1. Antonio sets his remote-controlled car at the start of the racetrack and starts its run on the track. His car stops 10m past the starting line. After he fixes a wheel, the car goes 2/5 the racetrack’s total distance and then stops for good. It is 38m from the starting line. How long is the racetrack? (a) Let x = the length of the racetrack. Write an equation that fits the story (b) Solve the equation from Part a. (How long is the racetrack?) Show your work. (c) How much farther did Antonio’s car have to go to the finish line? Show your work.

The solution of the function is as follows:

(f o g)(x) = 16x² + 17x + 1

(g o f)(x) = 4x² + 36x + 1

Composite functions are when the output of one function is used as the input of another.

Therefore, let's solve the composite functions as follows:

f(x) = x² + 9x

g(x) = 4x + 1

Therefore,

(f o g)(x) = f(g(x))

Therefore,

f(g(x)) = (4x + 1)² + 9x = (4x + 1)(4x + 1) + 9x = 16x² + 4x + 4x + 1 + 9x = 16x² + 17x + 1

(g o f)(x) = g(f(x))

g(f(x)) = 4(x² + 9x) + 1 = 4x² + 36x + 1

Therefore,

(f o g)(x) = 16x² + 17x + 1

(g o f)(x) = 4x² + 36x + 1

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The value of x is given as follows:

x = 18.

Similar triangles are triangles that share these following two features:

The similar triangles for this problem are listed as follows:

ABD and EDC.

The proportional side lengths are given as follows:

Hence the value of x is obtained as follows:

x/6 = 12/4

x/6 = 3.

x = 18.

(using the proportional relationship for the equivalent side lengths of the similar triangles).

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There are 6 goats and 17 ducks present in the farmhouse

Let x represent the number of goats

Let y represent the number of ducks

x + y= 23....equation1

Goats have 4 legs and ducks have 2 legs

4x + 2y= 58.......equation 2

From equation 1

x + y= 23

x= 23-y

Substitute 23-y for x in equation 2

4(23-y) + 2y = 58

92- 4y + 2y= 58

92-2y= 58

-2y= 58-92

-2y= -34

y- 34/2

y= 17

Substitute 17 for y in equation 1

x + y= 23

x + 17= 23

x= 23-17

x= 6

Hence there are 6 goats and 17 ducks in the farmhouse shelter

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Answer:

1,000

Step-by-step explanation:

Formula: An=8/5x5(4)

The quotient when 2x² + 4x – 14 is divided by 2x is [tex]x + 2 - \frac{7}{x}[/tex] .  

When two numbers are divided by each other, the result is known as a quotient. For instance, when we divide 8 by 2, the result will be 2, where 2 is the quotient.

As we know division is a mathematical operation that involves dividing items into equal groups. It is represented by the symbol (÷). When we divide a number, the result we ultimately get is the quotient.

Here we have

2x² + 4x – 14 is divided by 2x  

=> [tex](2x^{2} + 4x - 14) \div {2x}[/tex]  

=>  [tex]\frac{(2x^{2} + 4x -14)}{2x}[/tex]  

Take 2x as common

=>  [tex]\frac{2x(x + 2 - \frac{7}{x} )}{2x}[/tex]    

=>  [tex]x + 2 - \frac{7}{x}[/tex]    

Therefore,

The quotient when 2x² + 4x – 14 is divided by 2x is [tex]x + 2 - \frac{7}{x}[/tex] .  

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The function that represents how many more texts that were sent this week than last week is: B. (f − g)(x) = 3x + 16.

To find the value of a given functional expression, we have to apply the basic knowledge of how to solve algebraic expressions.

For example, if we are given that (f - g)(x), it means we are to subtract the function g(x) from f(x). That is (f - g)(x) = f(x) - g(x).

Given the following functions:

Number of texts sent this week is represented as the function, f(x) = 8x + 14

Number of texts sent last week is represented as the function, g(x) = 5x - 2.

Number of texts that were sent this week than last week = (f - g)(x)

(f - g)(x) = (8x + 14) - (5x - 2)

(f - g)(x) = 8x + 14 - 5x + 2

(f - g)(x) = 3x + 16

Therefore, the answer is: (f − g)(x) = 3x + 16.

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Therefore, the answer is: (f − g)(x) = 3x + 16.

The common ratio for the sequence is 1/2.

Sets of numbers that adhere to a pattern or norm are known as number sequences. A geometric sequence is one where the rule is to multiply or divide by the same number each time.

Formula : [tex]a_n[/tex] = a[tex]r^{n-1[/tex]

where a is the first term, r is the common ratio.

Given:

Sequence: 14, 7, 7/2, 7/4

To calculate the common ratio = [tex]a_1/ a_2[/tex]

Now, the common ratio

= 7/14

= 1/2

Hence, the common ratio is 1/2.

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The property of equality that was used to solve the equation is the division property of equality

From the question, we are to determine the property of equality that was used to solve the given equation

From the given information,

The equation is

-5x = 4

and

The solution is

(- 5x)/- 5 = 4/- 5

x = - 4/5

From the solution step, we can observe that both sides of the equation were divided by -5.

