The table below shows function r, which represents the total cost of a field trip to the science museum, in dollars, if n students return permission slips to attend. Which function represents the situation?

A.
r(n) = 15n + 55
B.
r(n) = 55n + 15
C.
r(n) = 30n - 20
D.
r(n) = 5n + 45

Answer:

$16.50

Step-by-step explanation:

If Jason spends $3.75 and Maya spends three times that amount, Maya spends 3.75(3) or $11.25. If Tia spends $5.25 more than Maya, Tia spends 11.25 + 5.25 or $16.50.

Answer:

$41.50

Step-by-step explanation:

He would have to give her $41.50 in order for them to have an equal amount of money, which would be $51.50.

Answer:

He should give her $41.50

Step-by-step explanation:

Answer:

f(x) = 70 + 2x

Step-by-step explanation:

In this problem, we first need to consider how much we get per day.

$70 is our constant.

$2 will be dependent on the number of books sold.

Here 'x' will represent the number of books sold.

f(x) = 70 + 2x

Now let's try it out.

Let's say we sold 0 books

f(0) = 70 + 2(0) = 70

This shows us that we only get our constant pay.

Now let's try for 1 or more books.

f(1) = 70 + 2(1) = 72

f(2) = 70 + 2(2) = 74

So the recursive formula f(x) = 70 + 2x is a good formula to model the situation.

Answer:

a1 = 70

an = an–1 + 2

Step-by-step explanation:

Each friend should get 1/5 of a bag of trail mix. Since there are three bags of trail mix, you would add those into it. Hope this helps!

Answer:

Step-by-step explanation:

(a) The Extreme Value Theorem.

(b)  We would differentiate the function and equate this to zero. The zeroes of the function will give us the values of the maxima / minima and we can find find the absolute maxima/minima from the results. Note we might have  multiple relative maxima/ minima  but only one absolute maximum and one absolute minimum.

Final answer:

The Extreme Value Theorem guarantees that a continuous function on a closed interval has an absolute maximum and minimum. To find these, one calculates the derivative to find critical points, analyzes the derivative's sign around these points, and evaluates the function at the critical points and the interval's endpoints.

Explanation:

Extreme Value Theorem and Finding Maximum and Minimum Values

The theorem that guarantees the existence of both an absolute maximum and minimum value for a continuous function defined on a closed interval a, b is known as the Extreme Value Theorem. This theorem plays a crucial role in calculus and mathematical analysis and is fundamental in understanding the behavior of continuous functions on closed intervals.

To find these maximum and minimum values, one would typically follow these steps:

Calculate f'(x), the derivative of the function f(x), to find the critical points.

Analyze the sign of f'(x) around the critical points to determine if they are local minima, local maxima, or saddle points.

Evaluate the function f(x) at each critical point as well as the endpoints of the interval [a, b] to determine the absolute extrema.

Moreover, if a function satisfies the criteria of being continuous on [a, b] and differentiable on (a, b), then by a related theorem called the Mean Value Theorem, there exists at least one c in (a, b) where f'(c) = 0.

These methods form the standard procedure for finding the extremal values that a continuous function may possess on a closed interval.

Answer:

Substitute the slope  and the coordinates of point P in the equation of the line  y=mx+b and then solve for b in each equation

Step-by-step explanation:

we know that

The first step is calculate the slopes of these two lines. Remember that if two lines are parallel then the slopes are the same (m1=m2) and if two lines are perpendicular then the slopes is equal to m1*m2=-1

The second step is substitute the slope m2 and the coordinates of point P in the equation of the line in slope-intercept form y=mx+b and then solve for b in each equation

Answer:

Step-by-step explanation: See answer below

Answer:

Final answer is [tex]x=0[/tex] and [tex]x=2\pi[/tex].

Step-by-step explanation:

Given equation is [tex]3\cdot\sec\left(x\right)-2=1[/tex]

Now we need to find the solution of  [tex]3\cdot\sec\left(x\right)-2=1[/tex] in given interval [tex][0, 2\pi ][/tex].

[tex]3\cdot\sec\left(x\right)-2=1[/tex]

[tex]3\cdot\sec\left(x\right)=1+2[/tex]

[tex]3\cdot\sec\left(x\right)=3[/tex]

[tex]\frac{3\cdot\sec\left(x\right)}{3}=\frac{3}{3}[/tex]

[tex]\sec\left(x\right)=1[/tex]

which gives [tex]x=0[/tex] and [tex]x=2\pi[/tex] in the given interval.

Hence final answer is [tex]x=0[/tex] and [tex]x=2\pi[/tex].

Answer:

x  = 0 and x = 2π

Step-by-step explanation:

We have given the equation.

3sec x -2 = 1

We have to solve it  interval [0,2pi].

3sec x -2 = 1

3secx = 1+2

3secx = 3

secx = 1

x= sec⁻¹(1)

x  = 0 and x = 2π is the answer in this interval.

if we distribute 2(2x+5), we'll end up with 4x + 10, so in short, the right-hand-side, is really the left-hand-side in disguise.

and since  4x + 10 = 4x + 10, both equations are equal, meaning in short, the system has an infinite number of solutions.

The width of the rectangle is 5 feet, its length is 20 feet. Thus, the perimeter is 50 feet.

To solve this problem, let's denote:

- Width of the rectangle as  w  (in feet)

- Length of the rectangle as 4w  (since it's four times the width)

Given that the area of the rectangle is 100 square feet, we can set up the equation for the area:

[tex]\[ \text{Area} = \text{Length} \times \text{Width} \][/tex]

[tex]\[ 100 = (4w) \times w \][/tex]

[tex]\[ 100 = 4w^2 \][/tex]

Now, let's solve for ( w ):

[tex]\[ 4w^2 = 100 \][/tex]

Divide both sides by 4:

[tex]\[ w^2 = 25 \][/tex]

Taking the square root of both sides:

[tex]\[ w = \sqrt{25} \][/tex]

[ w = 5 ]

So, the width of the rectangle is ( w = 5 ) feet.

Now, we can find the length:

[tex]\[ \text{Length} = 4w = 4(5) = 20 \] feet.[/tex]

Now that we have both the width and length, we can find the perimeter of the rectangle using the formula for perimeter:

[tex]\[ \text{Perimeter} = 2(\text{Length}) + 2(\text{Width}) \][/tex]

[tex]\[ \text{Perimeter} = 2(20) + 2(5) \][/tex]

[tex]\[ \text{Perimeter} = 40 + 10 \][/tex]

[tex]\[ \text{Perimeter} = 50 \][/tex]

So, the perimeter of the rectangle is ( 50 ) feet.

The possible coordinates for point B that are 10 units away from point A at (2,-3) could be (12, -3), (2, 7), (-8, -3), or (2, -13), moving directly horizontal or vertical.

If point A is located at (2,-3), and there are 10 points between A and B, we're likely looking at points equidistant from A, forming a sort of 'circle' with A at the centre. However, in a Cartesian plane, if the distance between two points is equal to 10 units, there are various possibilities depending on the orientation of the line connecting the two points.

For a point B to be 10 units away from point A at (2, -3), it can move horizontally to the right or left, or vertically up or down. Thus, the possible coordinates for point B would change by 10 units either in the x-coordinate (horizontally) or the y-coordinate (vertically).

Possible Coordinates for Point B

Horizontally to the right: (2+10, -3) = (12, -3)

Vertically up: (2, -3+10) = (2, 7)

Horizontally to the left: (2-10, -3) = (-8, -3)

Vertically down: (2, -3-10) = (2, -13)

Other options like (12, -6), (2, -7), (12, 3), or (-8, 3) do not adhere to being 10 units away from point A directly horizontally or vertically.

The complete question is:

If point A is located at (2,-3) , and there are 10 points between A and B, what * 1 could be the possible coordinates for point B? Choose all that apply. (12,-3) (2,7) (12,-6) (-8,-3) (2,-7) (-8,3) (2,-13) (12,3)

Answer:

[tex]\large\boxed{B.\ a^2+2ah+h^2+3a+3h+5}[/tex]

Step-by-step explanation:

[tex]f(x)=x^2+3x+5\\\\f(a+h)\to\text{substitute}\ x=a+h\ \text{to the equation:}\\\\f(a+h)=(a+h)^2+3(a+h)+5\\\\\text{Use}\ (x+y)^2=x^2+2xy+y^2\ \text{and the distributive property}\\\\f(a+h)=a^2+2ah+h^2+3a+3h+5[/tex]

Answer:

V = 960 ft^3

Step-by-step explanation:

The volume of a room can be found by

V = Area of base  time height

V = 120 * 8

V = 960 ft^3

QUESTION 1

We can use the cosine rule to find the missing side length.

Recall that the cosine rule for a triangle with sides a,b,c and an included angle A is

[tex]a^2=b^2+c^2-2bc\cos A[/tex]

Let the missing side length in the triangle with sides 6, 9 and the included angle of [tex]37\degree[/tex] be [tex]a[/tex] units.

We then substitute the values into the cosine rule to obtain;

[tex]a^2=6^2+9^2-2(6)(9)\cos 37\degree[/tex]

[tex]a^2=36+81-108\cos 37\degree[/tex]

[tex]a^2=30.747[/tex]

[tex]\Rightarrow a=\sqrt{30.747}[/tex]

[tex]\Rightarrow a=\sqrt{30.747}[/tex]

[tex]\Rightarrow a=5.5[/tex] units to the nearest tenth.

QUESTION 2

We again use the cosine rule: [tex]a^2=b^2+c^2-2bc\cos A[/tex]

We substitute the given values to obtain;

[tex]a^2=11^2+13^2-2(11)(13)\cos 108\degree[/tex]

[tex]a^2=121+169-286\cos 108\degree[/tex]

[tex]a^2=378.379[/tex]

[tex]\Rightarrow a=\sqrt{378.379}[/tex]

[tex]\Rightarrow a=19.5[/tex] to the nearest tenth

QUESTION 3

We again use the cosine rule :

[tex]|BA|^2=(\frac{1}{2})^2+(\frac{1}{3})^2-2(\frac{1}{2})(\frac{1}{3})\cos 100\degree[/tex]

[tex]|BA|^2=0.418899[/tex]

[tex]|BA|=\sqrt{0.418899}[/tex]

[tex]|BA|=0.65[/tex] to the nearest hundredth

The width of the rectangular park is 126 feet. This was found by setting up an equation based on the problem description and then solving for the width.

The subject of this question is Mathematics, specifically algebra. The problem states that the length of a rectangular park is 3 feet shorter than its width, with the length being given as 123 feet.

First of all, let's define the length with a variable L and the width with a variable W. From the problem, we can write the equation, L = W - 3. Since we know that L = 123 feet, we can substitute this value into the equation, getting 123 = W - 3.

To find W, all we need to do is add 3 to both sides of the equation. Hence, W = 123 + 3 = 126 feet. So, the width of the park is 126 feet.

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Answer:

8 < 6x

Step-by-step explanation:

Just write what you see by the letters

Answer:

the slope-intercept form:

y = 5x - 9

Step-by-step explanation:

y = 5x - 5, this line has slope = 5

parallel line, slope is the same so slope of the parallel = 5

equation

y - 6 = 5(x - 3)

y - 6 = 5x - 15

y = 5x - 9   <------the slope-intercept form

Answer: [tex]y=5x-9[/tex]

Step-by-step explanation:

The slope-intercept form of a equation of the line is:

[tex]y=mx+b[/tex]

Where m is the slope and b the y-intercept-

If the lines are parallel then they have the same slope:

m=5

Find b substitutin the point and the slope into the equation and solving for b:

[tex]6=3*5+b\\b=-9[/tex]

Then the equation is:

[tex]y=5x-9[/tex]

Answer: 250km

Step-by-step explanation:

Answer:

The cities are 35 cm apart in map.

Step-by-step explanation:

The scale on a map is 5 cm : 8 km.

[tex]\texttt{Scale = }\frac{5cm}{8km}\\\\\texttt{Scale = }\frac{5cm}{8\times 100000cm}=\frac{5}{800000}[/tex]

Now we need to find how much is the distance in map if the original distance is 56 km.

Distance in map = Scale x Original distance

[tex]\texttt{Distance in map = }\frac{5}{800000}\times 56km=\frac{5\times 5600000cm}{800000}=35cm[/tex]

The cities are 35 cm apart in map.

Answer: 9321 votes

Step-by-step explanation:

1- You can express the ratio as following:

[tex]5:8[/tex] or [tex]\frac{5}{8}[/tex]

2- Let's call the number of  no votes "N"

3- Therefore, if there were 3585 yes votes, then you can write the following expression to calculate the number of  no votes:

[tex]\frac{5}{8}=\frac{3585}{N}\\\\N=\frac{3585*8}{5}\\\\N=5736[/tex]

4- Then, the total number of votes is:

[tex]t=3585votes+5736votes=9321votes[/tex]

Answer:

9321

Step-by-step explanation:

We can simply make a ratio (fraction) to solve this. Let total number of  NO votes be N. Shown below is the ratio:

[tex]\frac{YesVotes}{NoVotes}=\frac{5}{8}=\frac{3585}{N}[/tex]

Now we can cross multiply and solve for N:

[tex]\frac{5}{8}=\frac{3585}{N}\\5N=8*3585\\5N=28,680\\N=\frac{28680}{5}=5736[/tex]

Hence, number of NO votes is 5736.

To get TOTAL number of votes, we add number of yes votes (3585) to that of number of no votes (5736).

Total votes = 3585 + 5736 = 9321

The cosine of the angle formed by the line whose equation is y=2x and the positive x axis cos =63.435 to the nearest thousandth

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Final answer:

To find the cosine of the angle formed by the line y=2x and the positive x-axis, we calculate the arctangent of the slope (2), which gives us an angle of approximately 63.435 degrees. The cosine of this angle is then approximately 0.447 to the nearest thousandth.

Explanation:

The question is asking for the cosine of the angle formed between the line y = 2x and the positive x-axis. To find this, we first need to determine the angle that the line makes with the x-axis. The slope of the line, given by m, is 2, which also represents the tangent of the angle the line forms with the x-axis (tan(θ) = m). We can calculate the angle using the arctangent function (tan⁻¹).

θ = tan⁻¹(2) = Approx. 63.435 degrees

The cosine of this angle can be found using the cosine function:

cos(θ) = cos(63.435 degrees) = Approx. 0.447

To the nearest thousandth, the cosine of the angle is 0.447.

7 * 0.055 = 0.385  

631.40 / 0.385 = $1,640

Answer:

a. No the particles will never collide.

b. The second particle is moving faster.

Step-by-step explanation:

We can tell they never collide based on the fact that they will never have the same two points. We can tell this because there is only one time in which they will have the same x value. To find this amount of time, set the two x values equal to each other and solve for t.

5t - 5 = 4t

-5 = -t

5 = t

So we know the x value will only be the same at 5 seconds. Now we can input that value and see if the y values are the same.

2t + 6 = t^2 - 2t - 1

2(5) + 6 = 5^2 - 2(5) - 1

10 + 6 = 25 - 10 - 1

16 = 14 (FALSE)

Therefore they do not collide.

For the second part of the question, we know that the second one is moving faster based on the fact that there is a squared value in the y formula. This shows that it is moving at an exponential rate, which always changes faster than a linear rate.

Particle A and particle B never collide.

The value of k where the particles collide is k = -4

The particle that is moving faster is Particle B since it has a square root in the y(t) = t² - 2t - 1.

An equation is a mathematical statement that is made up of two expressions connected by an equal sign.

We have,

Two particles:

Particle A:

x(t) = 5t - 5

y(t) = 2t - k

Particle B:

x(t) = 4t

y(t) = t² - 2t - 1

We see that,

The x(t) of particle A and x(t) of particle B are the same only at t = 5.

x(t) = 5t - 5 = 5 x 5 - 5 = 25 - 5 = 20

x(t) = 4t = 4 x 5 = 20

Now,

y(t) = 2t - k = 2 x 5 - k = 10 - k

y(t) = t² - 2t - 1 = 25 - 10 - 1 = 25 - 11 = 14

(a) If k = -6.

x(t) = 5t - 5 = 5 x 5 - 5 = 25 - 5 = 20

x(t) = 4t = 4 x 5 = 20

y(t) = 2t - k = 2 x 5 - k = 10 - k = 10 + 6 = 16

y(t) = t² - 2t - 1 = 25 - 10 - 1 = 25 - 11 = 14

In order to collide both the x(t) of particles A and B must be the same.

Similarly, y(t) must be the same.

So,

Particle A and particle B never collide.

(b)

The value of k where the particles collide.

k = -4

y(t) = 2t - k = 2 x 5 - k = 10 - k = 10 + 4 = 14

y(t) = t² - 2t - 1 = 25 - 10 - 1 = 25 - 11 = 14

(c)

The time at which the particles collide.

t = 5 and k = -4

x(t) = 5t - 5 = 5 x 5 - 5 = 25 - 5 = 20

x(t) = 4t = 4 x 5 = 20

y(t) = 2t - k = 2 x 5 - k = 10 - k = 10 + 4 = 14

y(t) = t² - 2t - 1 = 25 - 10 - 1 = 25 - 11 = 14

The particle that is moving faster is Particle B since it has a square root in the y(t) = t² - 2t - 1.

Thus,

Particle A and particle B never collide.

The value of k where the particles collide is k = -4

The particle that is moving faster is Particle B since it has a square root in the y(t) = t² - 2t - 1.

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She can buy 20 1$ items, or 10 $2 items, at most

Answer:

An Acute triangle

Step-by-step explanation:

It is an acute triangle, because the following characterization holds:

In this case,

[tex]6^2=36<5^2+4^2=25+16=41[/tex]

Answer:

The measure of the complement angle is [tex]18\°[/tex]

Step-by-step explanation:

Let

x-----> the angle

we know that

The complement of an angle is equal to [tex](90-x)\°[/tex]

The supplement of an angle is equal to [tex](180-x)\°[/tex]

we have

The complement of an angle is one-sixth the measure of the supplement of the angle

[tex](90-x)\°=(1/6)(180-x)\°[/tex]

solve for x

[tex](540-6x)\°=(180-x)\°[/tex]

[tex](6x-x)=(540-180)\°[/tex]

[tex](5x)=(360)\°[/tex]

[tex]x=72\°[/tex]

Find the measure of the complement angle

[tex](90-x)\°[/tex] ------> [tex](90-72)=18\°[/tex]

Answer:

18⁰

Step-by-step explanation:

Angle = x

Complement = 90 - x

Supplement = 180 - x

Given:

90 - x = 1/6 × (180 - x)

540 - 6x = 180 - x

5x = 360

x = 72

Complement = 90 - 72 = 18⁰

Answer:

12%

Step-by-step explanation:

Given,

% of leather sofas = p(A) = 30% = 30/100 = 0.3

% of leather sofas that are black = p(B) = 40% = 40/100 = 0.4

The percentage of total sofas made from leather = p(A) * p(B)

= 0.3 * 0.4

= .12

= 12%  

Answer:

The length of the arc is [tex]27\pi \ cm[/tex]

Step-by-step explanation:

step 1

Find the circumference

we know that

The length of a complete circle is equal to the circumference of the circle

The circumference is equal to

[tex]C=2\pi r[/tex]

we have

[tex]r=18\ cm[/tex]

substitute

[tex]C=2\pi (18)[/tex]

[tex]C=36\pi\ cm[/tex]

step 2

we know that

A central angle of  [tex]2\pi[/tex] radians subtends the circumference of [tex]36\pi\ cm[/tex]

so

by proportion

Find the length of the arc by a central angle of [tex]\frac{3\pi }{2}[/tex] radians

[tex]\frac{36\pi }{2\pi}\frac{cm}{radians}=\frac{x}{(3\pi/2)}\frac{cm}{radians} \\ \\x=18*(3\pi/2)\\ \\x=27\pi \ cm[/tex]

Answer: option b.

Step-by-step explanation:

To solve this exercise you must keep on mind the identities shown below:

1) [tex]sec\theta=\frac{1}{cos\theta}[/tex]

2) [tex]csc\theta=\frac{1}{sin\theta}[/tex]

3) [tex]tan\theta=\frac{sin\theta}{cos\theta}[/tex]

Therefore, to rewrite [tex]\frac{sec\theta}{csc\theta}=1[/tex] you must substitute identities and simplify the expression, as following:

[tex]\frac{sec\theta}{csc\theta}=1\\\\\frac{\frac{1}{cos\theta}}{\frac{1}{sin\theta}}=1\\\\\frac{sin\theta}{cos\theta}=1\\\\tan\theta=1[/tex]

Therefore, as you can see, the answer is the option b.

Answer:

Step-by-step explanation: B

Answer:

When b= 0.5, the period of orange graph is _4_ pi

When b= 2, the period of orange graph is _1_pi

Step-by-step explanation:

The period of the sinusoidal functions can be easily calculated by observing their graphs.

First, look at the orange graph when b = 0.5

Identify a point where the orange chart cuts the x-axis. For example at [tex]x = 0[/tex]. After completing the rise and fall cycle, the function cuts back to the x axis at [tex]x = 4\pi[/tex].

Then the period [tex]T = 4\pi[/tex].

Second, look at the orange graph when b = 2

Identify a point where the orange chart cuts the x-axis. For example at [tex]x = 0[/tex]. After completing the rise and fall cycle, the function cuts back to the x-axis at [tex]x = \pi[/tex].

Then the period [tex]T = \pi[/tex].

We also know that the period of a sinusoidal function is defined as [tex]T(b) = \frac{2\pi}{b}[/tex]

So:

[tex]T(0.5) = \frac{2\pi}{0.5} = 4\pi\\\\T(2) = \frac{2\pi}{2} = \pi[/tex]

Answer:

Part 2:

Based on this evidence,

When b > 1, the period

✔ decreases

.

When 0 < b < 1, the period

✔ increases

Step-by-step explanation:

This is correct for edge 2020. Hope this helps someone.