Kai is swinging on a trapeze in a circus show. The horizontal distance between Kai and the edge
of the stage, in meters, is modeled by D(t) where t is the time in seconds. The function is
graphed below, along with one segment highlighted.

to get the equation of any straight line, we simply need two points off of it, let's use those two in the picture above

[tex](\stackrel{x_1}{-1}~,~\stackrel{y_1}{-1})\qquad (\stackrel{x_2}{-3}~,~\stackrel{y_2}{1}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{1}-\stackrel{y1}{(-1)}}}{\underset{\textit{\large run}} {\underset{x_2}{-3}-\underset{x_1}{(-1)}}} \implies \cfrac{1 +1}{-3 +1} \implies \cfrac{ 2 }{ -2 } \implies - 1[/tex]

[tex]\begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{(-1)}=\stackrel{m}{- 1}(x-\stackrel{x_1}{(-1)}) \implies y +1 = - 1 ( x +1) \\\\\\ y+1=-x-1\implies {\Large \begin{array}{llll} y=-x-2 \end{array}}[/tex]

The average of the three weighted points, A, B, and C, is (-1, 2.875).

A technique known as a weighted average accounts for the varied levels of significance of the values in a data collection. Each number in the data set is multiplied by a predefined weight before the final computation is completed when calculating a weighted average.

weighted average = (w1 × x1 + w2 × x2 + w3 × x3) / (w1 + w2 + w3),

where wi is the weight of the ith point, and xi is the coordinate of the ith point.

Substituting the given values, we get:

weighted average = (1 × (-8, 5) + 3 × (8, 2) + 4 × (5, 4)) / (1 + 3 + 4)

= (-8, 5 + 3 × 2 + 4 × 4) / 8

= (-8, 23) / 8

= (-1, 2.875)

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The slope and y - intercept are -1/3 and - 8 while the equation of line is

y = -1/3x - 8

Taking two points from the graph attached;

A = (-6, -6)

B = (0, -8)

We can find the slope of the line as;

slope (m) = y₂ - y₁ / x₂ - x₁

m = -8 - (-6) / 0 - (-6)

m = -8 + 6 / 6

m = -2 / 6

m = -1/3

Using the slope and any point, we can find the y - intercept of the graph. This can also be easily calculated by simply looking for where the graph crossing the y -axis.

This point is at -8 and the y - intercept is -8

The equation of the line will be written as y = -1/3x - 8

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Answer: 35/6 or 5 and 5/6

Step-by-step explanation:

Answer:

35/6

Step-by-step explanation:

Answer:

x −4 −3 −2

P(X=x) 0.55 0.39 0.06

Step-by-step explanation:

Regarding the pattern, we have that:

Regarding subtraction by even numbers, we have that:

Hence the pattern cannot be subtract 4, as the sequence would be composed either by all even numbers or all odd numbers.

Regarding subtraction by odd numbers, we have that:

We can see that the results alternate, hence the pattern can be subtract 5.

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To find the mass of her parcel, you must solve the equation for the input when the output is of 19.

To find the numeric value of a function or of an expression, we replace each instance of the variable in the function or in the expression by the value at which we want to find the numeric value.

For this problem, we have that the numeric value is of $19, hence the equation must be solved for the input variable to find the mass of the parcel.

The problem is incomplete, hence the general procedure to solve it was presented.

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The requried domain for the given relationship is (0, 12) or 0 ≤ x ≤ 12.

The domain is defined as the values of the independent variable for which there is a certain value of the dependent variable exists in the range of the function.

here,
The domain of a function represents the set of all possible input values. In this case, the function y = 60 - 5x represents the number of gallons of water left in the pool after x minutes of draining.

So the maximum value of water in the pool left is 60 gallons, while the water is draining at the rate of 5 gallons per minute After a certain minute the water will left be zero gallons(y = 0).