Thus,

The property of equality that was used is the division property of equality

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Answer: y = 3x - 8

Step-by-step explanation: Given slope m = 3, and point x = 3, y = 1
y -1 = 3(x-3)
y = 3x - 9 + 1
y = 3x - 8

The cost of the class would be $740 x 3 = $2220. If you were to work instead, at $10 per hour for 10 hours per week for 15 weeks, you would earn $1500. So, the net cost of the class would be $2220 - $1500 = $720.

In general, a college degree can lead to higher earning potential and greater career opportunities, so in most cases, it can be considered a worthwhile investment. However, the decision to take a particular class should be based on individual circumstances and goals. If the class is relevant to your future career aspirations and you can afford the cost, it may be a good investment. However, if the class is not directly related to your career goals, or you are not able to afford the cost, it may not be worth it.

Answer:

Triangle VWX and triangle IGH are congruent

Step-by-step explanation:

Answer:

2.14 rounded ............

The solution to the inequality is z > 14

Mathematical expressions with inequalities include those in which the two sides are not equivalent. Contrary to equations, we assess and contrast two values in inequality. The inequality signs that are used in place of the equal signs are:

Given:

-6 > z/-2 + 1

Rewrite so "z" is on the left side of the inequality.

z/-2 + 1 <  -6

Move the negative in front of the fraction.

-z/2 + 1 < -6

Move all terms not containing z to the right side of the inequality.

-z/2 < - 6 - 1

-z/2 < - 7

Multiply both sides by 2

z < -14

Interchange the sign since z cannot be negative;

z > 14

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Answer:

x=0

y=0

z=0

Step-by-step explanation:

Answer:

So, Hypotenuse =  unit

Step-by-step explanation:

Answer: (-2,4) x (0,9)=36 simplified

y= 5/2x + 9 is the slope

Step-by-step explanation:

.

Answer: its -1

Step-by-step explanation:

[tex]30 + 60[/tex]

I am pretty sure I'm wrong

[tex]30 \div 60[/tex]

[tex]30 \times 60[/tex]

[tex]30 - 60[/tex]

Answer:

(115 x 5) / 100 = 0.575 grams of the pure drug.

Step-by-step explanation:

To make a 5% solution, the concentration of the pure drug is 5 grams per 100 mL of solution.

To find the amount of pure drug needed for a specific volume, such as 115 mL, you can use the formula:

(volume of solution x concentration of pure drug) / 100

So in this case:

(115 x 5) / 100 = 0.575 grams of the pure drug.

This calculation shows that 0.575 grams of the pure drug would be needed to make 115 mL of a 5% solution.

Answer:

5.75 g

Step-by-step explanation:

First calculate 5% of the total amount of solution, 115 ml:

[tex]\implies \sf 5\%\;of\;115\;ml=\dfrac{5}{100} \times 115=5.75\;ml[/tex]

As we have not been told any information about the drug, assume the standard equivalent that 1 ml equals 1 gram.  Therefore, the amount of pure drug needed to make 115 ml of solution at the 5% strength is 5.75 g.

Nya's average speed was 10 miles per hour.

What is the speed, distance, and time?

Speed, distance, and time are related concepts in physics that are used to measure and describe motion.

Speed is a measure of how fast an object is moving. It is typically measured in units of distance per unit of time, such as miles per hour or kilometers per hour. Speed can be calculated using the formula:

Speed = Distance / Time

To find Nya's average speed in miles per hour, we need to divide the distance she traveled by the time it took her to travel that distance.

Average speed = distance/time

We know that Nya rode her bike 2 ½ miles in ¼ hour. To use these measurements in the formula, we need to convert the units to the same unit of measurement. In this case, we need to convert the distance to miles and the time to hours.

Average speed = (2.5 miles) / (0.25 hour)

To find out the speed in miles per hour, you need to multiply by the conversion factor 1 hour/60 minutes, so

Average speed = (2.5 miles) / (0.25 hour) * (1 hour/60 minutes)

Average speed = (2.5 miles) / (0.25 hour) * (4)

Average speed = 10 miles per hour.

Hence, Nya's average speed was 10 miles per hour.

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1)  A sequence of real numbers is an ordered list of real numbers where each number in the list is assigned a specific position or index, usually denoted by n. The sequence is denoted by {xn} n>n0, where xn represents the nth term of the sequence and n ranges from 0 to infinity.

2)An example of a sequence of rational numbers is {2, 1.5, -0.5, -1.25, ...}, where each term in the sequence is a rational number.

3) The recursive definition of the summation of ai is a mathematical formula that defines the nth term of the summation in terms of the previous terms. It is typically denoted as:

∑(i=1)^ni = a1 + (a2 + (a3 + ....(an-1 + an)...)),

Where a1 is the first term of the summation, a2 is the second term, and so on and so forth. This recursive definition is used to calculate the nth term of the summation by adding the previous term to the current term. It is generally used in mathematical induction and recursion.

4)The associative property of summation notation states that the order in which the summation is performed does not affect the final result. In other words, for any three indices a, b, and c, and any sequence of numbers {xn}, the following equation holds:

∑(a=b)c xn = ∑(a=c)b xn = ∑(a=b)c xn

This property allows us to group the terms in a summation in different ways without changing the final result.

