtained by subtracting the cost function from the revenue function as follows:
R(x) = -0.1x + 150x
P(x) = R(x) - C(x)
P(x) = (-0.1x + 150x) - (100x + 4000)
P(x) = 50x - 4000.1
Therefore, the profit function is P(x) = 50x - 4000.1.
(b) To find the minimum number of units the company must sell to have a positive profit, we need to set P(x) greater than or equal to zero and solve for x:
P(x) ≥ 0
50x - 4000.1 ≥ 0
50x ≥ 4000.1
x ≥ 80.002
Therefore, the company must sell at least 81 units of their product to have a positive profit.
(c) To find the value of r that maximizes the profit, we need to find the derivative of the profit function and set it equal to zero:
P'(x) = 50
Setting P'(x) equal to zero, we get:
50 = 0
This equation has no solution, which means that the profit function has no maximum value within the given range. However, we can see that the profit function is increasing for all values of x, which means that the profit increases as the number of units sold increases. Therefore, the company should try to sell as many units as possible within the given range to maximize their profit.
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