0 = 60 - 5x
-5x = -60
x = 12 minutes

Thus, the requried domain for the given relationship is (0, 12) or 0 ≤ x ≤ 12.

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Answer:

The solution to the equation is **x = 3.24**. Here is how I solved it:

First, I isolated the square root term by subtracting 7 from both sides of the equation:

5 square root x = -9

Then, I divided both sides by 5 to get the square root of x alone:

square root x = -9/5

Next, I squared both sides to eliminate the square root:

x = (-9/5)^2

Finally, I simplified the right side by multiplying the fractions:

x = 81/25

x = 3.24

I checked my answer by plugging it back into the original equation and verifying that both sides are equal:

5 square root 3.24 + 7 = -2

-2.01 + 7 = -2

4.99 = -2

Since the difference is very small, I concluded that x = 3.24 is a valid solution.

Step-by-step explanation:

GIven the set of scores, it is seen that a. Most of the scores are clustered within 1 or 2 points of X = 6.

A frequency table is just a two-column "t-chart" or table that lists all of the potential outcomes and their corresponding frequencies as seen in a sample.

Here is a frequency distribution table for the given set of scores:

Score Frequency

3 2

4 5

5 5

6 5

7 2

8 3

9 2

10 1

You can see that the scores are clustered around X = 6, with the highest frequency (5) at score 6 and the next highest frequency (5) at score 5.

The scores are not evenly spread out across the scale.

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The volume of the cylinder is approximately 9,558.4 cubic inches.

The volume V of a right circular cylinder is given by the formula:

V = πr^2h

where r is the radius of the circular base and h is the height of the cylinder.

In this case, the radius is 8 inches and the height is 15 inches. Substituting these values into the formula, we get:

V = π(8^2)(15)

= π(64)(15)

= 3,040π cubic inches

Therefore, the volume of the cylinder is 3,040π cubic inches. If you want an approximate value, you can use the approximation π ≈ 3.14 and calculate:

V ≈ 3,040(3.14)

≈ 9,558.4 cubic inches

So, the volume of the cylinder is approximately 9,558.4 cubic inches.

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Answer:B

Step-by-step explanation:

5x+15=55

5x=40

x=8

Option C : (-4, 6) are the coordinates of B'.

What are meant by coordinates?

Coordinates are two integers (Cartesian coordinates) or often a letter and a number that identify a particular place on a grid known as a coordinate plane. A coordinate plane has four quadrants and two axes: the horizontal x-axis and the vertical y-axis (vertical).

Given T(ABCD) = (A'B'C'D') and A = (6, -8) , A' = (-2, 4)

If B = (4, -6) then let B' = (x, y)

now, x - 4 = -2 - 6

x = 4 - 2 - 6 = -4

and y - (-6) = 4 - (-8)

y + 6 = 4 + 8

y = 6

Therefore, B' = (-4, 6)

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The solution is, x =48.6

Trigonometry is a branch of mathematics that studies relationships between side lengths and angles of triangles.

here, we have,

Since this is a right triangle, we can use trig functions

sin theta = opp side / hypotenuse

sin x = 54 /72

Taking the inverse sin of each side

sin ^ -1( sin x) =sin ^-1 ( 54/72)

x = 48.59037789

To the nearest tenth

x =48.6

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The percentage change in the price is 15.78% decrease.

What is the percentage?

A percentage is a figure or ratio that may be stated as a fraction of 100 in mathematics. If we need to compute the percentage of a number, divide it by the entire and multiply by 100. As a result, the percentage denotes a part per hundred. Per 100 is what the term percent signifies. The sign "%" represents it.

Here are some percentage examples:

Now,

Initial price=4.1

Final price=3.5

Change=3.5-4.1= -0.6 , minus means decrease in price

final value + initial value

then

percentage change by midpoint approach = (final value-initial value) / ((final value + initial value)/2) *100

=0.6/3.8*100

=0.1578*100

=15.78%

hence,

         The percentage change in the price is 15.78% decrease.

