h(t)=3t-5
g(t)=t-2
Find (h·g)(t)

The point on the line 3x + y = 8 that is closest to the point (-3, 1) is (1.8, 2.6).

Quadratic equations are equations of the form a x² + b x + c = 0 where a, b and c are real numbers.

Given line is,

3x + y = 8 ⇒ y = -3x + 8

So we have any point (x, -3x + 8).

We have to find a point (x, -3x + 8) such that distance of this point from (-3, 1) is minimum.

Distance between (-3, 1) and (x, -3x + 8) = √[(x - -3)² + (-3x + 8 - 1)²]

                                                                  = √[(x + 3)² + (-3x + 7)²]

                                                                  = √(x² + 6x + 9 + 9x² - 42x + 49)

                                                                  = √(10x² - 36x + 58)

So the distance between (-3, 1) and (x, -3x + 8) is √(10x² - 36x + 58) should be minimum.

We minimize the function f(x) = 10x² - 36x + 58.

A quadratic function represents the equation of a parabola.

Coefficient of x² is positive, so parabola opens upward. So we will be having the minimum value.

Minimum value at the point x = -b / 2a for the quadratic equation a x² + b x + c.

Minimum value is at x = - (-36) / (2 × 10) = 36/20 = 9/5 = 1.8.

To find the y coordinate,

y = -3x + 8 = 2.6

So the required point is (1.8, 2.6).

Hence the point on the line closest to the point (-3, 1) is (1.8, 2.6)

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Answer: Pythagorean triplets.

Step-by-step explanation:

not sure but this is the triangle that helps at all.

The number of small rooms reserved is 4.

The number of large rooms reserved is 8.

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

We have,

Small rooms are denoted as S.

Large rooms are denoted as L.

Now,

We will make two equations.
L = 2S _____(1)

8L + 6S = 88 ______(2)

Putting (1) in (2) we get,

8L + 6S = 88

8 x (2S) + 6S = 88

16S + 6S = 88

22S = 88

S = 88/22

S = 8/2

S = 4

Now,

Putting S = 4 in (1) we get,

L = 2 x 4

L = 8

Thus,

The number of small rooms and large rooms reserved is 4 and 8.

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Reflection across the y-axis is the transformation used to the polygon ABCD.

Graph is a mathematical representation of a network and it describes the relationship between lines and points.

Given that Image of polygon ABCD and a second polygon A prime B prime C prime D prime to the right.

ABCD is the original image and  prime B prime C prime D prime is the transformed image.

ABCD rectangle is transformed to A'B'C'D'.

As we observe the diagram, the two rectangles look the same which means the shape remains constant.

The translation is applied to the original image which move to the right side.

A translation is a movement of the graph either horizontally parallel to the x axis or vertically parallel to the y axis.

We can reflect the graph of any function f about the x-axis by graphing y=-f(x) and we can reflect it about the y-axis by graphing y=f(-x).

Hence, Reflection across the y-axis is the transformation used to the polygon ABCD.

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Answer: reflection across the y axis

Step-by-step explanation: I hope that helps :)

Answer:

It will take 41.7 years for this investment to triple in value.

Step-by-step explanation:

Here is the formula for simple interest

[tex]A=P(1+rt)[/tex]

Lets solve for [tex]t[/tex].

Divide both sides by [tex]P[/tex].

[tex]\frac{A}{P}=rt+1[/tex]

Subtract 1 from both sides.

[tex]\frac{A}{P}-1=rt[/tex]

Divide both sides by [tex]r[/tex].

[tex]\frac{\frac{A}{P}-1}{r} =t[/tex]

To write [tex]-1[/tex] as a fraction with a common denominator, multiply by [tex]\frac{P}{P}[/tex] .

[tex]\frac{\frac{A}{P}-1*\frac{P}{P}}{r} =t[/tex]

[tex]\frac{\frac{A}{P}*\frac{-P}{P}}{r} =t[/tex]

Combine the numerators over the common denominator.

[tex]\frac{\frac{A-P}{P} }{r} =t[/tex]

Multiply the numerator by the reciprocal of the denominator.

[tex]t=\frac{A-P}{Pr}[/tex]

We are given

[tex]P=3800\\r=0.048[/tex]

We need to find what amount is triple his money.

[tex]3800*3=11400[/tex]

[tex]A=11400[/tex]

Substitute our numbers into the equation for [tex]t[/tex].

[tex]t=\frac{11400-3800}{3800*0.048}[/tex]

Factor 3800 out of 11400.

[tex]t=\frac{3800*3-3800}{3800*0.048}[/tex]

Factor 3800 out of −3800.

[tex]t=\frac{3800*3-3800*-1}{3800*0.048}[/tex]

Factor 3800 out of [tex]3800*3-3800*-1[/tex].

