Find the sum of a finite arithmetic sequence from n = 1 to n = 13, using the expression 3n + 3.

Final answer:

The function y = 2,000 - 90x shows Bradley's distance from home decreases by 90 meters for every minute walked, starting from 2,000 meters away. The graph of this equation is a straight line with a negative slope, indicating constant speed.

Explanation:

The equation y = 2,000 - 90x represents Bradley's distance from home in meters, y, as a function of the time x in minutes that he walks. This equation is a linear function where the initial value 2,000 meters represents the distance from home at the start, and -90 meters/minute is the rate at which this distance decreases as Bradley walks home. As time increases by 1 minute, the distance from home decreases by 90 meters. This relationship shows that Bradley travels at a constant speed since the slope of the line representing the equation (which is the rate of change of distance with respect to time) is constant.

If we graph this function, we would get a straight line that starts at 2,000 meters on the y-axis when t=0 and has a negative slope of 90. Therefore, the graph shows that as time passes, Bradley gets closer to home at a steady pace. This also reflects that the total distance Bradley would walk is 2,000 meters, and the time it would take for him to return home can be found by setting the function equal to zero and solving for x.

To prove that (m + dk) mod d = m mod d, we can use the definition of the modulo operation and properties of integers.

To prove that (m + dk) mod d = m mod d, we can use the definition of the modulo operation and properties of integers. Let's assume that m and d are integers, d > 0, and k is an integer.

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                The point that lie on the line is:

                        B.  (1,1)

We are given that a line passes through the point (0,-1) and has a slope of 2.

We know that the equation of a line passing through (a,b) and having slope m is given by:

           [tex]y-b=m(x-a)[/tex]

Here we have:  (a,b)=(0,-1) and m=2

This means that the equation of line is:

[tex]y-(-1)=2(x-0)\\\\y+1=2x\\\\y=2x-1[/tex]

Now we will check which option is true.

A)

                (2,1)

when x=2

we have:

[tex]y=2\times 2-1\\\\\\y=4-1\\\\\\y=3\neq 1[/tex]

Hence, option: A is incorrect.

B)

                     (1,1)

when x=1

we have:

[tex]y=2\times 1-1\\\\\\y=2-1\\\\\\y=1[/tex]

                Hence, option: B is correct.

C)

                      (0,1)

when x=0

we have:

[tex]y=2\times 0-1\\\\\\y=0-1\\\\\\y=-1\neq 1[/tex]

Hence, option: C is incorrect.

Answer:

length of the hypotenuse = 17 cm

Step-by-step explanation:

To find the length of the hypotenuse of any triangle we use pythagorean theorem

[tex]c^2 = a^2 + b^2[/tex]

Where c is the hypotenuse

a  and b are the legs of the triangle

Given : a= 8 cm, and b= 15 cm

We find hypotenuse C

[tex]c^2 = 15^2 + 8^2[/tex]

[tex]c^2 = 225 + 64[/tex]

[tex]c^2 = 289[/tex]

Take square root on both sides

c= 17

So length of the hypotenuse = 17 cm

To find the location of point R on the number line, you can use the formula for finding a point on a line segment given the endpoints and the ratio. In this case, the ratio is 3:5 and the endpoints are -14 and 2.

To find the location of point R, you can use the formula for finding a point on a line segment given the endpoints and the ratio. In this case, the formula is:

R = Q + r(QS)

where Q is the starting point, S is the ending point, r is the ratio between Q and S, and QS is the displacement vector from Q to S. In this problem, Q is -14, S is 2, and the ratio is 3:5. So we can substitute these values into the formula and solve:

R = -14 + (3/8)(2 - (-14)) = -14 + (3/8)(16) = -14 + 6 = -8

Therefore, the location of point R is -8 on the number line.

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The location of point R is -8. The correct answer is option a. [tex]\frac{3}{3+5}(2-(-14))+(-14)[/tex]

To find the location of point R which partitions the directed line segment from Q to S in a 3:5 ratio, we use the section formula. The formula is:

[tex]R =\frac{m}{m+n}(x_2-x_1)+x_1[/tex]

Here, m = 3 and n = 5, while Q is at [tex]x_1 =[/tex] -14 and S is at [tex]x_2 =[/tex] 2.

