Dude he played for 1/2 of the game half of 60 is 30.
50%.
I'm I missing something?
in the right triangle round to your nearest tenth. 18 15 X help please
The value οf the given angle x = 39.8 degree
What is Trigοnοmetric Functiοns?Trigοnοmetry uses six fundamental trigοnοmetric οperatiοns. Trigοnοmetric ratiοs describe these οperatiοns. The sine functiοn, cοsine functiοn, secant functiοn, cο-secant functiοn, tangent functiοn, and cο-tangent functiοn are the six fundamental trigοnοmetric functiοns.
The ratiο οf sides οf a right-angled triangle is the basis fοr trigοnοmetric functiοns and identities. Using trigοnοmetric fοrmulas, the sine, cοsine, tangent, secant, and cοtangent values are calculated fοr the perpendicular side, hypοtenuse, and base οf a right triangle.
In the figure tanx = p/h
[tex]x = tan^{-1(15/18)}[/tex]
x = 39.8
Hence the value οf the given angle x = 39.8 degree
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How many proper subsets are in {2,4,6,8...100}
Answer:
159 proper subsets.
Step-by-step explanation:
Given a set {2, 4, 6, 8...100}, how many proper subsets are there?
First, find how many subsets there are in 2 - 10:
That's 16.
Then because there are 10 10s in 100, multiply by 10:
16 x 10 = 160
Finally, because it says proper subsets, subtract by 1:
160 - 1 = 159 proper subsets.
Therefore, there are 159 proper subsets in {2, 4, 6, 8...100}
Increase £16470.45 by 3.5%
Give your answer rounded to 2 DP
Step-by-step explanation:
"increase" means to take the original 100% and put an additional 3.5% of these 100% on top of it.
so, we have to calculate
100% + 3.5% of £16470.45
100% of £16470.45 = £16470.45 × 100/100
3.5% of £16470.45 = £16470.45 × 3.5/100
the sum is therefore
£16470.45 × (100/100 + 3.5/100) =
= £16470.45 × (1 + 0.035) = £16470.45 × 1.035 =
= £17,046.91575 ≈ £17,046.92
Work out the size of angle x. 79°) 35
Answer: 66
Step-by-step explanation:
all 3 of them should equal to 180
so 79+35 is 114
180-114 will give us the answer which is 66
Assume that the readings at freezing on a batch of thermometers are normally distributed with a mean of 0°C and a standard deviation of 1.00°C. A single thermometer is randomly selected and tested. Find the probability of obtaining a reading between 0.59°C and 0.88°C.
The probability of obtaining a reading between 0.59°C and 0.88°C is 0.7224 and 0.8106.
What is mean?The sum of all possible values, weighted by the chance of each value, is equal to the mean of a discrete probability distribution of the random variable X. Each possible number of X must be multiplied by its probability P(x) before being added as a whole to determine the mean. In statistics, the mean is one measure of central trend in addition to the mode and median. The mean is simply the average of the numbers in the specified collection. It suggests that values in a specific data gathering are evenly distributed. In order to find the mean, the total values given in a datasheet must be added, and the result must be divided by the total number of values.
In this question, using the formula,
z-score = (x – μ) / σ
where:
x: individual data value
μ: population mean
σ: population standard deviation
for x=0.59
μ= 0
σ= 1
z-score= 0.59
Probability=0.7224
for x=0.88
z-score= 0.88
Probability=0.8106
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Give the interval(s) on which the function is continuous.
g(t) = 1/√16-t^2
The function g(t) is defined as:
g(t) = 1/√(16-t^2)
The function is continuous for all values of t that satisfy the following conditions:
The denominator is non-zero:
The denominator of the function is √(16-t^2). Therefore, the function is undefined when 16-t^2 < 0, or when t is outside the interval [-4,4].
There are no vertical asymptotes:
The function does not have any vertical asymptotes, because the denominator is always positive.
Thus, the function g(t) is continuous on the interval [-4,4].
