Write the equation of each line using the given information. Show all work.
1. Passing through the points (4, 5) and (-2, -7). Use fractions, not decimals.​

Answer:

(a) 44 (4 + 4 = 8), 62 (6 + 2 = 8)

(b) 165 (1 + 6 + 5 = 12)

Answer:

4(1/2)(3)(5) + (3^2) = 30 + 9 = 39 ft^2

A function that has a degree of 5 must have the following: d. at least one real 0.

In Mathematics and Geometry, a polynomial function is a mathematical expression which comprises intermediates (variables), constants, and whole number exponents with different numerical value, that are typically combined by using specific mathematical operations.

Generally speaking, the degree of a polynomial function is sometimes referred to as an absolute degree and it is the greatest exponent (leading coefficient) of each of its term.

In conclusion, a function that has a degree of 5 is referred to as a quintic function and it is characterized by an odd degree, can't have more than 4 turning points, and must have at least one real zero.

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Answer:

Step-by-step explanation:

76 RPMs

John will need 36 feet of fence to enclose his 9-foot square garden.

A sqaure is simply a polygon made up of four equal sides, with all the interior angles measuring 90 degrees.

The perimeter of a square is expressed as:

Perimeter = 4 × side length

Given that, the side length of the square garden is 9 ft.

To calculate the amount of fence John will need to enclose his 9-foot square garden, we need to determine the perimeter of the square.

Perimeter = 4 × side length

Plug in the side length

Perimeter = 4 × 9 ft

Perimeter = 36 feet

Therefore, the perimeter of the garden is 6 feet.

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These atomic formulas represent basic assertions or relationships between elements or sets.

In logic and mathematical logic, "atomic formulas" refer to the simplest, indivisible formulas that do not contain any logical connectives or quantifiers.

They are the building blocks from which more complex formulas are constructed.

In set theory, the atomic formulas are typically statements about membership or equality.

For example, "x belongs to y" (x ∈ y) and "x equals y" (x = y) are atomic formulas in set theory.

Using logical connectives (such as conjunction, disjunction, negation, implication, and equivalence) and quantifiers (such as "for all" (∀) and "there exists" (∃)), we can combine atomic formulas to form more complex formulas, enabling us to express a wide range of logical statements and mathematical properties.

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Answer: We can use the formula:

P(A or B) = P(A) + P(B) - P(A and B)

where A and B are two events.

In this case, we want to find the probability that a randomly selected senior is a varsity athlete or on the honor roll. We can define the events as follows:

A = the event that a senior is a varsity athlete

B = the event that a senior is on the honor roll

From the problem statement, we know:

P(A and B) = 10/200 = 1/20

P(A) = 70/200 = 7/20

P(B) = 68/200 = 17/50

Plugging these values into the formula:

P(A or B) = P(A) + P(B) - P(A and B)

= 7/20 + 17/50 - 1/20

= 21/50

Therefore, the probability that a randomly selected senior is a varsity athlete or on the honor roll is 21/50.

The equation of the quadratic function is y=x2−4x, the correct option is E

We are given that;

x=-4,-2,1

y=5,-5,0

Now,

The first three rows of the table, we get:

​y=−4 when x=−5y=5 when x=−2y=−3 when x=1​

Substituting into the general form, we get:

​−4=25a−5b+c5=4a−2b+c−3=a+b+c​

Solving this system of equations, we get:

​a=1b=−4c=0​

Therefore, the equation of the quadratic function is y=x2−4x.

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A quadrilateral is a polygon with four sides. All trapezoids have two pairs of parallel sides. Option (a) is correct.

A quadrilateral is a polygon with four sides, and there are several properties and characteristics that can be true about a quadrilateral. Some possible true statements about a quadrilateral include:

1. A quadrilateral has four angles.

2. A quadrilateral has four vertices.

3. The sum of the interior angles of a quadrilateral is always 360 degrees.

4. A quadrilateral can have sides of different lengths.

5. A quadrilateral can have parallel sides.

Compare each of the properties above to the given options.