Plugging in the values, we have:

Therefore, the location of point R is at -8 on the number line and it is calculated by the expression [tex]R =\frac{3}{3+5}(2-(-14))+(-14)[/tex].

The complete question is:
On a number line, the directed line segment from Q to S has endpoints Q at –14 and S at 2. Point R partitions the directed line segment from Q to S in a 3:5 ratio. Which expression correctly uses the formula [tex]R =\frac{m}{m+n}(x_2-x_1)+x_1[/tex] to find the location of point R?

a. [tex](\frac{3}{3+5})(2-(-14))+(-14)[/tex]

b. [tex](\frac{3}{3+5})(-14-2)+2[/tex]

c. [tex](\frac{3}{3+5})(2-14)+14[/tex]

d. [tex](\frac{3}{3+5})(-14-2)-2[/tex]

3x +3(x-10) = 360

3x +3x-30 =360

6x-30 =360

6x=390

X = 390/6 = 65

65-10 =55

 One car was driving 65 mph

 The other was 55 mph

A is located at (-3,-5) B is located at (1,-9)

-3 + 1 = -2/2 =-1

-5 + -9 = -14/2 = -7

 midpoint is (-1,-7)

The solution is, Option A. is correct.

F(x) = 1/ (x-1)(x+4)

A rational fraction is an algebraic fraction such that both the numerator and denominator are polynomials.

Here, we have,

a graph is given .

We need to find which of the given rational functions is graphed in image.

On x-axis, 1 unit = 2 units

Clearly, we can see the graph is not defined at point x = - 4 and at x = 1.

Corresponding to x = - 4, factor is (x+4) .

Corresponding to x = 1, factor is (x-1) .

So, this graph is of the rational fraction

F(x) = 1/ (x-1)(x+4)

Hence, Option A. is correct.

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For t=1.84 and t=0.76 the ball's height is 8 meters in equation h=1+13t-[tex]5t^{2}[/tex].

An equation is a relationship between two or more variables expressed in equal to form. Equation of two variables look like ax+ by=c. It is solved in order to find the values of variables.

We have been given an equation h=1+13t-[tex]5t^{2}[/tex] and we have to find the values of t for which h=8.

So,

1+13t-[tex]5t^{2}[/tex]=8

[tex]-5t^{2}[/tex]+13t+1-8=0

[tex]-5t^{2}[/tex]+13t-7=0

Removing negative signs.

[tex]5t^{2}[/tex]-13t+7=0

We cannot solve through factorization so e use the following formula:

x=-b+-[tex]\sqrt{b^{2} -4ac}/2a[/tex]

t=(13+-[tex]\sqrt{-13^{2} -4*5*7}[/tex])/2*5

t=(13+-[tex]\sqrt{169-140}[/tex])/10

t=(13+5.38)/10, (13-5.38)/10

t=1.838, 0.762

By rounding off to nearest hundred t=1.84, 0.76.

Hence value of t for which h=8 meters are 1.84 and 0.76.

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Answer: The starting population of the rabbits

Step-by-step explanation:

Given: The population of a species of rabbit triples every year. This can be modeled by f(x) = 4(3)x and f(5) = 972.

We know that the exponential growth function is given by :-

[tex]f(x)=Ab^x[/tex], where A is the initial amount and b is the multiplicative rate of change in time x.

As compared to the given exponential function, we have

A=4

Therefore, 4 represents the starting population of the rabbits

Answer:

Step-by-step explanation:

Part A: The two points that represent this situation are (3,43) and (8,113). To put this in standard form, I will first find the slope-intercept form. I did this in the image below.

Now, to find the standard form I will rearrange, 14x -y +1 = 0.

Part B: To put this in function notation I simply take my slope-intercept equation of y=14x+1, and replace y with f(x) so,   f(x)=14x+1

​

Part C: To graph this, put the y, intercept at 1, and give the line a slope of 14/1. Label the x-axis with hours, from 0 -  8, and the y-axis with dollars, from 0-150.