59, 60, 61, 62, 63, 64, 65, and 66 Find the values of x for which the series converges. Find the sum of the series for those values of x. 59. § (-5)".z" n=1 Answer + 00 60. Σ(α + 2)" n=1 61. (x - 2)" 3" n=0 Answer + 62. (-4)" (x - 5) n=0 00 63. 2" ch NO Answer
The values of x for which the series converges is x ∈ (-1/5, 1/5). The sum of the series for those values of x is (-5x)/(1 + 5x).
The series is [tex]\Sigma^{\infty}_{n=1}(-5)^nx^n[/tex].
We can write this series as [tex]\Sigma^{\infty}_{n=1}(-5x)^n[/tex].
This is a infinite geometric series with first term a = -5x and common ration r = -5x.
It is convergent when
|r| < 1
|-5x| < 1
|-5| |x| < 1
5|x| < 1
Divide by 5 on both side, we get
|x| < 1/5
The series is convergent when x ∈ (-1/5, 1/5).
Sum of the series is
Sₙ = a/1 - n
Sₙ = (-5x)/{1 - (-5x)}
Sₙ = (-5x)/(1 + 5x)
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The complete question is:
Find the values of x for which the series converges. Find the sum of the series for those values of x.
[tex]\Sigma^{\infty}_{n=1}(-5)^nx^n[/tex]
Mrs. Cabana has 8 pets total. Three of the pets are chameleons and the rest are fish. Select all the answers that are a ratio relationship for Mrs. Cabana's pets.
Question 1 options:
Multi choice
3/5
3 to 11
3:8
5 to 8
8:1
Answer: numbers 1,3 and 4
Step-by-step explanation:
A candy store owner used a cylindrical wooden log as a bench in their store. The height
of that log was 2
feet. The diameter of its base was 1.25
feet. If it costs $7.20
per square foot to paint that log at every side, how much approximately will it cost the
store owner? The total surface area of the right circular cylinder is 2nrh+2rr2
, where, r
is the radius of the base of the cylinder and, h
is the height of the cylinder
Answer:
Step-by-step explanation:
its 60
please help with finding the answer
The answer of the given question based on the transformation from its parent function the explanation part is given below and The equation of the function is y = -2(x+3)².
What is Function?In mathematics, function is relation between set of inputs and set of possible outputs with property that each input is related to exactly one output. It is rule that assigns to each input value exactly one output value. Functions can be represented in various ways, like algebraic expressions, graphs, tables, and words. They are used to model relationships between variables, to describe how one quantity depends on another, and to make predictions about future values. Functions are important concept in many fields of mathematics, as well as in science, engineering, economics, and other areas where quantitative analysis is used.
a. The graph appears to be a reflection of the parent function f(x) = x² over the x-axis followed by a vertical stretch by a factor of 2 and a horizontal shift to the left by 3 units.
b. The equation of the function is y = -2(x+3)².
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Question 13 (2 points)
Suppose you flip a coin and then roll a die. You record your result. What is the
probability you flip heads or roll a 3?
1/2
3/4
7/12
1
Step-by-step explanation:
a probability is always the ratio
desired cases / totally possible cases
we have 2 possible cases for the coin and 6 possible cases for the die.
so, we have 2×6 = 12 combined possible cases :
heads, 1
heads, 2
heads, 3
heads, 4
heads, 5
heads, 6
tails, 1
tails, 2
tails, 3
tails, 4
tails, 5
tails, 6
out of these 12 cases, which ones (how many) are desired ?
all first 6 plus (tails, 3) = 7 cases
so, the correct probability is
7/12
formally that is calculated :
1/2 × 6/6 + 1/2 × 1/6 = 6/12 + 1/12 = 7/12
the probability to get heads combined with the probability to roll anything on the die, plus the probability to get tails combined with the probability to roll 3.
2 bags of dog food. How many days will 3/4 last?
Answer:
3/4 of a bag of dog food will last 3 days.
spinner is divided into seven equal sections numbered 1 through 7 If the spinner is spun twice, what is the theoretical probability that it lands on 2 and then an odd number?
