consider the funciton f given by f(t)=log49(t7). find values for a and b such that f(t)=a+logb(t).

Therefore, the monthly payment for a student loan of $15,000 at a fixed APR of 12% for 20 years is $144.36.

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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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.

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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The required answers are 80.8, 79,and g(42) > f(42).

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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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)

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.

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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The value of x is 5

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

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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The value οf the given angle x = 39.8 degree

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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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

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.

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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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

Answer:

Step-by-step explanation:

i think it B

Neither anushka nor lukas are correct as both of their calculations are wrong.

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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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.

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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Therefore, the volume of the sphere to the nearest hundredth is 724,775.70 cubic millimeters.

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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a) The domain of the function is  [0, 7.1875], while the range is [0,23]. Considering the domain and the range, the graph of the function is given by the image presented at the end of the answer.

b) The trip had a duration of 5.1875 hours.

The function for this problem is defined as follows:

g(t) = 23 - 3.2t.

The domain is the set of input values that can be assumed by the function. The time cannot have negative measures, hence the lower bound of the domain is of zero, while the gas cannot be negative, hence the upper bound of the domain is given as follows:

23 - 3.2t = 0

3.2t = 23

t = 23/3.2

t = 7.1875 hours.

The range is given by the set of all output values assumed the function, which are the values of the gas, hence it is [0,23].

The graph is a linear function between points (0, 23) and (7.1875, 0).

At the end of the trip there were 6.4 gallons left, hence the length of the trip is obtained as follows:

23 - 3.2t = 6.4

t = (23 - 6.4)/3.2

t = 5.1875 hours.

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Answer:

3/4 of a bag of dog food will last 3 days.

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.

Answer:

Step-by-step explanation:

[tex]h(3)=3\times 3^2+3a-1 \rightarrow h(3)=26+3a[/tex]

But we cannot find [tex]a[/tex] unless we are told what [tex]h(3)[/tex] equals.

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

Answer:

They are similar

Step-by-step explanation:

It's because 27/18 = 12/8

27/18 = 1.5

12/8 = 1.5

Answer:

Step-by-step explanation:

its 60

The area to the right of x is 0.877.

In a normal distribution, the entire area under the curve is identical to 1. The area to the left of a specific value of x represents the possibility of observing a value largely lesser than or same tox.

However, we're capable to discover the area to the right of x with the aid of abating the left area from 1, If the place to the left of x is given.

In this case, the area to the left of x is 0.123. thus, the place to the right of x is

1-0.123 = 0.877

Thus, the area is 0.877 to the right of x.

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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].

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.

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.

A) Relative frequency is number of times an event happened over total number of events:

Answer is 21/50 = 0.42

B) On a 6 sides die, there are 3 even numbers and 3 odd numbers, so the theoretical probability of landing on odd would be 3/6 = 0.50

C) Because the die has an equal amount of chance landing on even or odd, both are 3/6, then the dice is not biased.

Because g(x) is continuous on the interval, we can see that the correct option is the last one (counting from the top)

Here we have the function g(x), and we know that it is continuous on the interval [1, 6], and that:

g(1) = 18

g(6) = 11

If it is continuous, then g(x) covers all the values between 18 and 11 in the given interval, this means that there must exist a value c in the given interval such that when we evaluat g(x) in that value c, we get the outcome 12, and we know this by the intermediate value theorem.

So the correct optionis the last one.

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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.

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}