The height (in meters) of the arch of a walkway relative to ground level can be modeled by f(x) = -0.05x", 4here * is measured in meters. The function g(x) = R represents the distance (in meters) an engineer plans to raise the walkway. Find and interpret h (x) = / (x) + 8 (x). (3 pts)

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Work Shown:

The notation f(-1) = 30 tells us that (x,y) = (-1,30) is a point on the exponential curve.

Plug in x = -1 and y = 30 and solve for 'a'

y = a*b^x

30 = a*b^(-1)

30 = a*(1/b^1)

30 = a/b

a = 30b

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

Now plug that into the equation to get

y = a*b^x

y = 30b*b^x

y = 30b^1*b^x

y = 30b^(1+x)

Afterward, plug in x = 4 and y = 85 to solve for b.

85 = 30b^(1+4)

85 = 30b^5

b^5 = 85/30

b = (85/30)^(1/5)

b = 1.23157122757057

We can then determine the value of 'a'

a = 30b

a = 30*1.23157122757057

a = 36.947136827117

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

To summarize,

both of which are approximate.

Since we want f(3) to the nearest hundredth, we don't really need to go too overboard with precision. Let's round 'a' and b to 4 decimal places.

We then get

This means

y = a*b^x

turns into

y = 36.9471*(1.2316)^x

As a check, plug in x = -1 to get y = 29.999269; we'll have rounding error since we rounded those 'a' and b values earlier. But when rounding 29.999269 to the nearest hundredth, we get y = 30.

Also, plug x = 4 into the equation to get y = 85.00785875 which rounds to 85.

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

The last step is to plug x = 3 into the equation

y = 36.9471*(1.2316)^x

y = 36.9471*(1.2316)^3

y = 69.0222951885528

y = 69.02 which is the final answer

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

If you were to use those more accurate values of 'a' and b, then x = 3 would lead to y = 69.0175266335801; that rounds to 69.02 when rounding to the nearest hundredth (aka 2 decimal places). So this shows we can stick with the less accurate 'a' and b values while still getting the correct final answer.

GeoGebra and Desmos are great tools to help quickly verify the answer.

Answer:

f(3) = 69.02  (nearest hundredth)

Step-by-step explanation:

An exponential function is used to calculate the exponential growth or decay of a given set of data.  In an exponential function, the variable is the exponent.

[tex]\boxed{\begin{minipage}{9 cm}\underline{General form of an Exponential Function}\\\\$y=ab^x$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the initial value ($y$-intercept). \\ \phantom{ww}$\bullet$ $b$ is the base (growth/decay factor) in decimal form.\\\end{minipage}}[/tex]

Given:

Substitute these values into the general form of an exponential function to create two equations in terms of a and b:

[tex]\begin{aligned}\implies f(-1)&=30\\ab^{-1}&=30\end{aligned}[/tex]

[tex]\begin{aligned}\implies f(4)&=85\\ab^{4}&=85\end{aligned}[/tex]

Divide the second equation by the first equation to eliminate a:

[tex]\begin{aligned}\implies \dfrac{ab^4}{ab^{-1}}&=\dfrac{85}{30}\\\\\dfrac{b^4}{b^{-1}}&=\dfrac{85}{30}\\\\\end{aligned}[/tex]

Solve for b:

[tex]\begin{aligned}\implies \dfrac{b^4}{b^{-1}}&=\dfrac{85}{30}\\\\b^{4-(-1)}&=\dfrac{17}{6}\\\\b^{5}&=\dfrac{17}{6}\\\\b&=\sqrt[5]{\dfrac{17}{6}}\end{aligned}[/tex]

Substitute the found value of b into ab⁴ = 85:

[tex]\implies a\left(\sqrt[5]{\dfrac{17}{6}}\right)^4=85[/tex]

Solve for a:

[tex]\implies a=\dfrac{85}{\left(\sqrt[5]{\dfrac{17}{6}}\right)^4}[/tex]

[tex]\implies a=\dfrac{85}{\sqrt[5]{\dfrac{17^4}{6^4}}}[/tex]

[tex]\implies a=\dfrac{85}{\sqrt[5]{\dfrac{83521}{1296}}}[/tex]

Substitute the found values of a and b into the exponential function formula to create an exponential equation for the given parameters:

[tex]f(x)=\left(\dfrac{85}{\sqrt[5]{\dfrac{83521}{1296}}}\right)\cdot \left(\sqrt[5]{\dfrac{17}{6}}\right)^x[/tex]

To find the value of f(3), substitute x = 3 into the equation:

[tex]\begin{aligned}\implies f(3)&=\left(\dfrac{85}{\sqrt[5]{\dfrac{83521}{1296}}}\right)\cdot \left(\sqrt[5]{\dfrac{17}{6}}\right)^3\\\\&=69.0175266...\\\\&=69.02\; \;\sf (nearest\;hundredth)\end{aligned}[/tex]

Therefore, the value of f(3) to the nearest hundredth is 69.02.

Answer:

2 ( 10 + 3 )

Step-by-step explanation:

The given expression equivalent to (5−4i)(3−2i)(5−4i)(3−2i) is -231 + 372i.

What is expression ?

In mathematics, an expression is a combination of numbers, variables, and operations that represents a value or a quantity. It can be a simple expression consisting of only one number or variable, or a complex expression consisting of several numbers, variables, and operations.

We can use the distributive property of multiplication to expand the given expression as follows:

(5−4i)(3−2i)(5−4i)(3−2i)

= (5−4i)(5−4i)(3−2i)(3−2i)

= (25 - 40i + 16i²)(9 - 12i + 4i²)

= (25 - 40i - 16)(9 - 12i - 4)

= (-231 + 372i)

Therefore, the given expression equivalent to (5−4i)(3−2i)(5−4i)(3−2i) is -231 + 372i.

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Answer:

72; Proofs attached to answer

Step-by-step explanation:

Proofs attached to answer

Answer:

The price of a hot dog is $2.00

Answer:

The 2nd graph

Step-by-step explanation:

We know

The solution is (0,5)

Looking at the options, we only see graph 2 has two lines intersecting at the solution point (0,5). So, the answer is graph 2.

Use the cookies to teach basic fractions, such as one-half or one-third of a cookie.

What is Activity ?

An activity is a task, exercise, or action that is carried out for a specific purpose or goal. It can be an organized event or a spontaneous action that requires some level of effort or engagement from participants.

Begin by gathering the Grade R learners together and introduce the theme "Sharing Treats with Friends" and the concept of sharing with remainders. Explain to the learners that sometimes when we share treats with our friends, we might not have enough for everyone to have an equal amount.

Place the bag of cookies on the plate and ask the learners to estimate how many cookies are in the bag. Write their guesses on the board or chart paper.

Open the bag of cookies and count how many there are. Record the actual number of cookies on the board or chart paper.

Explain that the learners will be sharing the cookies with their friends, but they must make sure that everyone gets the same number of cookies. Divide the class into small groups of three or four learners.

Instruct each group to count the number of cookies in the bag and divide them equally among the members of the group. If there are any remaining cookies, they should be placed on the plate.

Ask the learners to share their results with the class. Some groups may have exactly the same number of cookies, while others may have remainders. Write their results on the board or chart paper.

Discuss with the learners how they can share the remaining cookies equally among their group members. For example, they could break the remaining cookies into smaller pieces or take turns sharing the last cookie.

Give the learners a chance to practice sharing with remainders by repeating the activity with different amounts of cookies and different group sizes.

As a conclusion, ask the learners to reflect on the activity and discuss how it felt to share with their friends. Encourage them to think about why sharing is important and how it can make everyone feel happy.

Optional extension activities:

Have the learners draw pictures or write stories about sharing treats with their friends.

Therefore, Use the cookies to teach basic fractions, such as one-half or one-third of a cookie.

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The linear model would be a reasonable way to predict the temperature as temperature determines the number of ice cream that would be sold.

Temperature is a unit of hotness and coldness that can be described in terms of any number of arbitrary scales. It also represents the direction wherein heat energy will naturally flow, from either a hotter (i.e., higher temperature) body to a colder body.

Temperature is not the same as the energy of such a thermodynamic system; for instance, an iceberg has a significantly larger total heat energy than a match, despite the fact that a match is burning at a significantly higher temperature. The linear model would be a reasonable way to predict the temperature as temperature determines the number of ice cream that would be sold.

Therefore, the linear model would be a reasonable way to predict the temperature as temperature determines the number of ice cream that would be sold.

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The volume of the figure is 93π

From the question, we have the following parameters that can be used in our computation:

The composite figure

The volume is calculated as

Volume = Cone + Cylinder

Using the above as a guide, we have the following:

Volume = 1/3πr²(H - h) + πr²h

Substitute the known values in the above equation, so, we have the following representation

Volume = 1/3π * 3² * (15 - 8) + π * 3² * 8

Evaluate

Volume = 93π

Hence, the volume is 93π

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An arithmetic sequence with a common difference d = -3 include the following: D. 43,40,37,34,31.

Mathematically, the nth term of an arithmetic sequence can be calculated by using this expression:

aₙ =  a₁ + (n - 1)d

Where:

Next, we would determine the common difference as follows.

Common difference, d = a₂ - a₁

Common difference, d = 40 - 43

Common difference, d = -3

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Answer:

-csc^2 θ = dθ/dt = 1/5 (dx/dt)

or

3.49 km/min

Step-by-step explanation:

dx/dt

- 5cosec^2 θ * (rate of change of angle of elevation)

- 5 cosec^2 (π/3) * (-π/6)

(5π/6) * (4/3)

= 10π/9 km/min. = 3.49 km/min

Answer:

20=2g

g=10

20/2=10

g=10

The number of touchdowns scored by this professional football team could be: C. 10.

In Mathematics, a percentage simply refers to any number that is expressed as a fraction of hundred (100). This ultimately implies that, a percentage indicates the hundredth parts of any given number.

In order to determine the number of points scored by the defense of this professional football team, we would use the following mathematical expression:

Number of points = 24/100 × 242

Number of points = 0.24 × 242

Number of points = 58.08

For the number of touchdowns, we have:

Number of touchdowns = 58.08/6

Number of touchdowns = 9.68 touchdowns.

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Answer:

2289.12

Step-by-step explanation:

Answer:

2289.12

Step-by-step explanation:

the sum of 897.23 and 39.89 is 937.12

937.12+1352=2289.12

Answer:

[tex]3[/tex]×[tex]5[/tex][tex]=15[/tex]

[tex]15+1=16[/tex]

The required number line shows that the four numbers between 10 and 20 that have a remainder of 2 when divided by 3.

In math, a number line can be defined as a straight line with numbers arranged at equal segments or intervals throughout. A number line is typically shown horizontally and can be extended indefinitely in any direction.

To show the numbers between 10 and 20 that have a remainder of 2 when divided by 3, we need to start with the first number in that range that has a remainder of 2 when divided by 3, which is 11. Then, we can add 3 to that number to get the next number that has a remainder of 2 when divided by 3, which is 14.

We can continue adding 3 to each previous number to get the rest of the numbers that meet this condition:

11, 14, 17, 20

So, these are the four numbers between 10 and 20 that have a remainder of 2 when divided by 3. We can plot them on the number line as shown.

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The scaling factors need not be the same, and this transformation scales the x-axis by a different factor than the y-axis, resulting in a non-uniform scaling.

What is Transformation axes ?

Transformation axes refer to the new coordinate axes after a transformation is applied to a given set of coordinates or points.

The transformation (x, y) → (2x, 3y) is a linear transformation that stretches the x-coordinate by a factor of 2 and the y-coordinate by a factor of 3, while keeping the origin fixed.

This transformation is an example of a scaling transformation, where the scale factor along the x-axis is 2 and along the y-axis is 3.

Visually, this transformation will stretch out the graph horizontally and vertically, making objects appear larger or smaller depending on their position relative to the x and y axes.

Therefore, The scaling factors need not be the same, and this transformation scales the x-axis by a different factor than the y-axis, resulting in a non-uniform scaling.

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The requried measures of angles a, b, c, and d are 97°, 15°, 68°, and 68° respectively.

Orientation of the one line with respect to the horizontal or other respective line is known as a measure of orientation and this measure is known as the angle.

Here,
From the figure,
The calculation of the unknown angles is as follows,
112° and ∠d are supplementary angles.

So,
112 + ∠d = 180
∠d = 180 - 112
∠d = 68°

97° and angle a are corresponding angles, so
∠a = 97°.

112 and ∠c are also supplementary angles.
112 + ∠c = 180
∠c = 68°

Now,
∠a + ∠b  = 112
∠a + 97 = 112
∠a = 15°

Thus, the requried measure of angles a, b, c, and d are 97°, 15°, 68° and 68°.