Options:

a) All trapezoids have two pairs of parallel sides.

b) Only some rectangles have four right angles.

c) All squares are rectangles.

d) Some rhombuses are trapezoids.

We can conclude that option (a) is correct because it correspond to the property 5 of a quadrilateral.

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Answer:

Step-by-step explanation:

To solve the given linear programming problem graphically, we'll plot the feasible region determined by the given constraints and then identify the optimal solution by maximizing the objective function Z = 10x1 + 20x2.

Step 1: Plotting the Constraints

We'll start by plotting the equations representing each constraint.

Constraint 1: 5x1 + 3x2 ≤ 30

To plot this constraint, we'll first find the points on the line 5x1 + 3x2 = 30. We can do this by setting x1 to 0 and solving for x2, then setting x2 to 0 and solving for x1.

Setting x1 = 0, we get: 3x2 = 30

x2 = 10

So, one point is (0, 10).

Setting x2 = 0, we get: 5x1 = 30

x1 = 6

So, another point is (6, 0).

Plotting these two points and drawing a line passing through them, we get a boundary line for the first constraint.

Constraint 2: 3x1 + 6x2 ≤ 36

Similarly, we find two points on the line 3x1 + 6x2 = 36.

Setting x1 = 0, we get: 6x2 = 36

x2 = 6

So, one point is (0, 6).

Setting x2 = 0, we get: 3x1 = 36

x1 = 12

So, another point is (12, 0).

Plotting these points and drawing a line passing through them, we get a boundary line for the second constraint.

Constraint 3: 2x1 + 5x2 ≤ 20

Again, we find two points on the line 2x1 + 5x2 = 20.

Setting x1 = 0, we get: 5x2 = 20

x2 = 4

So, one point is (0, 4).

Setting x2 = 0, we get: 2x1 = 20

x1 = 10

So, another point is (10, 0).

Plotting these points and drawing a line passing through them, we get a boundary line for the third constraint.

Step 2: Finding the Feasible Region

Now, we shade the region of the graph that satisfies all the constraints. The feasible region is the intersection of the shaded regions determined by each constraint.

Step 3: Maximizing the Objective Function

To maximize the objective function Z = 10x1 + 20x2, we look for the highest possible value within the feasible region. The optimal solution will be at one of the corner points of the feasible region.

We calculate the coordinates of each corner point formed by the intersection of the constraint lines and evaluate Z = 10x1 + 20x2 at each of these points. The corner point with the highest Z value will be the optimal solution.

By evaluating Z at each corner point, we can determine the optimal solution and its corresponding values of x1 and x2.

Note: Without specific numerical values for the corner points, I am unable to provide the exact coordinates and the optimal solution for this problem. However, by following the steps outlined above and evaluating the objective function at each corner point, you can identify the optimal solution for the given linear programming problem.

Hi. First of all, please give me 5 stars. There is the response:

To solve this problem graphically, we first plot the lines corresponding to the constraints, and then find the feasible region by shading the region that satisfies all constraints. Finally, we evaluate the objective function at each corner point of the feasible region to find the optimal solution.

The constraints can be rewritten in slope-intercept form as follows:

5x1 + 3x2 ≤ 30 => x2 ≤ (-5/3)x1 + 10

3x1 + 6x2 ≤ 36 => x2 ≤ (-1/2)x1 + 6

2x1 + 5x2 ≤ 20 => x2 ≤ (-2/5)x1 + 4

We plot these lines on a graph:

```

| / 5x1 + 3x2 = 30

6 +-------/------------

| / 3x1 + 6x2 = 36

| /

4 +----/--------------

| / 2x1 + 5x2 = 20

| /

2 +-/----------------

| /

|/

+------------------

0 2 4 6 8

```

Next, we shade the feasible region:

```

| /

6 +-------/-----RR-----

| / -----RR-----

| / -----RR-----

4 +----/-----RRR------

| / --RRRRRR------

| / -RRRRR--------

2 +-/----RRRR----------

| / R--------------

|/ R--------------

+------------------

0 2 4 6 8

```

The feasible region is the shaded area. There are 4 corner points: (0, 4), (4, 2), (6, 0), and (8/7, 26/21).

