Write an exponential function for the graph that passes through the points (0, –3) and (4, –48)

Answer:

Step-by-step explanation:

To make x the subject of y = ax^2 + b, we can use the following steps:

1.Subtract b from both sides to isolate the term with x^2:

y - b = ax^2

2.Divide both sides by a to isolate x^2:

(y - b) / a = x^2

3.Take the square root of both sides to solve for x:

x = ± sqrt((y - b) / a))

Therefore, the expression for x as a subject of y = ax^2 + b is:

x = ± sqrt((y - b) / a))

Answer:

1/6 x (21x^2 -82)    

Step-by-step explanation:

1.)  Multiply the numbers.  

2.) Combine Exponents.

3.) Combine like terms.

4.) Common factor.

5.) Find one factor.

this should be right

Answer:

there are 120 different ways

Step-by-step explanation:

the factorial of 5

:)

If x is multiplied by 5 and then 3 is subtracted, then the function is f(x)=5x-3.  the inverse function of f(x) = 5x - 3 is f^(-1)(x) = (x + 3)/5.

To find the inverse function of f(x) = 5x - 3, we need to follow these steps:

Step 1: Replace f(x) with y:

y = 5x - 3

Step 2: Solve for x in terms of y:

y = 5x - 3

y + 3 = 5x

x = (y + 3)/5

Step 3: Replace x with f^(-1)(x) and y with x:

f^(-1)(x) = (x + 3)/5

Step 4: Check if the inverse function is valid by verifying if f(f^(-1)(x)) = x and f^(-1)(f(x)) = x:

f(f^(-1)(x)) = f((x+3)/5) = 5((x+3)/5) - 3 = x + 2

f^(-1)(f(x)) = f^(-1)(5x - 3) = (5x - 3 + 3)/5 = x/5

Since f(f^(-1)(x)) = x + 2 and f^(-1)(f(x)) = x/5, the inverse function is valid.

Therefore, the inverse function of f(x) = 5x - 3 is f^(-1)(x) = (x + 3)/5.

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The complete question is:

if x is multiplied by 5 and then 3 is subtracted, then the function is f(x)=5x-3.

What are the step to find the inverse to this function?

About the equation [tex]7 * 3 = 21[/tex], the generalization we can make is that dividing a whole number by a unit fraction is the same as multiplying a whole number by the denominator of the unit fraction.

We use generalizations to describe the pattern they see in the relationship of a specific group of numbers. It's a pattern that holds true every time.

In mathematics, generalization can be defined as a statement that is true for an entire class of objects, the process by which we obtain a general statement, or the method by which knowledge is transferred from one setting to another.

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The set of side lengths that can produce a triangle as required in the task content is; Choice E; 5, 7, 11.

It follows from the task content that the set of side lengths that could produce a triangle.

Recall from the triangle inequality theorem that the sum of any two side lengths is greater than the third side length. Also, the difference of any two side lengths is less than the third side length.

Hence, the correct answer choice is; Choice E; 5, 7, 11.

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Answer:

What is the question?

Step-by-step explanation:

N/A

Answer:

a. To find the perimeter of the first figure, we need to add up the lengths of all four sides. If the sides are equal, we can simply multiply the length of one side by 4. Therefore, the perimeter of a square with sides of 21 cm is:

Perimeter = 4 x side

Perimeter = 4 x 21 cm

Perimeter = 84 cm

b. To find the perimeter of the second figure, we need to add up the lengths of all three sides. Since all three sides are equal, we can simply multiply the length of one side by 3. Therefore, the perimeter of an equilateral triangle with sides of 35 cm is:

Perimeter = 3 x side

Perimeter = 3 x 35 cm

Perimeter = 105 cm

c. To find the perimeter of the third figure, we need to add up the lengths of all four sides. Therefore, the perimeter of a rectangle with sides of 6.75 cm and 2.25 cm is:

Perimeter = 2 x (length + width)

Perimeter = 2 x (6.75 cm + 2.25 cm)

Perimeter = 2 x 9 cm

Perimeter = 18 cm

Step-by-step explanation:

a. To find the perimeter of the first figure (a square with sides of 21 cm), we need to add up the lengths of all four sides. Since all four sides are equal, we can simply multiply the length of one side by 4. Therefore:

Perimeter = 4 x side

Perimeter = 4 x 21 cm

Perimeter = 84 cm

So the perimeter of the square is 84 cm.

b. To find the perimeter of the second figure (an equilateral triangle with sides of 35 cm), we need to add up the lengths of all three sides. Since all three sides are equal, we can simply multiply the length of one side by 3. Therefore:

Perimeter = 3 x side

Perimeter = 3 x 35 cm

Perimeter = 105 cm

So the perimeter of the equilateral triangle is 105 cm.

c. To find the perimeter of the third figure (a rectangle with sides of 6.75 cm and 2.25 cm), we need to add up the lengths of all four sides. Therefore:

Perimeter = 2 x (length + width)

Perimeter = 2 x (6.75 cm + 2.25 cm)

Perimeter = 2 x 9 cm

Perimeter = 18 cm

So the perimeter of the rectangle is 18 cm.

A possible comment from Prudence math instructor would be Great job solving the rational equation! It's not an easy task, and you managed to find the solutions x = 1 and x = -4, which is correct.

Here are some other possible comments that Prudence's math instructor might make about her solution:

Make sure to check your solutions by plugging them back into the original equation. It's always a good practice to do so, as it helps you avoid mistakes and verify that your solutions are indeed valid.

When dealing with rational equations, you need to be careful about the restrictions on the domain. In this case, the denominator cannot be equal to zero, so x = -4 and x = 2 are not valid solutions. Make sure to mention this in your solution.

When simplifying the expression, make sure to distribute the negative sign (-) properly. In this case, you should have -7x - 20 instead of -7x + 20. Double-check your calculations to avoid such errors.

Make sure to show all the steps in your solution, including how you obtained the common denominator and how you simplified the expression. This helps the reader follow your thought process and understand your solution better.

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Answer:

20/81

Step-by-step explanation:

The total number of brownies is 2 + 2 + 5 = 9. The probability of Gavin taking a brownie with nuts on the first pick is 5/9. After putting it back, the total number of brownies is still 9, but the number of brownies with nuts is now 4. Therefore, the probability of Gavin taking a brownie without nuts on the second pick is 4/9. The probability of both events happening together is the product of the individual probabilities: (5/9) x (4/9) = 20/81. Therefore, the probability that Gavin will randomly take a brownie with nuts, put it back, and grab one without nuts is 20/81.

Answer:

if you add the whole amount, then 2+2+5 is 9. then divide by 4 (the number without nuts) 4/9 is 44 percent

Step-by-step explanation:

From the list of options, the expression that is correctly evaluated is: (a.) 6² = 36

The expression that is correctly evaluated is:

a.) 6² = 36

This is because 6² means 6 raised to the power of 2, which is the same as 6 multiplied by itself. So, 6² equals 6 times 6, which is 36.

For other expressions, we have

b.) 12¹ means 12 raised to the power of 1, which is simply 12. Therefore, 12¹ equals 12, not 0.

c.) 3^4 means 3 raised to the power of 4, which is 81, not 12.

d.) 10⁰ means 10 raised to the power of 0, which is always equal to 1, not 10.

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The value of the functions are;

x = 1, f(x) = 1/2

x = 2, f(x) = -1

x = 3 , f(x) = -5/2

x = 3, f(x) = -4

Note that functions are expressions or equations that shows the relationship between variables.

The variables are;

From the information given, we have the function as;

f(x) = -3/2x + 2

Now, substitute the value of x as 1

f(1) = -3/2(1) + 2

f(1) = -3/2 + 2 = -3 + 4 /2 = 1/2

f(2) = -3/2(2) + 2

expand the bracket

f(2) = -6/2 + 2 =-6 + 4  /2 = -2/2 = -1

For the value of x = 3

f(3) = -3/2(3) + 2

expand the bracket'

f(3) = -9/2 + 2

f(3) = -5/2

For the value of x = 4

f(4) = -3/2(4) + 2

f(4) = -8/2

f(4) = -4

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Math instructor may provide feedback on Prudence's solution by advising her to check her solutions for extraneous solutions, showing more detailed work, and continuing to practice solving challenging problems to improve her skills for equation.