1 1/4 per month

12 months per year

1 1/4 * 12 =

5/4 * 12/1 = 60/4 = 15

 they get 15 days per year

Answer:

The translation function g(x) is given as:

[tex]g(x)=\dfrac{1}{2^x}+4[/tex]

step-by-step explanation:

The parent function is f(x) and its representation is given as:

[tex]f(x)=\dfrac{1}{2^x}[/tex]

Now the graph g*x) is obtained by translation of the graph f(x) by some units.

Now as the graph of g(x) is a shift of the graph f(x) or the graph g(x) is translated by 4 units upwards.

hence the function g(x) is represented by:

g(x)=f(x)+4.

Hence the translation function g(x) is given as:

[tex]g(x)=\dfrac{1}{2^x}+4[/tex]

To solve the problem we should know about the Equation of a line and slope of a line.

The equations are (y≤ x+2) and (y< -2x+6).

Given to us

Given the points (0, 2), and (-2, 0), therefore,

[tex]x_2=0\\y_2=2\\x_1=-2\\y_2=0[/tex]

Substituting the values in the formula of the slope,

[tex]m=\dfrac{(y_2-y_1)}{(x_2-x_1)}[/tex]

[tex]m=\dfrac{2-0}{0-(-2)} = \dfrac{2}{2} = 1[/tex]

Substitute the value of slope and a point in the formula of line,

[tex]y = mx+c\\y_2 = mx_2+c\\2 = (1)0 +c\\c = 2[/tex]

Thus, the equation of the line is y=x+2, but as given the line is solid and is shaded below the line. therefore,

y≤ x+2

Given the points (0, 6) and (3, 0), therefore,

[tex]x_2=0\\y_2=6\\x_1=3\\y_2=0[/tex]

Substituting the values in the formula of the slope,

[tex]m=\dfrac{(y_2-y_1)}{(x_2-x_1)}[/tex]

[tex]m=\dfrac{6-0}{0-3} = \dfrac{6}{-3} = -2[/tex]

Substitute the value of slope and a point in the formula of line,

[tex]y = mx+c\\y_2 = mx_2+c\\6 = (-2)0 +c\\c = 6[/tex]

Thus, the equation of the line is y=-2x+6, but as given the line is shaded below the line. therefore,

y< -2x+6

Hence, the equations are (y≤ x+2) and (y< -2x+6).

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Answer:

Possible outcomes in tossing a 10-faced die 5 times is:

100,000

Step-by-step explanation:

A computer simulation tossed a 10-faced die 5 times.

Number of outcomes in each throw=10

Number of outcomes in 2 throws

=Number of outcomes in first throw×number of outcomes in second throw

=10×10

= 100

Hence, Number of outcomes in 5 throws

=10×10×10×10×10

=100,000

Possible outcomes in tossing a 10-faced die 5 times is:

100,000

The equation y = -1/2x^2 - 1 represents a downward opening parabola with the vertex at (0, -1), making it a maximum point.

Graph: The given equation is y = -1/2x^2 - 1. This represents a downward opening parabola.

Vertex: The vertex of the parabola is at (0, -1), making it a maximum point.

Conclusion: The correct answer is option 4: (0, -1); minimum.

Answer:

Step-by-step explanation:

We know that there are a total 52 cards out of which:

There are 12 face cards ( 4 kings,4 queen and 4 jack)

There are 4 pack:

13- spades     13- club     13-heart     13-diamond.

Out of which there are 26 black cards( 13 spade and 13 club)

There are 26 red cards( 13 heart and 13 diamond)

Now , we are asked to find the probability of each of the following,

1)

P(face card)

Since there are total 12 face cards out of 52 playing cards.

Hence,

P(face card)=12/52=3/13

2)

P( seven of hearts)

As there is just 1 seven of heart out pf 52 cards.

Hence,  P(seven of hearts)=1/52

3)

P(no black)

This means we are asked to find the probability of red card.

As there are 26 red card.

Hence P(no black)=26/52=1/2

4)

P(king)

As there are 4 kings out of 52 cards.

Hence, P(king)=4/52=1/13

5)

P(diamond)

As there are total 13 cards of diamond.

Hence,

P(diamond)=13/52=1/4

48 degrees remains

divide minutes by 60

36/60 = 0.6

then divide seconds by 3600

12/3600=0.003 ( to nearest thousandth)

now add them all up

48 +0.6 + 0.003 = 48.603 degrees