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Answer:

Bases

Step-by-step explanation:

Therefore, the stamp's estimated value is $21.54 considering that when he first got it, it was only worth $14.50.

A mathematical assertion that two expressions are equivalent is known as an equation. The equal symbol (=) separates the two sides, which are labeled as the left-hand side (LHS) and the right-hand side (RHS). The equation must have identical values on both sides, and it is frequently used to determine an unknown variable's value.

Numerous mathematical processes, such as addition, subtraction, multiplication, division, exponents, logarithms, and more, can be used in equations.

given

a. The formula for the stamp's worth V after t years is as follows:

[tex]V = 14.50(1 + 0.02)^t[/tex]

where t is the period of time since Leroy got the stamp, and 0.02 denotes a 2% annual increase in value.

b. We enter t = 18 into the equation to estimate the stamp's value after 18 years:

[tex]V = 14.50(1 + 0.02)^{18[/tex]

[tex]V = 14.50(1.02) (1.02)^{18[/tex]

V = 14.50(1.485) (1.485)

V = 21.54

Therefore, the stamp's estimated value is $21.54 considering that when he first got it, it was only worth $14.50.

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Answer:

The volume V of a rectangular prism is given by the formula V = lwh, where l, w, and h are the length, width, and height, respectively.

In this case, we have l = 2, w = 7/3, and h = 2. Therefore, the volume is:

V = (2)(7/3)(2)

V = 28/3

V = 9.33 (rounded to two decimal places)

So the volume of the rectangular prism is approximately 9.33 cubic units.

Answer:

The practical domain of the function is the set of integers from 0 to 8, inclusive, which is option B.

Step-by-step explanation:

The practical domain of the function is limited by the given constraints: Joe can purchase up to 8 boxes of hot dogs, and he cannot purchase partial boxes. Therefore, the number of boxes he can purchase is a whole number between 0 and 8, inclusive.

Substituting the values from 0 to 8 into the function, we get:

f(0) = 48(0) + 20 = 20

f(1) = 48(1) + 20 = 68

f(2) = 48(2) + 20 = 116

f(3) = 48(3) + 20 = 164

f(4) = 48(4) + 20 = 212

f(5) = 48(5) + 20 = 260

f(6) = 48(6) + 20 = 308

f(7) = 48(7) + 20 = 356

f(8) = 48(8) + 20 = 404

Therefore, the practical domain of the function is the set of integers from 0 to 8, inclusive, which is option B.

The Simple Interest is $10

Simple Interest is  the method of calculating the interest amount for some principal amount of money. It is also finding the product of the principal amount borrowed or lent, the rate of interest and the term or repayment period of the loan.

Where the Principal is the borrowed money which is given for a specific period.

Simple Interest = P * R * T/100

Where P = Principal = $1000

R = Rate = 2%

T = Time = 6 months

When 6 months converted to year = 6/12 = 1/2 = 0.5 year

SI = 1000 * 2 * 0.5/100

SI = 1000/100

SI = $10

Therefore, the Simple Interest is $10

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Answer:

Step-by-step explanation:

Simple Interest formula = P * R * T/100

p  = $1000

r = 2%

t = 6 months (0.5 year)

so, SI = 1000 * 2 * 0.5/100

SI = $10

hope this helps :))

The standard deviation is 0.67 metric tonnes, and the mean of the sampling distribution of the sample mean impact force is 30 metric tonnes.

What is the mean and standard deviation of the sampling distribution of the sample mean impact force?

a. The following formulas can be used to determine the mean and standard deviation of the sampling distribution of the sample mean impact force:

30 metric tonnes are the population's mean and the sampling distribution's mean.

Standard deviation of the sample distribution is equal to the square root of the population's standard deviation.

= [tex]2 / sqrt(9) (9)[/tex]

= 0.67 tonnes in metric

As a result, the standard deviation is 0.67 metric tonnes, and the mean of the sampling distribution of the sample mean impact force is 30 metric tonnes.

What is the probability of a sample mean impact force being less than 28.7 metric tons?

b. Using the z-score calculation and common normal distribution tables, we can determine the likelihood that the sample mean impact force will be less than 28.7 metric tonnes. The z-score equation is:

(Sample Mean - Population Mean) = Z (standard deviation of sampling distribution)

= (28.7 - 30) / 0.67 = -2

We discover that the probability of a z-score less than -2 is 0.0228 using the usual normal distribution table. Consequently, 0.0228 or 2.28% of the time, the sample mean impact force will be less than 28.7 metric tonnes.