One possible comment the math instructor could make about Prudence's solution is that it is important to check the solutions she obtained to make sure they are valid. When solving rational equations, it is possible to obtain extraneous solutions, which are solutions that do not work in the original equation. Therefore, the instructor may advise Prudence to substitute her solutions back into the original equation and verify that they satisfy the equation.

Another comment the instructor could make is that Prudence should show her work and steps in more detail, especially if she is turning in her solution for a graded assignment. By showing more detailed work, Prudence can demonstrate a deeper understanding of the problem and how she arrived at her solution.

The instructor may also commend Prudence on her ability to solve a complex rational equation with multiple variables and parentheses. This shows that she has a strong grasp of algebraic concepts and can apply them effectively. However, the instructor may also suggest that Prudence continue to practice solving more challenging problems to further improve her skills.

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The value of angle B is 68 degrees and the length of the segments b is 27.81 and c is 11.28.

A triangle's sides and angles are related by the Law of Sines, a trigonometric formula. It specifically specifies that for all three sides and angles of the triangle, the ratio of the length of one side to the sine of the angle opposite that side is the same. This may be expressed as the following equation:

sin A/a = sin B/b = sin C/c

where a, b, and c are the lengths of the sides that are on each side of the triangle's three angles, A, B, and C.

For the give triangle we can use the Law of sines to find the missing values.

The law of sine is given as:

sin A/a = sin B/b = sin C/c

Given A = 90 degrees and C = 22 degrees,

sin 90 / 30 = sin 22 / c

c = 30 sin 22 / sin 90

c = 30 (0.37) / 1

c = 11.28

Now, Using the sum of triangles:

90 + 22 + B = 180

112 + b = 180

B = 68

Now,

sin A/a = sin B/b

Sin 90 / 30 = sin (68) / b

b = 30 sin (68)  sin 90

b = 27.81

Hence, the value of angle B is 68 degrees and the length of the segments b is 27.81 and c is 11.28.

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The solution of x using the graphing tool for the expression [tex]-3^{-x} -6 = -3^{x} + 10[/tex] is found as: x = 2.5.

A line represents the graph of a linear equation. The equation has a solution at each point along the line. We shall plot two lines for a set of two equations.

The given expression is-

[tex]-3^{-x} -6 = -3^{x} + 10[/tex]

The graph of the equation for this expression if plotted using graphing tool.

From the graph, the solution of x is found as: x = 2.5.

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Answer:

-349

Step-by-step explanation:

1)[-6-22]+[-321]

2)(-28)-321

3)-349

Hence, they are different because they drawn from the different scales values.

Data in math is a collection of facts and figures that can be in any form—numerical or non-numerical.

Numerical data is the one you can calculate, and it is always collected in number form, such as scores of students in class, wages of workers in an organization or height of players on a football team, etc.

As per the given graphs they are different because they drawn with the help of different scales.

Hence, they are different because they drawn from the different scales values.

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Rounding to the nearest 50 dollars, we get:

value = $12,550.

The annual rate of change refers to the percentage change in a quantity over a period of one year. It is commonly used to express the growth or decline of economic indicators, such as GDP, inflation, or employment, as well as the performance of financial assets, such as stocks or bonds. To calculate the annual rate of change, you need to take the difference between the initial and final values of the quantity, divide it by the initial value, and multiply the result by 100 to express it as a percentage. For example, if a company's revenue was $1 million in 2021 and $1.2 million in 2022, the annual rate of change in revenue would be (1.2-1)/1 * 100 = 20%.

A) The car depreciated from $44,000 in 1992 to $15,000 in 2006, a period of 14 years.

The annual rate of change can be found using the formula:

[tex]r = (V2/V1)^{(1/n)} - 1[/tex]

where V1 is the initial value, V2 is the final value, and n is the number of years.

Substituting the given values, we get:

[tex]r = ($15,000/$44,000)^{(1/14)} - 1[/tex]

[tex]r = 0.0576[/tex]

So, the annual rate of change (or depreciation) is 0.0576.