A) 1/49
B) 4/49
C) 1/7
D) 4/7
Answer:
B is correct
Step-by-step explanation:
If it is spun twice then the probability of it landing on 2 and then an odd number is:
Pr(2,1) or Pr(2,3) or Pr(2,5) or Pr(2,7)
1/49 * 4
4/49
Let f(x) = x? - 6x + 8 and g (x) = x - 5.
Find (f + g) (x) and (f - g) (x) .
Samir's statement shows a previous balance of $5,336.22, a payment of $607, and a
new transaction totaling $186. What is his new balance if his APR is 29.0%? Round
answer to hundredths place if answer does not have a hundredths place this use
zeros so it does. Do not include the units. Be sure to attach work for credit
Your Answer:
Samir's new balance is $5,044.17.
To calculate Samir's new balance, add the previous balance, subtract the payment, add the new transaction, and multiply by the interest rate for one period. The following formula can be used to calculate the interest for a single period:
balance * APR / 12 = interest
where APR stands for annual percentage rate and 12 represents the number of months in a year.
When we apply this formula to Samir's balance and APR, we get:
5336.22 * 0.29 / 12 = 128.95 in interest
As a result, the total new balance is:
5336.22 - 607 + 186 + 128.95 = 5044.17
We get the following when we round to the nearest hundredth:
$5,044.17
As a result, Samir now has a balance of $5,044.17.
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Find the missing length indicated
The value of x is 5
Define the term Similar triangles?Triangles with the same shape but different sizes are said to be similar triangles. To be more specific, two triangles are comparable if their respective sides are proportionate and their corresponding angles are congruent.
Two triangles are similar if corresponding angles are congruent and corresponding sides are proportional.
from the below figure, both the triangles are similar, ∆ABC ≈ ∆EFB
By using Thales's theorem, the ratio of the sides of triangles are;
BE/EA = BF/FC
15/30 = x/10
x = 5
Therefore the value of x is 5
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The ratio of triangle sides can be calculated using Thales' theory, and it is the value of x is 5
Define the term Similar triangles?Similar triangles are those with the same shape but varying sizes. To be more precise, two triangles are comparable if their matching angles and respective sides are congruent.
If matching sides are proportional and corresponding angles are congruent, two triangles are similar.
Both triangles in the following figure are comparable ∆ABC ≈ ∆EFB
The ratio of triangle sides can be calculated using Thales' theory, and it is;
BE/EA = BF/FC
15/30 = x/10
x = 5
Therefore the value of x is 5
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PLEASE HELP
The linear function f(x) = 0.9× + 79 represents the average test score in your math class, where x is the number of the test taken. The linear function g(x) represents the average test score in your science class, where x is the number of the test taken.
The required answers are 80.8, 79,and g(42) > f(42).
How to find average of equation?Part A:
To determine the test average for the math class after completing test 2, we need to evaluate the function f(x) at x=2. That is,
[tex]$$f(2) = 0.9(2) + 79 = 80.8$$[/tex]
Therefore, the test average for the math class after completing test 2 is 80.8.
Part B:
To determine the test average for the science class after completing test 2, we need to find the equation of the linear function g(x) that passes through the given points (1,78) and (2,79). The slope of the line passing through these points is
[tex]$m=\frac{y_2-y_1}{x_2-x_1}=\frac{79-78}{2-1}=1$$[/tex]
We can use the point-slope form of a line to find the equation of the line passing through the point (1,78) with slope m=1. That is,
[tex]$$y-78 = 1(x-1)$$[/tex]
Simplifying, we get
y = x + 77
Therefore, the test average for the science class after completing test 2 is
g(2) = 2 + 77 = 79
Part C:
To determine which class had a higher average after completing test 42, we need to evaluate f(42) and g(42) and compare the results. We have
[tex]$$f(42) = 0.9(42) + 79 = 117.8$$[/tex]
To find (42), we need to extend the linear function g(x) beyond the given data points by assuming that the function is linear and continues with the same slope m=1. That is,
g(x) = x + 77
for all [tex]$x\geq 1$[/tex]. Therefore,
[tex]$$g(42) = 42 + 77 = 119$$[/tex]
Since g(42) > f(42), we conclude that the science class had a higher average after completing test 42.
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solve please and thank you it’ll help a lot. 15 points.
Parallelogram (Opposite sides have the same length). Parallelogram (Area is one-half the base times the height). Parallelogram (Opposite sides are parallel). Parallelogram (Angles can be right angles)
What is the assertion of the parallelogram?According to the parallelogram law, the sum of the squares of a parallelogram's four sides is equal to the sum of the squares of its two diagonals. It is essential for the parallelogram to have equal opposite sides in Euclidean geometry.
Are a parallelogram's opposing sides parallel?A parallelogram is a particular sort of polygon. It is a quadrilateral in which the opposite side pairs are parallel to one another. There are six crucial parallelogram characteristics to be aware of: Congruent sides are those when AB = DC.
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PLEASE HELP FAST!!
Find the slope of a line perpendicular to the line whose equation is
4x−6y=−24. Fully simplify your answer.
Answer: -3/2
Step-by-step explanation:
FIrst rearrange the equation in y = mx + b form.
4x - 6y = -24
-6y = -4x - 24
y = 2/3x + 4
If the line is perpendicular, the slope must be the negative reciprocal of the current line.
The negative reciprocal of 2/3 is -3/2.
Given that (1, 2, 3] System{1, 4, 7,6] for a system known to be LTI, compute the system's impulse response h[n] without using z-transforms.
Given that (1, 2, 3] System{1, 4, 7,6] for a system known to be LTI, the impulse response of the system: h[n] = (1/2)*δ[n] + δ[n-1] + (3/2)*δ[n-2]
To compute the impulse response h[n] of a linear time-invariant (LTI) system given its input-output relationship, we can use the convolution sum:
y[n] = x[n] * h[n]
y[n] = (1/2)*(x[n] + 2x[n-1] + 3x[n-2])
y[n] = (1/2)*(δ[n] + 2δ[n-1] + 3δ[n-2])
y[n] = (1/2)*δ[n] + δ[n-1] + (3/2)*δ[n-2]
Thus, the impulse response of the system is:
h[n] = (1/2)*δ[n] + δ[n-1] + (3/2)*δ[n-2],where δ[n] is the impulse signal.
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To the nearest hundredth, what is the volume of the sphere? (Use 3.14 for pie.)
Therefore, the volume of the sphere to the nearest hundredth is 724,775.70 cubic millimeters.
What is volume?Volume is a measurement of the amount of space occupied by a three-dimensional object. It is often expressed in units such as cubic meters (m³), cubic centimeters (cm³), cubic feet (ft³), or gallons (gal), depending on the context. The volume of a solid object can be calculated by multiplying its length, width, and height or using a specific formula depending on the shape of the object. For example, the volume of a rectangular box can be calculated as length x width x height, while the volume of a cylinder can be calculated as π x radius² x height. In general, volume is an important concept in many fields, including physics, chemistry, engineering, and architecture. It is often used to describe the capacity of containers, the displacement of fluids, and the amount of material used in construction or manufacturing.
Here,
The formula for the volume of a sphere is given as V = (4/3)πr³, where r is the radius of the sphere and π is approximately 3.14.
Substituting the given value of the radius, we get:
V = (4/3) x 3.14 x 48³
V ≈ 724,775.68 cubic millimeters
Rounding this value to the nearest hundredth, we get:
V ≈ 724,775.68 ≈ 724,775.70 cubic millimeters (rounded to two decimal places)
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Someone plezzz help me
Neither anushka nor lukas are correct as both of their calculations are wrong.
How are linear equations solved?Basic arithmetic operations like addition, subtraction, multiplication, and division are used to isolate the variable on one side of a linear equation and solve it. The objective is to make the equation as simple as possible until the variable can be identified and its value calculated. In order to solve a linear equation, you must first combine like terms to simplify the expressions on both sides of the problem.
Then, you can use inverse operations to get rid of constants and coefficients. The value of the variable can be ascertained by solving for it once it has been isolated. By looking at the coefficients and constants of the equation, it can be established if the equation has no solution or infinite solutions. In various disciplines, such as science, engineering, and finance, linear equations are used to represent connections between variables.