Finally, we evaluate the objective function at each corner point:

- (0, 4): Z = 10(0) + 20(4) = 80

- (4, 2): Z = 10(4) + 20(2) = 80

- (6, 0): Z = 10(6) + 20(0) = 60

- (8/7, 26/21): Z = 10(8/7) + 20(26/21) = 170/7

The optimal solution is (8/7, 26/21) with a maximum value of 170/7.

The measure of angle A in the right triangle is 75.52°

The figure in the image is a right triangle.

Measure of angle A = ?

Adjacent to angle A = 2

Hypotenuse = 8

To solve for the measure of angle A, we use the trigonometric ratio.

Note: cosine = adjacent / hypotensue

Hence:

cos( A) = adjacent / hypotensue

cos( A ) = 2/8

cos( A ) = 1/4

Take the cos inverse

A = cos⁻¹( 1/4 )

A = 75.52°

Therefore, the measure of the angle is 75.52 degree.

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A graph of the image of polygon DGF after a dilation centered at the origin with a scale factor of 1.5 is shown below.

In Geometry, a dilation is a type of transformation which typically changes the dimension (size) or side lengths of a geometric figure, but not its shape.

This ultimately implies that, the side lengths of the dilated geometric figure would be enlarged or reduced based on the scale factor applied.

In this scenario an exercise, we would dilate the coordinates of the polygon by applying a scale factor of 1.5 that is centered at the origin as follows:

D (-2, 0) → (-2 × 1.5, 0 × 1.5) = D' (-3, 0).

G (0, 2) → (0 × 1.5, 2 × 1.5) = G' (0, 3).

F (2, -2) → (2 × 1.5, -2 × 1.5) = F' (3, -3).

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Using the z-score values given, the area under the standard normal distribution curve is 0.123 squared units

The total area under the standard normal distribution curve is 1. Therefore, if we have a z-score, i.e., how far a value is above or below the mean, we can determine the probability of a value less than or greater than that.

The points are

A = z₂ - z₁

z₂ = 0.6230

z₁ = 0.5000

A = 0.6230 - 0.5000

A = 0.123 squared units

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It would take Arron 3 hours to travel 18 miles upstream (against the river)

An equation is an expression that shows how numbers and variables are related to each other using mathematical operations.

Speed is the ratio of total distance to total time taken.

Arron boat has a speed of 9 mph in still water and the speed of the river is 3 mph. Since he is travelling upstream, we subtract both speed:

He needs to travel 18 miles upstream at time t, hence:

(9 mph - 3 mph) = 18 miles / t

6t = 18

t = 3 hours

It would take Arron 3 hours

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Answer:

3)  A = 35.2°, B = 38.6°, C = 106.2°

4)  B = 70.4°, C = 31.6°, a = 18.7

Step-by-step explanation:

To solve for the remaining angles of the triangle ABC given its side lengths, use the Law of Cosines for finding angles.

[tex]\boxed{\begin{minipage}{7.6 cm}\underline{Law of Cosines (for finding angles)} \\\\$\cos(C)=\dfrac{a^2+b^2-c^2}{2ab}$\\\\\\where:\\ \phantom{ww}$\bullet$ $C$ is the angle. \\ \phantom{ww}$\bullet$ $a$ and $b$ are the sides adjacent the angle. \\ \phantom{ww}$\bullet$ $c$ is the side opposite the angle.\\\end{minipage}}[/tex]

Given sides of triangle ABC:

Substitute the values of a, b and c into the Law of Cosines formula and solve for angle C:

[tex]\implies \cos(C)=\dfrac{a^2+b^2-c^2}{2ab}[/tex]

[tex]\implies \cos(C)=\dfrac{12^2+13^2-20^2}{2(12)(13)}[/tex]

[tex]\implies \cos(C)=\dfrac{-87}{312}[/tex]

[tex]\implies \cos(C)=-\dfrac{29}{104}[/tex]

[tex]\implies C=\cos^{-1}\left(-\dfrac{29}{104}\right)[/tex]

[tex]\implies C=106.191351...^{\circ}[/tex]

To find the measure of angle B, swap b and c in the formula, and change C for B:

[tex]\implies \cos(B)=\dfrac{a^2+c^2-b^2}{2ac}[/tex]

[tex]\implies \cos(B)=\dfrac{12^2+20^2-13^2}{2(12)(20)}[/tex]

[tex]\implies \cos(B)=\dfrac{375}{480}[/tex]

[tex]\implies B=\cos^{-1}\left(\dfrac{375}{480}\right)[/tex]

[tex]\implies B=38.6248438...^{\circ}[/tex]

To find the measure of angle A, swap a and c in the formula, and change C for A:

[tex]\implies \cos(A)=\dfrac{c^2+b^2-a^2}{2cb}[/tex]

[tex]\implies \cos(A)=\dfrac{20^2+13^2-12^2}{2(20)(13)}[/tex]

[tex]\implies \cos(A)=\dfrac{425}{520}[/tex]

[tex]\implies A=\cos^{-1}\left(\dfrac{425}{520}\right)[/tex]

[tex]\implies A=35.1838154...^{\circ}[/tex]

Therefore, the measures of the angles of triangle ABC with sides a = 12, b = 13 and c = 20 are:

[tex]\hrulefill[/tex]

Given values of triangle ABC:

First, find the measure of side a using the Law of Cosines for finding sides.

[tex]\boxed{\begin{minipage}{6 cm}\underline{Law of Cosines (for finding sides)} \\\\$c^2=a^2+b^2-2ab \cos (C)$\\\\where:\\ \phantom{ww}$\bullet$ $a, b$ and $c$ are the sides.\\ \phantom{ww}$\bullet$ $C$ is the angle opposite side $c$. \\\end{minipage}}[/tex]

As the given angle is A, change C for A in the formula and swap a and c:

[tex]\implies a^2=c^2+b^2-2cb \cos (A)[/tex]

Substitute the given values and solve for a:

[tex]\implies a^2=10^2+18^2-2(10)(18) \cos (78^{\circ})[/tex]

[tex]\implies a^2=424-360\cos (78^{\circ})[/tex]

[tex]\implies a=\sqrt{424-360\cos (78^{\circ})}[/tex]

[tex]\implies a=18.6856038...[/tex]

Now we have the measures of all three sides of the triangle, we can use the Law of Cosines for finding angles to find the measures of angles B and C.

To find the measure of angle C, substitute the values of a, b and c into the formula:

[tex]\implies \cos(C)=\dfrac{a^2+b^2-c^2}{2ab}[/tex]

[tex]\implies \cos(C)=\dfrac{(18.6856038...)^2+18^2-10^2}{2(18.6856038...)(18)}[/tex]

[tex]\implies \cos(C)=0.852040063...[/tex]

[tex]\implies C=31.565743...^{\circ}[/tex]

To find the measure of angle B, swap b and c in the formula, and change C for B:

[tex]\implies \cos(B)=\dfrac{a^2+c^2-b^2}{2ac}[/tex]

[tex]\implies \cos(B)=\dfrac{(18.6856038...)^2+10^2-18^2}{2(18.6856038...)(10)}[/tex]

[tex]\implies \cos{B}=0.334888270...[/tex]

[tex]\implies B=\cos^{-1}(0.334888270...)[/tex]

[tex]\implies B=70.434256...^{\circ}[/tex]

Therefore, the remaining side and angles for triangle ABC are:

The trigonometric ratios are expressed as;

sin θ = 3/5

cos θ = 4/5

tan θ = 3/4

To determine the trigonometric ratios, we need to know the six identities.