B) To express this rate as a percentage, we can multiply it by 100 and add a percent sign:

[tex]r = 0.0576 x 100%[/tex]

[tex]r = 5.76%[/tex]

So, the car depreciated at an annual rate of 5.76%.

C) If we assume that the car value continues to drop by the same percentage, we can use the formula for exponential decay:

[tex]V = V0*(1-r)^t[/tex]

where V0 is the initial value, r is the rate of change, and t is the time elapsed.

Substituting the given values, we get:

[tex]V = $15,000*(1-0.0576)^3[/tex]

[tex]V = $12,527.31[/tex]

Rounding to the nearest 50 dollars, we get:

[tex]value=12,550[/tex]

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Answer:

e. 22 years

Step-by-step explanation:

Lyla pays  $ 24.14 in total.

What is the total amount?

To add two or more numbers and determine the final total, use the sum. A series of any form of numbers, known as addends or summands, is added; the outcome is their sum or total.

Here, we have

Given: Lyla goes out to dinner with her friends where the subtotal is $23.89. A 7% tax and 18% tip are added to the subtotal.

we have to find out how much did she pay in total.

= $23.89 + 7% + 18%

= $23.89 + 7/100 + 18/100

= $23.89 + 0.07 + 0.18

= $ 24.14

Hence, Lyla pays  $ 24.14 in total.

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now, we could use the compound interest equation using a compounding period of 365 for a year with 365 days, so a daily compounding, OR we can just use the continuously compounding equation, which is equivalent.

[tex]~~~~~~ \textit{Continuously Compounding Interest Earned Amount} \\\\ A=Pe^{rt}\qquad \begin{cases} A=\textit{accumulated amount}\dotfill & \$ 5000\\ P=\textit{original amount deposited}\dotfill & \$1000\\ r=rate\to 2\%\to \frac{2}{100}\dotfill &0.02\\ t=years \end{cases} \\\\\\ 5000 = 1000e^{0.02\cdot t} \implies \cfrac{5000}{1000}=e^{0.02t}\implies 5=e^{0.02t} \\\\\\ \log_e(5)=\log_e(e^{0.02t})\implies \log_e(5)=0.02t\implies \ln(5)=0.02t \\\\\\ \cfrac{\ln(5)}{0.02}=t\implies \boxed{80\approx t}[/tex]

We can long divide (24x^3 - 2x^2 - 8x + 30) by (x - 2) as follows:

            24x^2  +  46x  +  86

         _______________________

x - 2   |   24x^3  -  2x^2  - 8x  + 30

        -(24x^3  - 48x^2)

        _______________

                 46x^2 - 8x

                 46x^2 - 92x

                 ___________

                           84x + 30

                           84x - 168

                           ________

                               198

Therefore, (24x^3 - 2x^2 - 8x + 30) ÷ (x - 2) = 24x^2 + 46x + 86 with a remainder of 198/(x - 2).

Answer:

the quotient is 24x² + 46x + 86 and the remainder is 198. Thus, we can write:

(24x³ - 2x² - 8x + 30) ÷ (x - 2) = 24x² + 46x + 86 + 198/(x - 2)

Step-by-step explanation:

We can perform polynomial long division to divide (24x³ - 2x² - 8x + 30) by (x - 2).

24x² + 46x + 86

----------------------

x - 2 | 24x³ - 2x² - 8x + 30

- (24x³ - 48x²)

------------------

46x² - 8x

- (46x² - 92x)

--------------

84x + 30

- (84x - 168)

--------------

198

Therefore, the quotient is 24x² + 46x + 86 and the remainder is 198. Thus, we can write:

(24x³ - 2x² - 8x + 30) ÷ (x - 2) = 24x² + 46x + 86 + 198/(x - 2)

The ball will reach its maximum height after 3.5 seconds.

What is projectile motion?

Projectile motion is the motion of an object that is launched into the air and then moves along a curved path under the influence of gravity.

In projectile motion, the object follows a parabolic trajectory, which is a curve described by a quadratic function. The path of the object depends on the initial velocity, angle of projection, and acceleration due to gravity.