[tex]t=\frac{3800*(3-1)}{3800*0.048}[/tex]

Cancel the common factor of 3800.

[tex]t=\frac{3-1}{0.048}[/tex]

Simplify.

[tex]t=\frac{2}{0.048}[/tex]

[tex]t=41.666[/tex]

Merry Christmas!

The slope of the line is 7/9

It is important to note that the equation of a line is represented as;

y = mx + c

Where;

The formula for calculating the slope of a line is expressed as;

Slope, m = y₂ - y₁/x₂ - x₁

Now, let's substitute the values into the formula from the points given we have;

Slope, m =1 -(-6)/ -3 - (-6)

expand the bracket

Slope, m = 1 + 6/ 3 + 6

add the values

Slope, m = 7/9

Hence, the value is 7/9

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The maximum height of the firework is; 387.78 ft

The time for which the firework is in the air before it explodes is; 1 second

We are given the function;

h = -(500/9)t² + (1000/3)t + 10

where;

h is height after time of t seconds

At height of zero, the firework would either be about to launch or when it has come down.

Thus, let us set h = 0 to find the times.

0 = -(500/9)t² + (1000/3)t + 10

Using quadratic equation calculator, we have;

t ≈ 2 seconds

Now the  firework explodes at the highest point which will be the mid point of where the launching started and the endpoint where it stopped.

This means time at midpoint = (0 + 2)/2 = 1 sec

It explodes after 1 second.

Thus maximum height at this time is;

h = -(500/9)(1)² + (1000/3)(1) + 10

h = 387.78 ft

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To solve the given system of equations using elimination, we can add the two equations together to eliminate one of the variables. Doing this, we get 3x + 9y + (-10x - 6y) = 33 + (-14), which simplifies to -7x + 3y = 19. We can then solve for y by dividing both sides of the equation by 3, giving us y = 19/3 = 6.33. We can substitute this value for y in either of the original equations to solve for x. For example, if we substitute 6.33 for y in the first equation, we get 3x + 9 * 6.33 = 33, which simplifies to 3x = 3.67. Dividing both sides of the equation by 3, we get x = 1.22. Therefore, the solution of the system is (1.22, 6.33), which is approximately (1, 6). The answer choice that is closest to this solution is (d) (-1, 4). Thus, the solution of the system is (-1, 4).

The answers are a) FR = 365 N b) u = 70.8° d and c) (MR)O = 2364 N m (counterclockwise)

When an object is subject to several forces, the resultant force is the force that alone produces the same acceleration as all those forces.

Given is a figure, we need to find equivalent resultant force and couple moment at point O

Equivalent Resultant Force And Couple Moment At O.

+→ (FR)x = ΣFx; (FR)x = 600 cos 60° - 455(12 / 13)

= -120 N = 120 N ←

+ ↑ (FR)y = ΣFy; (FR)y = 455(5/13) - 600 sin 60° = -344.62 N = 344.62 N ↓

As indicated in Fig (attached)

a) FR = √F(R)x²+F(R)y²

= √2120² + 344.62² = 364.91 N = 365 N

And

b) θ = tan-1{(FR)y / (FR)x}

= tan-1 {344.62/120}

= 70.80°

Also,

c) a+(MR)O = ΣMO; (MR)O = 455(12/13)(2) + 600 cos 60° (0.75) + 600 sin 60° (2.5)

= 2364.04 N m

= 2364 N m (counterclockwise)

Hence, The answers are a) FR = 365 N b) u = 70.8° d and c) (MR)O = 2364 N m (counterclockwise)

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With the help of Linear relationships The difference in the rate of change between the two containers is 1/5 gallon per minute.

what is Linear relationships?

A statistical word used to express a straight-line relationship between two variables is a linear relationship (or linear association). Graphs and mathematical equations of the form y = mx + b can both be used to represent linear relationships. In everyday life, linear relationships are quite prevalent.

Given:

Volume of water in each container.

To find:

Difference in the rate of change.

Solution:

Take any two points on container 1.

Let the points are (10, 2) and (20, 4).

m = (y2 - y1) / ( x2 - x1 )

   = ( 4 - 2 ) / ( 20 - 10 )

   = 2/ 10

   = 1/ 5

Rate of change for container 1 is .

Take any two points on container 2.

Let the points are (5, 2) and (10, 4).

m = (y2 - y1) / ( x2 - x1 )

   = ( 4 - 2 ) / ( 10 - 5)

   = 2 / 5

Rate of change for container 2 is 2/5

Difference = (2 /5 ) - ( 1/ 5 )

                  = 1 / 5  

Hence, The difference in the rate of change between the two containers is 1/5 gallon per minute.

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Answer:

Step-by-step explanation:

Ss, variance and the standard deviation for the population of scores is 2.83 and 8 respectively

What is standard deviation and variance?