These trigonometric identities are listed thus;

From the information given, we have that;

sin θ = opposite/hypotenuse

cos θ = adjacent/hypotenuse

tan θ = opposite/adjacent

We have that;

The hypotenuse is labeled as 5.

The adjacent leg is 4

The angle opposite the side labeled 3 is theta

Now, substitute the value to determine the ratios for each identity, we have that;

sin θ = 3/5

cos θ = 4/5

tan θ = 3/4

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Answer: To find the gradient of a curve at a given point, we need to find the derivative of the function with respect to x and then evaluate it at the desired x-coordinate.

Given the function y = 3x^4 - 2x^2 + 5x - 2, we can find its derivative dy/dx as follows:

dy/dx = d/dx(3x^4) - d/dx(2x^2) + d/dx(5x) - d/dx(2)

Taking the derivative of each term:

dy/dx = 12x^3 - 4x + 5

Now, let's evaluate the derivative at the given points:

Point (0, -2):

Substituting x = 0 into the derivative:

dy/dx = 12(0)^3 - 4(0) + 5

dy/dx = 0 - 0 + 5

dy/dx = 5

Therefore, the gradient at (0, -2) is 5.

Point (1, 4):

Substituting x = 1 into the derivative:

dy/dx = 12(1)^3 - 4(1) + 5

dy/dx = 12 - 4 + 5

dy/dx = 13

Therefore, the gradient at (1, 4) is 13.

Answer:

Step-by-step explanation:

let the number of apples represent x

3x^2 - 22x + 35

The mixed number4 13/20  is equal to the decimal 4.65.

By Converting the mixed number to an improper fraction

To convert 4 13/20 to an improper fraction, we multiply the whole number (4) by the denominator (20) and add the numerator (13) to get the new numerator. The denominator remains the same.

4 13/20 = (4 * 20 + 13) / 20 = 93/20

Dividing the numerator by the denominator:

Divide the numerator (93) by the denominator (20) to get the decimal representation.

93 ÷ 20 = 4.65

Therefore, the decimal representation of 4 13/20 is 4.65.

Complete question: How do you write 4 13/20 as a decimal

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The solution is, the values are, -3, -2, [tex]\left[\begin{array}{ccc}-2&0\\0&-3\\\end{array}\right][/tex], 6.

Here, we have

given that,

the matrices are:

A = [tex]\left[\begin{array}{ccc}-1&0\\0&3\end{array}\right][/tex]

B = [tex]\left[\begin{array}{ccc}2&0\\0&-1\\\end{array}\right][/tex]

now, we have to find the determinant of them

|A| = -1*3 - 0

    = -3

|B| = -1*2 - 0

    = -2

Now, we have to find AB ,

we get,  A×B = [tex]\left[\begin{array}{ccc}-2&0\\0&-3\\\end{array}\right][/tex]

So, we have, the determinant value is :

|AB| = -2 * -3 - 0

      = 6

Hence, The solution is, the values are, -3, -2, [tex]\left[\begin{array}{ccc}-2&0\\0&-3\\\end{array}\right][/tex], 6.

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Answer:

-10

Step-by-step explanation:

The required y-intercept of the line y = 4x - 10 is -10.

What is the intercept in the equation?

In the equation, the intercept is the value of the linear function where either of the variables is zero.

Here,

The equation y = 4x - 10 is in slope-intercept form, where the coefficient of x is the slope of the line, and the constant term is the y-intercept.

So, the y-intercept of the line y = 4x - 10 is -10. This means that the line intersects the y-axis at the point (0,-10). When x is 0, the value of y is -10, which is the y-intercept.

Thus, the required y-intercept of the line y = 4x - 10 is -10.