2/5b + 1 = -11
2/5b = -12
b= -12 x 5/2
b = -30
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What is the end behavior of the polynomial function?
Answer: D. As x → -∞, y → -∞.
Step-by-step explanation:
The graph shows the function approaching negative infinity on the x-axis (left side). When the x-axis is decreasing, the y-axis is also decreasing towards negative infinity.
Which construction is shown in the diagram below?
Answer:
Step-by-step explanation:
i think it B
consider a student loan of $15000 at a fixed APR of 12 % for 20 years
Therefore, the monthly payment for a student loan of $15,000 at a fixed APR of 12% for 20 years is $144.36.
What is interest?Interest is the cost of borrowing money or the return on investing money. When you borrow money, you usually have to pay back more than you borrowed, and the additional amount you pay is the interest. The interest rate is expressed as a percentage of the borrowed amount, and it can vary depending on factors such as the borrower's credit score, the term of the loan, and the lender's policies.
Given by the question.
Assuming the loan has a fixed interest rate of 12% per annum, the amount of interest charged each year will be:
12% of $15,000 = $1,800
The total interest charged over 20 years will be:
$1,800 x 20 = $36,000
The total amount to be repaid (principal + interest) will be:
$15,000 + $36,000 = $51,000
If the loan is being repaid in equal monthly installments over the 20-year term, the monthly payment can be calculated using the following formula:
M = P * (r[tex](1+r)^{n}[/tex]) / ([tex](1+r)^{n}[/tex]- 1)
Where:
M = Monthly payment
P = Principal amount (in this case, $15,000)
r = Monthly interest rate (12% per annum / 12 months = 1% per month)
n = Total number of payments (20 years x 12 months per year = 240)
Plugging in the values:
M = $15,000 * (0.01[tex](1+0.01)^{240}[/tex]) / ([tex](1+0.01)^{240}[/tex] - 1)
M = $144.36
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(a) Show that if λ is an eigenvalue of A, then λ is an eigenvalue of [tex]A^{T}[/tex]. Show with an example that the eigenvectors of A and [tex]A^{T}[/tex] are not the same.
(b) Show that if λ is an eigenvalue of A, and A is invertible, then λ^-1 is an eigenvalue of A^-1.
If λ is an eigenvalue of A, then λ is an eigenvalue of [tex]A^T[/tex]. Show with an example that the eigenvectors of A and [tex]A^T[/tex] are not the same.
What are eigenvalues and eigenvectors?The equation Av = λv, where v is a non-zero vector, is satisfied by an eigenvector v and an eigenvalue given a square matrix A. In other words, the eigenvector v is multiplied by the matrix A to produce a scalar multiple of v. Due to their role in illuminating the behaviour of linear transformations and differential equation systems, eigenvectors play a crucial role in many branches of mathematics and science. When the eigenvector v is multiplied by A, the eigenvalue indicates how much it is scaled.
The eigenvalue and eigenvector states that, let v be a non-zero eigenvector of A corresponding to the eigenvalue λ.
Then, we have:
Av = λv
Taking transpose on both sides we have:
[tex]v^T A^T = \lambda v^T[/tex]
The above equations thus relates transpose of vector and transpose of A to λ.
Now, consider a matrix:
[tex]\left[\begin{array}{cc}1&2\\3&4\\\end{array}\right][/tex]
Now, the eigen values of this matrix are λ1 = -0.37 and λ2 = 5.37.
The eigenvectors are:
[tex]v1 = [-0.8246, 0.5658]^T\\v2 = [-0.4159, -0.9094]^T[/tex]
Now, for transpose of A:
[tex]A^T=\left[\begin{array}{cc}1&3\\2&4\\\end{array}\right][/tex]
The eigen vectors are:
[tex]u1 = [-0.7071, -0.7071]^T\\u2 = [0.8944, -0.4472]^T[/tex]
Hence, we see that, if λ is an eigenvalue of A, then λ is an eigenvalue of [tex]A^T[/tex]. Show with an example that the eigenvectors of A and [tex]A^T[/tex] are not the same.
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given natural numbers a and b not both equal to 0, we know that there exist integers k and l with ak bl
The equation can be rearranged to the form y = -qx + r. This is the equation of a straight line, which can be graphed. The point of intersection of the two lines, ak + bl = 0 and y = -qx + r, is the solution for the two variables (k and l).
The equation ak + bl = 0 is a linear equation in two variables and is solved using the method of elimination. The equation can be written in the form ax + by = c, where a, b, c are constants. To solve this equation, both sides of the equation should be divided by the coefficient of one of the variables (a or b). This will result in a equation of the form x + qy = r, where q and r are constants. Then, the equation can be rearranged to the form y = -qx + r. This is the equation of a straight line, which can be graphed. The point of intersection of the two lines, ak + bl = 0 and y = -qx + r, is the solution for the two variables (k and l). The two variables can then be calculated using the point of intersection by substituting the x and y values into the two equations. In this way, the two variables k and l can be found such that ak + bl = 0.
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What are the integers k and l such that ak + bl = 0?
which statemnt is ture when the dimensions of a two-dimensional figures are dilated by a scale factor of 2
When a shape is dilated, the size of the shape changes. The true statement is (d) The scale factor is 2.5.
Dilation:
Dilation is the process of changing the size of an object or shape by reducing or increasing its size by a specific scale factor. For example, a circle with a radius of 10 units shrinks to a circle with a radius of 5 units. Applications of this method are in photography, arts and crafts, sign making and more.
According to the Question:
How to determine the scale factor
In figure A, we have:
Length = 0.6
In figure B, we have:
Length =1.5
The scale factor is then calculated as:
K = 1.5/0.6
Dividing the equation:
k = 2.5
Hence, the true statement is (d) The scale factor is 2.5.
Complete Question:
The first figure is dilated to form the second figure. Which statement is true?
The scale factor is 0.4.
The scale factor is 0.9.
The scale factor is 2.1.
The scale factor is 2.5.
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The following joint probability density function for the random variables Y1 and Y2, which represent the proportions of two components in a somaple from a mixture of insecticide.
f(y1,y2) = { 2, 0 <= y1 <= 1, 0 <= y2 <= 1, 0 <= y1+y2 <=1
{ 0, elsewhere
For the chemicals under considerationm an important quantity is the total proportion Y1 +Y2 found in any sample. Find E(Y1+Y2) and V(Y1+Y2).
The joint probability density function for the random variables Y1 and Y2 E(Y1+Y2) and V(Y1+Y2) is 41/144.
To find E(Y1+Y2), we need to integrate the sum of Y1 and Y2 over their joint probability density function:
E(Y1+Y2) = ∫∫ (y1 + y2) f(y1,y2) dy1 dy2
= ∫∫ (y1 + y2) (2) dy1 dy2, where the limits of integration are 0 to 1 for both y1 and y2 and y1+y2 <=1
= ∫[tex]0^1[/tex] ∫[tex]0^{(1-y1)}[/tex](y1 + y2) (2) dy2 dy1
= ∫[tex]0^1[/tex] (2y1 + 1) (1-y1)² dy1
= 5/12
To find V(Y1+Y2), we can use the formula V(Y1+Y2) = E[(Y1+Y2)²] - [E(Y1+Y2)]².
First, we need to find E[(Y1+Y2)^2]:
E[(Y1+Y2)²] = ∫∫ (y1+y2)² f(y1,y2) dy1 dy2
= ∫∫ (y1² + y2² + 2y1y2) (2) dy1 dy2, where the limits of integration are 0 to 1 for both y1 and y2 and y1+y2 = 1
= ∫[tex]0^1[/tex] ∫[tex]0^{(1-y1)}[/tex] (y1² + y2² + 2y1y2) (2) dy2 dy1
= ∫[tex]0^1[/tex] (1/3)y1³ + (1/2)y1² + (1/2)y1
(1/3)y1 + (1/4) dy1
= 7/12
Next, we need to find [E(Y1+Y2)]²:
[E(Y1+Y2)]² = (5/12)² = 25/144
Therefore, V(Y1+Y2) = E[(Y1+Y2)²] - [E(Y1+Y2)]² = (7/12) - (25/144) = 41/144.
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A firm is a monopoly for the good it produces. Its average cost function is AC = 9+(3/10)q+30/q, where q is the quantity produced. The demand equation for its good is given by q = 40 - (4/3)p where p is the price.
(a) Find expressions, in terms of q, for the total revenue.
(b) What is the equation for the Total cost?
(c) Find the expression for profit. (d) Find the total output and revenue at the break even point.
(e) Find the profit when 20 units are produced.
(f) Find the profit when 7 units are produced.
(g) Find the output required to obtain a profit of RM100.
The answer of the given question is (a) TR = p(40 - (4/3)p), or TR = 40p - (4/3)p² , (b) TC = 9q + (3/10)q² + 30 , (c) π = 40p - (4/3)p² - 9q - (3/10)q² - 30 , (d) TR ≈ 342.67 , (e) the profit when 20 units are produced is approximately RM188.27 , (f) the profit when 7 units are produced is approximately -RM24.44, indicating a loss , (g) the output required to obtain a profit of RM100 is approximately 8.78 units.
What is Equation?An equation is mathematical statement that asserts yhe equality of two expressions. It typically consists of variables, constants, and mathematical operations like addition, subtraction, multiplication, and division, among others. Equations are often used to solve problems, to model real-world phenomena, and to describe mathematical relationships.
(a) The total revenue is given by TR = p x q. Substituting the demand equation q = 40 - (4/3)p, we get TR = p(40 - (4/3)p), or TR = 40p - (4/3)p².
(b) The total cost is given by TC = q x AC. Substituting the given average cost function, we get TC = 9q + (3/10)q² + 30.
(c) The profit is given by π = TR - TC. Substituting the expressions we found in parts (a) and (b), we get π = 40p - (4/3)p² - 9q - (3/10)q² - 30.
(d) At the break even point, the firm earns zero profit, so we set π = 0 and solve for q. Substituting the expression we found in part (a) for p, we get:
0 = 40p - (4/3)p² - 9q - (3/10)q² - 30
0 = 40(40/3 - (3/4)q) - (4/3)(40/3 - (3/4)q)² - 9q - (3/10)q² - 30
0 = 533.33 - 51.25q - 0.22q^2
Solving for q using the quadratic formula, we get:
q = (51.25 ± sqrt(51.25² - 4(-0.22)(533.33))) / 2(-0.22)
q ≈ 22.75 or q ≈ 206.58
We reject the solution q ≈ 206.58 because it is outside the relevant range of output, which is between 0 and 40. Therefore, the total output at the break even point is approximately 22.75 units. To find the total revenue at the break even point, we substitute q = 22.75 into the demand equation from part (a) and get:
p = (40/3) - (3/4)q
p ≈ 15.08
TR = p x q
TR ≈ 342.67
(e) To find the profit when 20 units are produced, we substitute q = 20 into the expression for profit we found in part (c) and get:
π = 40p - (4/3)p² - 9q - (3/10)q² - 30
π ≈ 188.27
Therefore, the profit when 20 units are produced is approximately RM188.27.
(f) To find the profit when 7 units are produced, we substitute q = 7 into the expression for profit we found in part (c) and get:
π = 40p - (4/3)p² - 9q - (3/10)q² - 30
π ≈ -24.44
Therefore, the profit when 7 units are produced is approximately -RM24.44, indicating a loss.
(g) To find the output required to obtain a profit of RM100, we set the profit equation equal to 100 and solve for q:
Profit = TR - TC
100 = pq - ACq
100 = (40-(4/3)p)*q - (9+(3/10)q+30/q)*q
100 = (40-(4/3)p - 9q - 3q²/10)
Multiplying by 10 and rearranging terms, we get a quadratic equation in q:
3q² + 91q - 310 = 0
Solving for q using the quadratic formula, we get:
q = (-91 ± sqrt(91² - 43(-310)))/(2*3)
q ≈ 8.78 or q ≈ -29.44
Since the quantity produced cannot be negative, the output required to obtain a profit of RM100 is approximately 8.78 units.
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