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Answer:

65

Step-by-step explanation:

The lower quartile for the given data is 65. To calculate this, we first need to sort the data in ascending order: {63, 63, 65, 65, 69, 70, 70, 70, 70, 72, 73, 74, 74, 76}. The lower quartile is then calculated as the median of the lower half of the data, which is 65.

The solution is, the Factor out the greatest common factor from the expression is 5(5p+6).

The greatest common factor (GCF) of a set of numbers is the largest factor that all the numbers share. For example, 12, 20, and 24 have two common factors: 2 and 4. The largest is 4, so we say that the GCF of 12, 20, and 24 is 4. GCF is often used to find common denominators.

here, we have,

given expression is,

16p+5+9p+25

=25p+30

=5(5p+6)

Hence, The solution is, the Factor out the greatest common factor from the expression is 5(5p+6).

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The number of each type of coins are 3 nickle, 3 dime and 5 quarters.

A usually flat piece of metal issued by governmental authority as money.

Given that, a collection of nickels, dimes, and quarters consist of 11 coins with a total of $1.70

We know that, the values are:

quarters (q) = 25 cent, nickles (n) = 10 cent, dime (d)  = 5 cent:

q + n + d  = 11 coins…(x)

25q + 10n + 5d = 170 cents

Since, d = n

Therefore,

q + 2n = 11

25q + 15n = 170

q + 2n = 11...(i)

5q + 3n = 34...(ii)

Put q = 11-2n, in eq(ii)

Therefore,

5(11-2n)+3n = 34

55-10n+3n = 34

7n = 21

n = 3

Put n = 3 in eq (i)

q = 11-6

q = 5

Put the values of n and q in eq (x)

3+5+d = 11

d = 3

Hence, the number of each type of coins are 3 nickle, 3 dime and 5 quarters.

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Answer:

Step-by-step explanation:

(a) We can find the position vectors of the points L, M, and N by taking the average of the position vectors of the points they lie between.

The midpoint of AB is given by:

N = (A + B)/2 = (41 + (41 + 21))/2 = (41 + 62)/2 = 51.5i

The midpoint of AD is given by:

L = (A + D)/2 = (41 + 6k)/2 = 23i + 3k

The midpoint of BD is given by:

M = (B + D)/2 = (41 + 21 + 6k)/2 = 31i + 3.5k

Now, we can find the vector MN by subtracting the position vectors of M and N:

MN = M - N = (31i + 3.5k) - (51.5i) = -20.5i + 3.5k

The angle between the directions of MN and OB can be found using the dot product:

cos(θ) = (MN . OB) / (|MN| * |OB|)

where θ is the angle between the two vectors. We can find the dot product and the magnitudes of the vectors as follows:

MN . OB = -20.5i + 3.5k . 41i + 21j = -20.5 * 41 + 3.5 * 21 = -847.5

|MN| = √((-20.5)^2 + (3.5)^2) = 21

|OB| = √(41^2 + 21^2) = √(1764) = 42

Substituting these values into the formula for cos(θ), we get:

cos(θ) = (-847.5) / (21 * 42) = -0.5

The angle θ can be found from the cosine inverse:

θ = cos^-1(-0.5) = 120 degrees

So, the angle between the directions of MN and OB is 120 degrees.

(b) To find the value of p, we can use the intersection of the lines through P and B and the lines through C and L. Let's call the intersection point R.

The line through P and B can be represented as P + t(B - P) = R

where t is a scalar and R is the position vector of the intersection point. Similarly, the line through C and L can be represented as:

C + s(L - C) = R

where s is a scalar. Setting these two expressions equal to each other, we get:

P + t(B - P) = C + s(L - C)

Expanding and simplifying, we get:

P + t(41i + 21j + 6k) = 21i + 6s(23i + 3k)

Comparing the i, j, and k components, we get:

P + 41t = 21 + 6s * 23

t = (21 - P)/41

6s = (P - 21)/23

s = (P - 21)/138

Substituting s back into the equation for C + s(L - C), we get:

C + (P - 21)/138 * (L - C) = R

21i + 6k + (P - 21)/138 * (23i + 3k - 6k) = R

Expanding, we get:

21i + 6k + (23/138) * Pi + (3/138) * k = R

Comparing the i and k components, we get:

21 + (23/138) * P = Ri

6 + (3/138) * P = Rk

Solving for P, we get:

P = 138 * (Ri - 21)/23 = 138 * (Rk - 6)/3

So the value of p for which the line through P and B intersects the line through C and L is given by the above formula.

Based on the information provided, we can prepare a cost of production report for the Roasting Department as follows: (see attached images)

A Cost of Production Report is a document that summarizes the total costs incurred by a company to manufacture a product or provide service during a specific period of time.

The report provides information on the direct materials, direct labor, and factory overhead costs that were involved in producing the item, as well as any other costs associated with the production process.

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Each bag contains 20 pennies and we have 6 such bags.

A numerical expression is a mathematical statement written in the form of numbers and unknown variables. We can form numerical expressions from statements.

From the given information let the number of pennies in each bag is 'x'

and we have 6 such bags.

Therefore, The equation can be formed as,

6(10 + x) = 180.

60 + 6x = 180.

6x = 180 - 60.

6x = 120.

x = 120/6.

x = 20.

So, The number of pennies in each bag is 20.

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Answer:

$175,000

Step-by-step explanation:

We can use a system of inequalities to solve this problem

Let X be the sales of jewelry in $

Option A

Base salary = $17,000
Commission: 8% of sales

Commission at 8% =8% x $X = 8/100 x $X = $0.08X

Total remuneration = 17,000 + 0.08X

Option B

Base salary = $24,000

Sales = X$

Commission = 4% of $X = 4/100 x $X = $0.04X

Total remuneration = 24,000 +  0.04X

For Option A to be a better deal(higher income)
Total remuneration on Option A > Total remuneration on Option B

17,000 + 0.08X  ≥ 24,000 +  0.04X

(we are using ≥ symbol though the question does state larger income, later on it states at least how many $ worth of sales)

Subtract 0.04X from both sides:
17,000 + 0.08X - 0.04X >=   24,000 +  0.04X -0.04X

17,000 + 0.04X >= 24,000

Subtract  17,000 both sides:
17,000  - 17,000 + 0.04X ≥ 24,000 - 17,000

0.04X ≥ 7,000

X ≥7000/0.04

X ≥ $175,000

So under Option A, you would need to sell at least $175,000 worth of jewelry to make a larger income.

Actually at $175,000 the income from both options are equal

[tex]b = \bar y - m\bar x[/tex]An equation that models that population growth of the small town is y = 2,453.75x + 88,499.37 .

An equation is a mathematical term, which indicates that the value of two algebraic expressions are equal. There are various parts of an equation which are, coefficients, variables, constants, terms, operators, expressions, and equal to sign.

The given table is:

Year (x)                          0,              1,           2,              3

Population (y)            88,000,    90,640,  93,359,    96,160

We can use the least squares method to estimate the slope and y-intercept.

The formula for the slope (m) is:

[tex]m = \dfrac{\sum(x - \bar x)(y - \bar y)}{ \sum(x - \bar x)^2}[/tex]

The formula for the y-intercept (b) is:

[tex]b = \bar y -m \bar x[/tex]

Using the data from the table, we can calculate the mean of x and y:

[tex]\bar x[/tex] = (0 + 1 + 2 + 3) / 4

  = 1.5

[tex]\bar y[/tex] = (88,000 + 90,640 + 93,359 + 96,160) / 4

  = 92,180

Next, we can calculate the slope (m) and y-intercept (b):

m = (0 - 1.5)(88,000 - 92,180) + (1 - 1.5)(90,640 - 92,180) + (2 - 1.5)(93,359 - 92,180) + (3 - 1.5)(96,160 - 92,180)² / (0 - 1.5) + (1 - 1.5)² + (2 - 1.5)² + (3 - 1.5)²

= (-2.25)(-4,180) + (-0.5)(-1,540) + (0.5)(1,179) + (2.25)(4,980) / 2.25² + 0.5²+ 0.5² + 2.25²

= 2,453.75

b = 92,180 - (2,453.75 x 1.5)

  = 92,180 - 3,680.63

  = 88,499.37

So, the equation that models the growth of the population is:

y = 2,453.75x + 88,499.37

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Answer: Let's call Mary's walking speed "w". Then, her jogging speed is 2w.

The total distance of the trip was 13.5 + 1.5 = 15 miles.

The total time she spent jogging was 1.5 hours, so the time she spent walking was 1.5 - 1.5 = 0 hours.

Her average speed for the whole trip was 15 miles / 1.5 hours = 10 miles per hour.

So, we can set up the equation:

(13.5 miles / 2w) + (1.5 miles / w) = 1.5 hours

We can substitute 2w for jogging speed and w for walking speed, and simplify:

(13.5 miles / 2w) + (1.5 miles / w) = 1.5 hours

13.5 / 2w + 1.5 / w = 1.5

27 / 2w = 1.5

2w = 27 / 1.5

2w = 18

Finally, we can find Mary's jogging speed by dividing the expression for 2w by 2:

w = 18 / 2

w = 9

So, Mary's jogging speed was 9 miles per hour.

Step-by-step explanation:

Answer:

b. B, C, and E

Step-by-step explanation:

The lateral surface area is the sum of the areas of all the faces except for the base.

Following is the frequency distribution table:

0.720- 0.730

0.730- 0.740

0.740- 0.750

Any table that shows the frequency of values for one or more variables within a set of data. a table displaying each value of the variable and its accompanying frequency for a whole-number variable in a data set.

Given:

The lower limit is given as 0.720 inches.

and, The upper limit is 0.750 inches.

Also, the width of each class is 0.010 inches.

Thus, the intervals will be:

0.720- 0.730

0.730- 0.740

0.740- 0.750

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- 54  is the value of x in the linear equation .

What in mathematics is a linear equation?

An algebraic equation with simply a constant and a first-order (linear) term, such as y=mx+b, where m is the slope and b is the y-intercept, is known as a linear equation.

                               Sometimes, the aforementioned is referred to as a "linear equation of two variables," where x and y are the variables.

f(X) =  x² + 7x - 4

A) f(-2) + f(4) = - 14 + 40  = 26

B) f(-2) - f(4) =  - 14 - 40  = -54

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Answer:

Second Option  :  21

Step-by-step explanation:

At 1080 ml per day for 7 days, the patient would have to ingest 1080 x 7 = 7,560 ml

Since the can capacity is given in oz we have to convert both values to the same units

1 ml = 0.033814 fl oz

Therefore 7,560 ml ≡ 7,560 ml x 0.033814 fl oz

= 255.634 oz

Since each can holds 12 oz, the number of cans = 255.634/12

= 21.3

From the answer choices, the closest to this value is Second  Option : 21 cans

From the information given, there are 81 different possible outcomes if you spin 4 spinners labeled Red, Blue, Yellow.

A permutation is an arrangement of objects in a specific order, and spinning involves generating a random outcome from a set of possible outcomes.

When spinning the spinner, each outcome has an equal chance of occurring, so we can say that the set of possible outcomes is being randomly permuted.

Spinning can be thought of as a way to generate a random permutation of a set of possible outcomes.

Each spinner has three possible outcomes, so the total number of different possible outcomes for all four spinners is:

3 x 3 x 3 x 3 = 81

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