The key characteristics of projectile motion are that the object moves in two dimensions, typically horizontally and vertically, and that the motion in each dimension is independent of the other. This means that the horizontal and vertical components of motion can be analyzed separately.

The height function of the ball thrown The key characteristics of projectile motion are that the object moves in two dimensions, typically horizontally and vertically, and that the motion in each dimension is independent of the other. This means that the horizontal and vertical components of motion can be analyzed separately. is given by h(t) = 112t - 16t^2, where t is the time in seconds and h(t) is the height of the ball at time t.

To find the maximum height, we need to find the vertex of the parabolic function h(t). The vertex of a parabola given by the equation y = ax^2 + bx + c is located at x = -b/2a.

In the case of our height function h(t) = -16t^2 + 112t, the coefficient of the squared term is -16, and the coefficient of the linear term is 112. Therefore, the time at which the ball reaches its maximum height is given by:

t = -b/2a = -112/(2*(-16)) = 3.5

So, the ball will reach its maximum height after 3.5 seconds.

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Answer:

Here are the angles in each quadrant with a common reference angle of 165°:

First quadrant: angle is 15° (subtract 165° from 180°)

Second quadrant: angle is 195° (subtract 165° from 180° and add the result to 180°)

Third quadrant: angle is 195° (subtract 165° from 180° and then subtract the result from 180°)

Fourth quadrant: angle is 195° (subtract 165° from 360°)

The cost of the fence for a rectangular field of 1,500 square yards will be 150k√50 dollars.

Let the length of the rectangular field be L and the width be W. Since the area of the field is 1,500 square yards, we have:

L × W = 1,500

P = 2L + W

C = kP + (1/2)k(L/2)

C = (2k)L + (k/2)W

W = 1500/L

Substituting this into the expression for C, we get:

C = (2k)L + (k/2)(1500/L)

dC/dL = 2k - (k/2)(1500/[tex]L^2[/tex]) = 0

Solving for L, we get:

L = 30√50 yards

Substituting this back into the equation for W, we get:

W = 50/√50 yards

Therefore, the dimensions of the rectangular field that minimize the cost of the fence are L = 30√50 yards and W = 50/√50 yards.

P = 2L + W = 60√50 + 50/√50 yards

C = (2k)L + (k/2)(1500/L) = 90k√50 dollars

k/2 = k/2

Substituting this into the expression for C, we get:

C = 150k√50 dollars

Therefore, to minimize the cost of the fence, the rancher should build a rectangular field with dimensions of L = 30√50 yards and W = 50/√50 yards, and the total cost of the fence will be 150k√50 dollars.

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Answer:1

Step-by-step explanation:

then, x should be smaller than 3

in this canse it is 1;

Answer: 27 hours

Step-by-step explanation:

The E-cοli grοwth rate οf 26% per hοur. At the end οf the experiment there are 7693 E-cοli bacteria. The experiment was dοne last tο 27 hοurs.

Grοwth rates are the percentage changes in a given measure οver a given span οf time. Depending οn whether the size οf the variable is grοwing οr declining οver time, grοwth rates can be either pοsitive οr negative.

Grοwth rates have been used tο study ecοnοmic activity, business management, and financial yields since they were first used by biοlοgists tο study pοpulatiοn sizes.

Using the fοrmula mentiοned belοw-

[tex]A = P(1 + r) \wedge t[/tex]

Grοwth rate (r)= 26%= 26/100= 0.26

Initial vοlume (P)= 15

Final vοlume [tex](A)= 76937693= 15(1+0.26)^t[/tex]

Or, [tex](1.26)^t= 7693/15[/tex]

Or, t ln(1.26)= ln(7693/15)

Or, t≅27

Hence the experiment was dοne at 27 hοurs.

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The difference  between the most expensive and the cheapest material is £1.78​

The term "difference" can also be used more generally to refer to the amount by which two quantities differ.

The concept of difference is closely related to the concept of distance, which refers to the numerical value of the separation between two points, or to prices as in the case we have here.

The difference  between the most expensive and the cheapest material is; £9.77 -  £7.99 =  £1.78​

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the answer is in the attachment below: