3 problems for 1 final answer. fairly easy 8th grade math.

The equation of the line parallel is y = mx + (1 + 2m).

We have,

The equation of the line.

y = mx + c

Slope = m

And,

(-2, 1) = (x, y)

So,

1 = m x -2 + c

c = 1 + 2m

Now,

y = mx + c

y = mx + (1 + 2m)

Thus,

The equation of the line parallel is y = mx + (1 + 2m).

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The quadratic equation is solved and discriminant is 5, so it has two real solutions

Given data ,

The given equation is a quadratic equation in the standard form ax² + bx + c = 0, where a = 1, b = -1, and c = -1

The discriminant of a quadratic equation is given by b² - 4ac. So, the discriminant of the given equation is

(-1)² - 4(1)(-1) = 1 + 4 = 5

Since the discriminant is positive (not zero or negative), the equation has two real solutions.

Hence , its discriminant is 5, so it has two real solutions

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Answer:

70.7°

350 feet

Im not too sure about the last one

a) Based on the first derivative test, h(x) has a relative minimum at x = -2 and a relative maximum at x = 0.

b)  For x = -2: h''(-2) = 6(-2) + 6 = -6 < 0, so h(x) has a relative maximum at x = -2.

For x = 0: h''(0) = 6(0) + 6 = 6 > 0, so h(x) has a relative minimum at x = 0.

Calculus is a branch of mathematics that deals with the study of rates of change, accumulation, and the properties and behavior of functions.

(a) First derivative test:

The first derivative test involves finding the critical points of the function, where the first derivative is equal to zero or undefined, and then checking the sign of the first derivative in the intervals between the critical points to determine whether the function has relative extrema at those points.

Find the first derivative of h(x):

h'(x) = 3x² + 6x

Set h'(x) = 0 and solve for x to find the critical points:

3x² + 6x = 0

x(x + 2) = 0

x = 0 or x = -2

Test the intervals between the critical points using the sign of the first derivative:

For x < -2: Choose x = -3, h'(-3) = 27 + (-18) = 9 > 0, so h(x) is increasing.

For -2 < x < 0: Choose x = -1, h'(-1) = 3 - 6 = -3 < 0, so h(x) is decreasing.

For x > 0: Choose x = 1, h'(1) = 3 + 6 = 9 > 0, so h(x) is increasing.

Based on the first derivative test, h(x) has a relative minimum at x = -2 and a relative maximum at x = 0.

(b) Second derivative test:

The second derivative test involves finding the critical points of the function using the first derivative, and then checking the sign of the second derivative at those points to determine whether the function has relative extrema at those points.

Find the second derivative of h(x):

h''(x) = 6x + 6

Evaluate the second derivative at the critical points found in step 2 of the first derivative test:

For x = -2: h''(-2) = 6(-2) + 6 = -6 < 0, so h(x) has a relative maximum at x = -2.

For x = 0: h''(0) = 6(0) + 6 = 6 > 0, so h(x) has a relative minimum at x = 0.

Hence, the ease of a test may vary for different individuals and their familiarity with calculus concepts.

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Answer:

A

Step-by-step explanation:

a. The dependent variable is the number of unit sold. The independent variable is price.

b. The value of r is -0.9965

c. ŷ = -0.68688X + 56.95837

X Values

∑ = 301

Mean = 50.167

∑(X - Mx)2 = SSx = 920.833

Y Values

∑ = 135

Mean = 22.5

∑(Y - My)2 = SSy = 437.5

X and Y Combined

N = 6

∑(X - Mx)(Y - My) = -632.5

R Calculation

r = ∑((X - My)(Y - Mx)) / √((SSx)(SSy))

r = -632.5 / √((920.833)(437.5)) = -0.9965

Meta Numerics (cross-check)

r = -0.9965

c. Regression line calculation

Sum of X = 301

Sum of Y = 135

Mean X = 50.1667

Mean Y = 22.5

Sum of squares (SSX) = 920.8333

Sum of products (SP) = -632.5

Regression Equation = ŷ = bX + a

b = SP/SSX = -632.5/920.83 = -0.68688

a = MY - bMX = 22.5 - (-0.69*50.17) = 56.95837

ŷ = -0.68688X + 56.95837

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Answer:

All lines are parallel.

Step-by-step explanation:

Get each equation in Slope-Intercept form:

  1. Divide both sides by 3: [tex]y=-\frac{4}{3}x+\frac{7}{3}[/tex]

  2. No change

  3. Subtract 8x and divide by 6 on both sides: [tex]y=-\frac{4}{3}x-\frac{2}{3}[/tex]

Notice:

a. All slopes are -4/3

b. All y-intercepts are different

The equation of the line AB,with slope of -1/2 is  y = (-1/2)x + 7/2. We then use the dot product to show that the direction vectors of the lines AB and AC are orthogonal, and the coordinates of point C is (6,7).