Standard deviation is a measure of the distribution of statistical data, whereas variance is a measure of how data points differ from the mean. The fundamental distinction between the two is that although the variance is expressed in squared units, the standard deviation is expressed in the same units as the data's mean.

Given,

SS = 40

Variance = 10

Standard Deviation = 3.16

Using Population Variance Formula :

= [tex]\frac{1}{n}[/tex]∑( x-xbar)²  where i is 1

For the complete solution see the attachment of table

sum ( x - x bar)² = 40

so, variance = 40/5

= 8

standard deviation = [tex]\sqrt{8}[/tex]

= 2.83

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Answer:  2x-4

Step-by-step explanation:

Answer:

solution

you must multiply a difference of x and 2 which is. >{x-2}

mathematical

4 x {x-2}

Answer:

(a)  $897.72:  26 yr

     $1526.49:  10 yr

(b)  See below.

(c)  x = 705

     more, $705, longer it will take

Step-by-step explanation:

Given equation:

[tex]t=16.708 \ln \left(\dfrac{x}{x-705}\right)[/tex]

where:

[tex]\begin{aligned}x=897.72 \implies t& =16.708 \ln \left(\dfrac{897.72}{897.72-705}\right)\\& =16.708 \ln \left(4.65815691...\right)\\&=16.708(1.53861985...)\\&=25.70726...\\&=26\; \rm years\end{aligned}[/tex]

[tex]\begin{aligned}x=1526.49 \implies t& =16.708 \ln \left(\dfrac{1526.49}{1526.49-705}\right)\\& =16.708 \ln \left(1.85819669...\right)\\& =16.708(0.619606496...)\\&=10.352385...\\&=10 \; \rm years\end{aligned}[/tex]

To approximate the total amounts paid (in dollars) over the term of the mortgage, multiply the monthly payment by the term.

Please note I have provided two calculations per monthly payment:

(1) by using the exact term, and (2) using the rounded term from part (a).

[tex]\implies \$897.72 \times 25.7072605... \times 12 =\$276935.06[/tex]

[tex]\implies \$897.72 \times 26 \times 12=\$280088.64[/tex]

[tex]\implies \$1526.49 \times 10.3523853... \times 12=\$189633.75[/tex]

[tex]\implies \$1526.49 \times 10 \times 12 =\$183178.80[/tex]

To calculate the amount of interest costs (in dollars) in each case, subtract $150,000 from the total amounts paid:

[tex]\$897.72\implies 276935.06-150000=\$126935.06[/tex]

[tex]\$897.72 \implies 280088.64-150000=\$130088.64[/tex]

[tex]\$1526.49 \implies 189633.75-150000=\$39633.75[/tex]

[tex]\$1526.49\implies 183178.80-150000=\$33178.80[/tex]

The natural logarithm of a negative number cannot be taken.

Therefore, x > 705.

So the vertical asymptote for the model is:

The monthly payment must be more than $705, and the closer to this value the payment is, the longer it will take to pay off the mortgage.

Answer:

either 8.6 or 9. check if it's correct?

7² + 5² = 74

square root 74 = 8.6

Answer: no solutions

Step-by-step explanation:

Answer:

1

Step-by-step explanation:

A)

[tex]( \sqrt{3 } - \sqrt{2} )( \sqrt{3} + \sqrt{2} )( \sqrt{5 } - \sqrt{4} )( \sqrt{5} + \sqrt{4} )....[/tex]

using difference of two squares

[tex]( \sqrt{a} + \sqrt{b} )(\sqrt{a} - \sqrt{b} ) = a - b[/tex]

we can rewrite the A) to 1×1×1×1.... so the answer would be 1

The histogram that best represents the distribution of job applications for graduating seniors is given as follows:

a. Histogram III.

An histogram shows the number of times that each observation appears in a data-set.

Data on college graduates suggests that most will send out a large number of applications before getting their first job, hence the left bins on the histogram should have small numbers, while the right bins on the histogram should have greater numbers.

The only histogram that follows this pattern is Histogram III, hence option A gives the histogram that best represents the distribution of job applications for graduating seniors.

The histograms are given by the image shown at the end of the answer.

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The measure of the largest exterior angle of the triangle will be 162°.

The polygonal shape of a triangle has a number of sides and three independent variables. Angles in the triangle add up to 180 °.

The exterior angles of the triangle are 3x°, 8x°, and 9x°. Then the measure of the interior angle of the triangle will be 180° - 3x°, 180° - 8x°, and 180° - 9x°.

The sum of the interior angle of the triangle is 180°. Then the equation is given as,

180° - 3x° + 180° - 8x° + 180° - 9x° = 180°

360° - 20x° = 0

x = 18°

Then the measure of the exterior angles of the triangle will be given as,

3x° = 3 × 18° = 54°

8x° = 8 × 18° = 144°

9x° = 9 × 18° = 162°

The measure of the largest exterior angle of the triangle will be 162°.

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Answer:

1. A(t) = 65t + 700

2.B(t) = 40t + 900

3. High School A and High School B will both have 1,220 students in the 8th year

Step-by-step explanation:

1. The equation for the number of students in High School A represents a linear function.

⭐ What is a linear function?

Let's write the equation for High School A in the [tex]y = mx+b[/tex] format, known as slope-intercept form:

∴ High School A: [tex]y = 65x + 700[/tex]

2. Let's write the equation for High School B in the [tex]y = mx + b[/tex] format, known as slope-intercept form:

∴ High School B: [tex]y = 40x + 900[/tex]

3. To find at what year High School A and High School B will have the same number of students, we need to solve a system of linear equations.

⭐What is a system of linear equations?

For this problem, let's set both linear equations equal to each other to see at what point will the high school populations be the same.

[tex]A(t) = B(t)[/tex]

[tex]65t + 700 = 40t + 900[/tex]

[tex]25t = 200[/tex]

[tex]t = 8[/tex]

Now we know that in the 8th year, High School A will have the same population as High School B.

We need to find what the population will be in year 8.

Thus, substitute the value of t into one of the functions and solve.

I am choosing to substitute t into A(t), but you can also do B(t).

[tex]A(8) = 65(8) + 700[/tex]

[tex]A(8) = 520 + 700[/tex]

[tex]A(8) = 1,220[/tex]

⚠️!!! CAUTION !!! ⚠️

Some people may stop at this point and write that in the 8th year, both high schools will have a population of 1,220 students.

However, you should also substitute 8 into the other function you didn't substitute it into to make sure that 8 is correct.

[tex]B(8) = 40(8) + 900[/tex]

[tex]B(8) = 320 + 900[/tex]

[tex]B(8) = 1,220[/tex]

∴ In the 8th year, both high schools will have a population of 1,220 students.

No, A is not defective.

The rank–nullity theorem is a theorem in linear algebra, which asserts that the dimension of the domain of a linear map is the sum of its rank (the dimension of its image) and its nullity (the dimension of its kernel).

Given here  A be a 4 x 4 matrix and let λ be an eigenvalue of multiplicity 3 and rank 1, then by rank nullity theorem we have

    Rank(A-λI) + Nullity((A-λI) =4

⇒dimN(N-λI) = Nullity(A-λI)

⇒dimN(N-λI) = 4 - Rank(A-λI)

⇒dimN(N-λI)  = 4-1

⇒dimN(N-λI) = 3

Therefore 3 linearly independent eigenvectors belong to eigenvalue λ. Since λ is an eigenvalue of multiplicity 3, this ensures that there is a different eigenvalue different from λ. Along with 3 linearly independent eigenvectors in the eigenspace of λ, this gives us 4 linearly independent eigenvectors in the matrix A. Now we An n × n matrix A is diagonalizable if and only if A has n linearly independent eigenvectors. Thus A is nondefective

Hence, matrix A is not defective.

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Answer: 177.8

Step-by-step explanation: To find the percent increase, we have to use this formula:

New value-old value/ old value x 100

So, following this formula, we would do 25-9, which, is 16, divided by 9. That would be 1.777... So now let's round that to 1.78, so 1.78 x 100 = 177.8. I hope this helped.

Answer:

Prove that for any x0∈X and any r>0, the open ball Br(xo) is open. My attempt: Let y∈Br(x0). By definition, d(y,x0)<r

Step-by-step explanation:

Prove that for any x0∈X and any r>0, the open ball Br(xo) is open. My attempt: Let y∈Br(x0). By definition, d(y,x0)<r

Answer:

22

There are 22 multiples that end in 22.

The Equation of line is y= 16.1x + 210. 8.

The approximate price by 2025 is 452.3.

A line's slope is determined by how its y coordinate changes in relation to how its x coordinate changes. y and x are the net changes in the y and x coordinates, respectively. Therefore, it is possible to write the change in y coordinate with respect to the change in x coordinate as,

m = Δy/Δx where, m is the slope

Given:

The points are (1, 226.9) and (4, 275.2).

So, Slope for line

m = ( 275.2- 226.9)/( 4-1)

m = 48.3/ 3

m= 16.1

Now, using the slope- intercept form

y= mx+ b

y = 16.1x + b

Now, x= 1 and y= 226.9

By, y = 16.1 + x

226.9 = 16.1(1)+ b

b= 210.8

So, the equation of line is y= 16.1x + 210. 8

Now,  x is the number of years since 2010 and by 2025

x= 2025 - 2010 = 15

Then, the approximate price by 2025

y= 16.1(15) + 210.8

y= 452.3

Hence, the approximate price by 2025 is 452.3.

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