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Step-by-step explanation:

factors of 14: 1, 14, 2, 7

factors of 22: 1, 22, 2, 11

factors of 30: 1, 30, 2, 15, 3, 10, 5, 6

factors of 25: 1, 25, 5, 5

Answer:

Factors 14 : 1, 14, 2, 7
Factors 22 : 1, 22, 2, 11
Factors 30 : 1, 30, 2, 15, 3, 10, 5, 6
Factors 25 : 1, 25, 5

In response to the query, we have that Hence, the following are the answers to this equation: 5x - 2 = 0 or x - 1 = 0 Thus, x = 2/5 or x = 1.

A quadratic equation in a single variable is x ax2+bx+c=0, a form of quadratic equation. a 0. The fact that this polynomial is of second sample guarantees that it comprises at least one solution according to the Basic Theorem of Algebra. Solutions might be straightforward or difficult. A quadratic equation is one that has four variables. This suggests that at least single word in it has to be rounded. The formula "ax2 + bx + c = 0" constitutes one of the widely used solutions for complex numbers. where the undefined variable "X" is represented by the numerical values or constants a, b, and c.

2x² - 9x - 18 = 0

Secondly, we must identify two integers whose total is -9 and whose product is 2(-18)=-36. These are the numbers -12 and +3. Hence, we may say:

2x² - 9x - 18 = (2x + 3)(x - 6) = 0

Hence, the following are the answers to this equation:

2x + 3 = 0 or x - 6 = 0

Thus, x = -3/2 or 6.

8x² + 2x - 3 = 0

8x² + 2x - 3 = (4x - 3) (4x - 3)

(2x + 1) = 0

Hence, the following are the answers to this equation:

4x - 3 = 0 or 2x + 1 = 0

Thus, either x = 3/4 or x = -1/2.

5x² - 11x + 2 = 0

5x² - 11x + 2 = (5x - 2)(x - 1) = 0

Hence, the following are the answers to this equation:

5x - 2 = 0 or x - 1 = 0

Thus, x = 2/5 or x = 1.

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Answer: To find how much greater the beetle's length is than its width, we need to subtract the width from the length:

Length - Width

Substituting the given values, we get:

2/5 inch - 1/5 inch

Simplifying the expression by finding the common denominator of 5, we get:

(2/5 - 1/5) inch

= 1/5 inch

Therefore, the beetle's length is 1/5 inch greater than its width.

Step-by-step explanation:

Therefore , the solution of the given problem of function comes out to be the ball can  reach a height of 74 feet, to the closest foot.

The study of mathematics includes topics such as numbers, symbols, and their component components, as well as anatomy, building, and that both true and fictitious geographic locations. A function describes the relationships between different components that all lead to the same result. A function is made up of several variable  distinct components that, when placed together, produce specific outputs for every input.

Here,

The starting height (h0) is 4 feet, and the initial speed (v0) is 75 feet per second.

The ball's vertical movement is modelled by the formula h(t) = 16t2 + v0t + h0.

When we change the numbers, we obtain:

h(t) = −16t2 + 75t + 4

The vertex of the parabolic function marks the height of the spheroid at its greatest height.

The vertex's t-value is determined by:

t = −v0/2a

where a represents the t2 component in the h-function (t).

When we change the numbers, we obtain:

t = −75/(2(−16)) = 2.34375

2.34375 seconds pass after the ball is thrown before it hits its highest point.

Given is the utmost height:

h(2.34375) = −16(2.34375) (2.34375)

2 + 75(2.34375) + 4 = 74 feet

The ball can therefore reach a height of 74 feet, to the closest foot.

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The following function won't produce the intended composite function:

f(x) = 8/x + 4

g(x) = 1/x²

A method or connection known as a function connects each element of a non-empty set A to at least one element of a second non-empty set B. A function in mathematics is a relation f between a set A (the function's domain) and a different set B (the function's co-domain).

In the given question,

When f(x) = 8/x + 4 and g(x) = 1/x²

Then (f o g(x)) = 8/1/x² + 4

= 8x² + 4

As a result, the following function would not produce the intended composite function:

f(x) = 8/x + 4

g(x) = 1/x²

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