In an orthonormal frame (o; i; j), the points A and B are given as A(3;1) and B(1;2), and the line (AC) is given by the equation 2x-y-5=0.

To determine the equation of line (AB), we need to find the slope (or gradient) of the line passing through A and B. We can find this by using the formula

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

where (x1, y1) and (x2, y2) are the coordinates of two points on the line. Substituting the values of A and B, we get

slope = (2 - 1) / (1 - 3) = -1/2

Now that we have the slope, we can use the point-slope form of a line to find the equation of line (AB). Let (x,y) be any point on the line, then we have

y - 1 = (-1/2)(x - 3)

Simplifying, we get

y = (-1/2)x + 7/2

So the equation of line (AB) is y = (-1/2)x + 7/2.

To show that the direction vectors AB and AC of the lines (AB) and (AC) are orthogonal, we need to find the direction vectors of these lines and show that their dot product is zero. The direction vector of a line is just the vector that "points" in the direction of the line. To find it, we can take any two points on the line and subtract their coordinates to get a vector.

For line (AC), we can take A and C as two points on the line. Since we don't know the coordinates of C yet, we can solve the equation of line (AC) for x to get

x = (y + 5) / 2

Now substituting x by the above expression in the coordinates of point A we get

C((y + 5)/2, y)

Thus, the direction vector of line (AC) is

AC = C - A = ((y + 5)/2 - 3, y - 1) = (y/2 - 7/2, y - 1)

Similarly, we can take A and B as two points on the line (AB) to get the direction vector of line (AB):

AB = B - A = (1-3, 2-1) = (-2, 1)

Now, the dot product of AB and AC is

AB · AC = (-2)(y/2 - 7/2) + (1)(y - 1) = -y + 7

We can see that the dot product is zero when y=7. Hence, direction vectors AB and AC are orthogonal at point C(3,7).

To determine the coordinates of C, we substitute y=7 in the equation of line (AC) we obtained earlier

2x - y - 5 = 0

2x - 7 - 5 = 0

2x = 12

x = 6

Therefore, C has coordinates (6,7).

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The exponential function that gives the amount A(t) that the plant earns per hour t years after it opens is [tex]A(t) = 55(1.07)^t[/tex]

A manufacturer earned $55 per hour of labor when it opened.

Each year the manufacturer earns an additional 7% per hour.

Since the manufacturer earns an additional 7% per hour each year, the amount earned per hour can be represented as follows:

[tex]A(t) = 55(1 + 0.07)^t[/tex]

where t is the number of years after the plant opened. T

This can be simplified as function:

[tex]A(t) = 55(1.07)^t[/tex]

Therefore, the exponential function that gives the amount A(t) that the plant earns per hour t years after it opens is [tex]A(t) = 55(1.07)^t[/tex]

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Answer:

Step-by-step explanation:

To predict the number of bacteria after 10 hours using the formula P = Aekt, where P is the final population, A is the initial population, k is the growth rate, and t is the time elapsed, we need to first find the value of k.

dw

We know that after 4 hours, the population grew from 3,000 bacteria to 3,600 bacteria. So we can set up an equation:

3,600 = 3,000e^(4k)

Dividing both sides by 3,000 gives:

1.2 = e^(4k)

Taking the natural logarithm of both sides gives:

ln(1.2) = 4k

Solving for k, we get:

k = ln(1.2)/4

k ≈ 0.051

Now that we have the value of k, we can use the formula to predict the number of bacteria after 10 hours:

P = 3,000e^(0.051*10)

P ≈ 5,426

Therefore, we predict that after 10 hours, there will be approximately 5,426 bacteria present in the culture.

To calculate the percentage of questions Danielle got correct on her math test, we can divide the number of questions she got correct by the total number of questions on the test and then multiply by 100. In this case, Danielle got 23 out of 25 questions correct, so we can calculate her percentage as follows: (23/25) * 100 = 92%. Therefore, Danielle got 92% of the questions correct on her math test.

Percent = amount per 100

23/25 = amount per 25

25 x 4 = 100

23 x 4 = 92

92/100 = 92%

Answer = 92%

Hope this helps!

Answer:

The quotient is

[tex]8 {x}^{2} + 15x - \frac{1}{8} [/tex]

and the remainder is

[tex] \frac{63}{64} [/tex]

Answer:  (5, -1)

Step-by-step explanation:

format:  f(x)

in  f(5)=-1       x=5

also sometimes f(x) is called y so  y= -1

put into ordered pair format

(5, -1)

The surface area of the triangular prism is approximately 44.29 square millimeters.

What is surface area?

Surface area is the total area that the surface of a three-dimensional object covers. It is a measure of the amount of space that the surface of an object occupies. Surface area is usually measured in square units such as square meters, square centimeters, square inches, or square feet.

What is the area?

The total space occupied by a flat (2-D) surface or the shape of an object is defined as its area.

The area of a plane figure is the space enclosed by its boundary.

According to the given information:

To find the surface area of a triangular prism, we need to add up the area of all the faces. A triangular prism has three rectangular faces and two triangular faces.

Let's start by finding the area of the triangular faces. The base of the triangular prism is a right triangle with base 4 millimeters, height 4 millimeters, and hypotenuse 5.7 millimeters. We can use the Pythagorean theorem to find the missing side:

[tex]a^2 + b^2 = c^2\\4^2 + 4^2 = 5.7^2[/tex]

16 + 16 = 32.49

32 = 32.49 - 0.49

32 = 32

So the missing side has length [tex]\sqrt{(5.7^2 - 4^2)} =3.69 millimeters.[/tex] This is the base of each triangular face.

The height of the triangular prism is 3 millimeters, so the height of each triangular face is also 3 millimeters.

The area of each triangular face is:

(1/2) × base × height

= (1/2) × 3.69 × 3

≈ 5.54 square millimeters

So the total area of the two triangular faces is:

2 × 5.54= 11.08 square millimeters

Now let's find the area of each rectangular face. The length of each rectangular face is the same as the base of the triangular face, which is 3.69 millimeters. The width of each rectangular face is the height of the triangular prism, which is 3 millimeters.

The area of each rectangular face is:

length × width

= 3.69 × 3

≈ 11.07 square millimeters

So the total area of the three rectangular faces is:

3 × 11.07

= 33.21 square millimeters

To find the surface area of the triangular prism, we add up the area of all five faces:

11.08 + 33.21

= 44.29 square millimeters

Therefore, the surface area of the triangular prism is approximately 44.29 square millimeters.

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The probability of rolling a 4 on a single throw of a fair 6-sided die is 1/6, since there is only one way to roll a 4 and there are 6 equally likely outcomes in total.

The probability of rolling an even number on a single throw of a fair 6-sided die is 3/6, or 1/2, since there are three even numbers (2, 4, and 6) out of six possible outcomes.

To find the probability of rolling a 4 or an even number on a single throw, we can add the probabilities of these two events:

P(4 or even) = P(4) + P(even) - P(4 and even)

where P(4 and even) is the probability of rolling a 4 and an even number on the same throw. Since there is only one outcome (rolling a 4), which is not even, this probability is 0.

Therefore:

P(4 or even) = P(4) + P(even) - P(4 and even)

             = 1/6 + 1/2 - 0

             = 2/3

So the probability of rolling a 4 or an even number on a single throw of a 6-sided die is 2/3.

If the die is thrown 2 times, the probability of rolling a 4 or an even number on both throws is the product of the probabilities of rolling a 4 or an even number on each throw:

P(4 or even on both throws) = P(4 or even) × P(4 or even)

                           = (2/3) × (2/3)

                           = 4/9

Therefore, the probability of rolling a 4 or an even number on both throws of a 6-sided die is 4/9.

Answer: w = –27i + 35j

Step-by-step explanation:

If the line passes through the point (2,8) that cuts off the least area from the first quadrant, the slope is 8/3 and the y-intercept is 0.

To find the equation of the line that passes through the point (2, 8) and cuts off the least area from the first quadrant, we need to first determine the slope of the line. The slope-intercept form of a line is y = mx + b, where m is the slope and b is the y-intercept.

We can use the point-slope form of the equation of a line to find the slope. The point-slope form is y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line. Plugging in (2, 8) as the point, we get:

y - 8 = m(x - 2)

Next, we want to minimize the area that the line cuts off in the first quadrant. Since the line passes through the origin (0, 0), the area cut off by the line in the first quadrant is equal to the product of the x- and y-intercepts of the line.

We can express the x-intercept in terms of y by setting y = 0 in the equation of the line and solving for x:

0 - 8 = m(x - 2)

x = 2 + 8/m

The y-intercept is simply the y-coordinate of the point where the line intersects the y-axis, which is given by:

y = mx + b

8 = 2m + b

b = 8 - 2m

We can now express the area cut off by the line as:

A = x*y

A = (2 + 8/m)*8 - (8 - 2m)*2/m

A = (16 + 64/m) - (16 - 4m)/m

A = 64/m + 4m/m

To minimize the area, we can take the derivative of A with respect to m and set it equal to zero:

dA/dm = -64/m² + 4/m² = 0

64 = 4

m = 8

Plugging m = 8 into the equation for the x-intercept, we get:

x = 2 + 8/8 = 3

So the equation of the line is y = 8x/3.

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Comparing each pair of expressions using >, <, or = is given below:

-32 < |-32| (because -32 is negative and |-32| is positive)

5 - 5 = 0 (because subtracting the same number results in zero)

15 > |___15| (because |___15| is equal to 15)

|5| = |___|-5|| (because both expressions are equal to 5)

2 - 17 < ▾ (because 2 - 17 equals -15, which is less than the square root symbol)

2 > |____|-17|| (because 2 is positive and |-17| is also positive)

|-27| > ||-45|| (because |-27| is 27 and ||-45|| is 45)

-27 > -45 (because -27 is closer to zero than -45)

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Compound interest is the interest earned on the principal and the interest previously accumulated. It is given by

[tex]A=P(1+r/n)^{nt}[/tex]

where P = Principal, r = annual rate of interest, n = the number of times interest is compounded per year & t = time in years.

The given principal is $1600 for 11 years & amount is $2400 compounded quarterly.

To find the rate of interest compounded quarterly for 11 years substitute the given values in the above formula i.e

[tex]2400=1600(1+r/4)^{11*4}[/tex]

r=3.70%.

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A set of data that could be represented by the box-and-whisker plot include the following: D.

The median of the set of data is equal to 10.

In Mathematics and Statistics, a box-and-whisker plot is a type of chart that can be used to graphically or visually represent the five-number summary of a data set with respect to locality, skewness, and spread.

Based on the information provided about the data set (2, 2, 7, 10, 10, 11, 13, 15, 17, 20, 20, 24, 26, 27, 27, 28), the five-number summary for the given data set include the following:

Minimum (Min) = 2.

First quartile (Q₁) = 16.

Median (Med) = 10.

Third quartile (Q₃) = 25.5.

Maximum (Max) = 28.

In conclusion, we can logically deduce that the median of the data set is 10 and it has no outlier.

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"The line has a positive slope and passes through the x-axis at -2. Additionally, it crosses through point (2, 1).

The formula in this case is 2x-y = -4. When we set the value of y to 0, we can use this equation to calculate the x-intercept: 2x0=42x=4. When we multiply both sides by 2, we obtain 2x2=42x=2. The x-intercept is therefore -2.

We may use the slope-intercept version of the equation, y = mx + b, where m is the slope and b is the y-intercept, to graph the line y = 4x - 2.

We can observe that the slope is m = 4 and the y-intercept is b = -2 by comparing y = 4x - 2 to y = mx + b.

Starting with the y-intercept of -2 on the y-axis, we can graph line by finding other points on it using the slope of 4.

To get to the point, if we move two units to the right, we must move up eight units. (2, 6). To get to the point, if we move two units to the left, we must move down eight units. (-2, -10).

According to the description and choices given, "The line has a positive slope and passes through the x-axis at -2" is the option that most closely matches our graph. Additionally, it crosses through point (2, 1).

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Answer:

D. its a two dimensional object

Answer:

A. It is a polygon

C. It is a one-dimensional object

The shape is a polygon in two dimensions since a polygon must have at least three straight sides.

If x + 5 is a factor of the given polynomial, then (x + 5) must divide the polynomial evenly, meaning that the remainder is 0 when the polynomial is divided by x + 5.

We can use polynomial long division or synthetic division to find the quotient and remainder, but it's easier to use the fact that if x + 5 is a factor, then (-5) must be a root of the polynomial.

So, we can substitute x = -5 into the polynomial and set it equal to 0 to find m:

-3(-5)^4 - 10(-5)^3 + 20(-5)^2 - 22(-5) + m = 0

Simplifying and solving for m:

-3(625) + 10(125) + 20(25) + 110 + m = 0

-1875 + 1250 + 500 + 110 + m = 0

m = 1015

Therefore, m = 1015 so that x + 5 is a factor of - 3x^4 - 10x^3 + 20x^2 - 22x + m.

Answer:

Step-by-step explanation:

8+(6-3)-9

The first action is addition in parentheses

(6-3) = 3

The second action is addition and then subtraction, you can subtract first and then add, it makes no difference because the answer will be the same in all cases

8 + 3 - 9 = 11 - 9 = 2

The airplane weigh if it is carrying 52 gallons of fuel will be, 2338 pounds

Let,

Weight of airplane = y

Amount of fuel = x

The general linear equation can be written as, y=mx + c, where, (m) is slope, and (c) is a constant.

It is given,

2091 = 14m + c ..... (1)

2312 = 48m + c ..... (2)

On subtracting, (1) with (2), we get

221 = 34m

m= (221/34)= 6.5

Now, on putting the value of (m) in (1), we get,

2091 = 14(6.5) + c

So,

c = 2000

Therefore, the linear equation can be written as,

y = (6.5)mx + 2000

Now to find the airplane weight at 52 gallons of fuel, we put x = 52, to find y.

That is,

y = (6.5)(52) + 2000

y = 2338

So, the airplane weigh if it is carrying 52 gallons of fuel will be, 2338 pounds.

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Answer:

Step-by-step explanation:

decimal: 0.44444444444

fraction: 4/9

x = 3.6 units

First, some definitions before working the problem:

The three standard trigonometric functions, cosine, tangent, and sine, are defined as follows for right triangles:

[tex]sin(\theta)=\dfrac{opposite}{hypotenuse}[/tex]

[tex]cos(\theta)=\dfrac{adjacent}{hypotenuse}[/tex]

[tex]tan(\theta)=\dfrac{opposite}{adjacent}[/tex]

One memorization tactic is "Soh Cah Toa" where the first capital letter represents one of those three trigonometric functions, and the "o" "a" and "h" represent the "opposite" "adjacent" and "hypotenuse" respectively.

The triangle must be a right triangle, or there wouldn't be a "hypotenuse", because the hypotenuse is always across from the right angle.

Working the problem

For the given triangle, the right angle is in the top right, so the side on the bottom left is the hypotenuse.

We know the angle in the lower right corner (angle S), so the side touching it (side ST) with unknown length is the adjacent side.  (notice that the points that form the side include the vertex of the angle -- so, it's the adjacent side).

For this triangle, the "adjacent" leg is unknown, our "goal to find" side.  Additionally, the "hypotenuse" is known.

Therefore, the two sides of the triangle that are known or are a "goal to find" are the "adjacent" & "hypotenuse".

Out of "Soh Cah Toa," the part that uses "a" & "h" is "Cah".  So, the desired function to use for this triangle is the Cosine function.

[tex]cos(\theta)=\dfrac{adjacent}{hypotenuse}[/tex]

[tex]cos(69^o)=\dfrac{x}{10}[/tex]

To isolate "x", multiply both sides by 10...

[tex]10*cos(69^o)=x[/tex]

Make sure your calculator is set to degree mode, and calculate:

[tex]10*(0.3583679495453...)=x[/tex]

[tex]x=3.583679495453...[/tex] units

Rounded to the nearest tenth...

x = 3.6 units

Answer:

well 10^4 is 10000, and 4x10^2 is 1600 and if we divide 10000 by 1600 we can find out how many times greater 10000 is than 1600, and the anwser is 6.25

Step-by-step explanation:

Answer:

Step-by-step explanation:

y=-3x-1

format for formula:

y=mx+b

b=1  that is your y-intercept.  where it hits the y-axis

m= -3    this is your slope    [tex]\frac{rise}{run} =\frac{-3}{1}[/tex]  

from a point you have, the y-intercept, you go down 3 (because of the negative in front of it), this is your rise,
and to the right 1, this is your run

The speed limit for the highway in rounded to the nearest whole is 33 mi/h.

The speed is the ratio of the distance in a given time interval. The distance is represented by a unit of length and the time is represented by a unit of time.

There are different units for the length or distance. In the International System Units (SI), the standard unit of distance is the meter (m) and the standard unit of time is the second (s). Nonetheless, there are others units, for example: inches (in), miles (mi) and  yards (yd).

For solving this exercise, you need to know the relation between the given units for distance and time.

The question gives - 100 km/h. The kilometer (km) is a multiple of the  standard unit of distance -  the meter (m) and the hour is a submultiple of the  standard unit of time  - the second (s)

For solving this question, it is necessary that you know the relation between km/h and mi/h. See  below.

[tex]\text{1 km/h}= 0.6213712 \ \text{mi/h}[/tex]

Now you can solve the question from a math tool - Rule of three. Thus,

[tex]\text{1 km/h}= 0.6213712 \ \text{mi/h}[/tex]

[tex]100 \ \text{km/h}= \text{x mi/h}[/tex]

[tex]\text{x}= 100 \times 0.6[/tex]

[tex]\text{x}=33 \ \text{mi/h}[/tex]

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