5) The translation invariance property of summation notation states that the value of a summation is unchanged when the index of summation is translated (shifted) by a constant value. In other words, for any indices a and b, any constant k, and any sequence of numbers {xn}, the following equation holds:

∑(a=b)xn = ∑(a+k=b+k)xn

This property allows us to shift the index of summation without affecting the final result of the summation.

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Answer:

a = 3.31ft.

Step-by-step explanation:

The opposite is 3.31 feet.

Please let me know if this helped!!!

The solution to the equation -9 = 6m + 15 is m = -4.

Given the equation in the question;

-9 = 6m + 15

Rewrite the equation as 6m + 15 = -9

Move all terms not containing m to the right side of the equation by subtracting 15 from both sides

6m + 15 - 15 = -9 - 15

6m = -9 - 15

Add -9 and -15

6m = -24

Divide both sides by 6

6m/6 = -24/6

m = -24/6

m = -4

Solution check

Plug m = -4 into the equation and check

-9 = 6m + 15

-9 = 6( -4 ) + 15

-9 = -24 + 15

-9 = -9

Correct.

Therefore, the numerical value of m is -4.

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Using substitution method to solve the system of linear equations, the granola and drinks cost $1.25 and $2.5 respectively while the modelled system of linear equation is attached below

A grouping of two or more linear equations that share the same set of variables is known as a system of linear equations. When a, b, and c are constants and x and y are variables, the equations can be written as ax + by = c. Elimination, substitution, and graphing are just a few of the techniques that can be used to solve these problems.

In this given problem, we have to define our variables and then set our system of equations.

a)

Let;

2x + 5y = 15 ...eq(i)

4x + 3y = 12.50 ...eq(ii)

b)

Using substitution method;

From equ(i)

2x + 5y = 15

2x = 15 - 5y

x = (15 - 5y) / 2 ...eq(iii)

Put eq(iii) into eq(ii)

4x + 3y = 12.50

4(15 - 5y) /2 + 3y = 12.50

y = 2.5

Put y = 2.5 into eq(i)

2x + 5(2.5) = 15

x = 1.25

c)

Granola = 1.0

drinks = 3.0

The system of linear equations will be

x + y = 4

2x + 3y = 11

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The answers to each part is mentioned below..

The area of a square is -

Area = Side x Side

A = S²

Given is that the function {A} gives you the area in square inches of a square with side length {X} inches

We can complete the table as -

[x] :     0     1      2      3      4     5     6

A[x]  :  (0)²    (1)²   (2)²   (3)²  (4)²  (5)²  (6)²

A[x]  :   0       1       4      9     16    25   36

We can Represent the function A using the equation as -

A(x) = x²

The graph of the function is attached.

Therefore, the answers to each part is mentioned above.

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On solving the provided question, we can say that by the property of the triangle, if a line made from 2 mid points of a triangle the length of line of 2 mid points = 2 times the base ; AB = 80m

A triangle is a polygon since it has three sides and three vertices. It is one of the basic geometric shapes. The name given to a triangle containing the vertices A, B, and C is Triangle ABC. A unique plane and triangle in Euclidean geometry are discovered when the three points are not collinear. Three sides and three corners define a triangle as a polygon. The triangle's corners are defined as the locations where the three sides converge. 180 degrees is the result of multiplying three triangle angles.

here,

by the property of the triangle, if a line made from 2 mid points of a triangle the length of line of 2 mid points = 2 times the base

so DC = 2*AB

AB = 80m

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The function f(x)=⌊x⌋−3 in interval [−3,3] is represented by following graph:

The correct option is D.

The floor function floor(x) is defined as the function that gives the highest integer less than or equal to x.

Given function

f(x)=⌊x⌋−3 in interval [−3,3]

Above function is floor function.

f(x) = ⌊x⌋−3

f(3) = 3 - 3

f(3) = 0

f(2) = 2 - 3

f(2) = -1

f(1) = 1 - 3

f(1) = -2

f(0) = 0 - 3

f(0) = -3

f(-1) = -1 -3

f(-1) = -4

f(-2) = -2 -3

f(-2) = -5

f(-3) = -3 -3

f(-3) = -6

Plotting above points the graph we get is as follows:

Hence, Graph that represents the function f(x)=⌊x⌋−3 in interval  [−3,3] is below.

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Answer:

31.80

Step-by-step explanation:

20.53

× 1.55=

31.82

-.02

32.80

Answer: The missing value is 86

Step-by-step explanation: The mean of a data set is the same as the average. The formula for mean is the sum of the data values/(the number of data values). Right now there are only 6 data values and we are looking for a seventh. If we sum the 6 current values and divide by 6, we get a data set with a mean of 96.5. We need a lower mean. The data set we need has 7 points so the sum of all 7 points should be 7(95) since if we divide 7(95) by the number of points which is 7, we get 95 which is the mean we want. 7(95) is 665. We can subtract the sum of the current 6 points from 665 to find the value we want.