Answer:

----------------

Use midpoint equation to find each coordinate:

and

So the midpoint is (12, 16).

The missing probability is P(A|B) is 12/25 if P(B) is 1/4P and (A and B) is 3/25P.

P(B) = 1/4

P(A and B) = 3/25

The conditional probability is used to find the P(A|B):

P(A|B) = P(A and B) / P(B)

Conditional probability is defined as the number of possible outcomes from a given event based on the previous outcomes of the events.

By substituting the given values in the given probability, we get:

P(A|B) = (3/25) / (1/4)

Divide the fraction and multiply it with its reciprocal:

P(A|B) = (3/25) * (4/1)

P(A|B) = 12/25

Therefore, we can conclude that the missing probability is P(A|B) is 12/25.

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The complete question is:

Find the missing probability.

P(B)=1/4P

(AandB)=3/25P

P(A|B)=?

Answer:

The correct answer is:

(A) enlargement

Answer:

A

Step-by-step explanation:

If you look at the angles or use something to measure it you can tell that a and b are opposite sides that are parallel to each other. So they are equal.

Toheebah covered a distance of 9600 meters by running around the playground 12 times.To

The perimeter of a rectangle can be calculated by adding the lengths of all four sides. In this case, the playground has dimensions of 240 meters by 160 meters, so the perimeter is:

Perimeter = 2 * (Length + Width)

Perimeter = 2 * (240m + 160m)

Perimeter = 2 * 400m

Perimeter = 800m

Therefore, the perimeter of the playground is 800 meters.

To find the distance Toheebah covered by running around the playground 12 times, we multiply the perimeter by 12:

Distance covered = Perimeter * Number of laps

Distance covered = 800m * 12

Distance covered = 9600m

So, Toheebah covered a distance of 9600 meters by running around the playground 12 times.

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MRP Scheme: Determine demand, calculate gross requirements, consider lead time, determine scheduled receipts, calculate planned order receipts, determine planned order release, monitor inventory levels, repeat process periodically. Lot-for-lot ordering, no safety stock.

MRP (Material Requirements Planning) Scheme:

Determine the Demand:

Gather the demand forecast or sales orders for the upcoming period.

Calculate the net demand by subtracting any existing inventory from the total demand.

Calculate the Gross Requirements:

Gross requirements are the total quantity of materials needed to fulfill the net demand.

Consider the lead time of two weeks when calculating the gross requirements.

Determine the Scheduled Receipts:

Check if there are any open purchase or production orders scheduled to arrive within the lead time.

Include the quantities of materials expected to be received during this period.

Calculate the Planned Order Receipts:

Subtract the scheduled receipts from the gross requirements to determine the planned order receipts.

If the planned order receipts are negative, no action is required as the scheduled receipts are sufficient.

If the planned order receipts are positive, create a purchase or production order for that quantity.

Determine the Planned Order Release:

The planned order release specifies when the purchase or production order should be initiated.

It is typically based on the lead time, taking into account any constraints or preferences.

The planned order release should ensure that the materials arrive in time to meet the net demand.

Monitor Inventory Levels:

Keep track of inventory levels regularly to identify any deviations from the plan.

If the actual inventory falls below the net demand, adjust the planned order releases accordingly.

If the actual inventory exceeds the net demand, consider reducing the planned order releases.

Repeat the Process:

Review and update the MRP scheme periodically based on the changing demand and inventory levels.

Continuously adjust the planned order releases to align with the updated requirements.

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Answer:

130 blackberries

Step-by-step explanation:

[tex]\displaystyle 600\biggr(\frac{78^\circ}{360^\circ}\biggr)=130[/tex]

The difference of the two polynomials is [tex]2x^3 + 2x[/tex]. Therefore option no 2 is correct from the image given.

A polynomial is a mathematical expression along with variables and coefficients, combined using only addition, subtraction, multiplication, & non-negative integer exponents.

To discover the distinction of those two polynomials, we want to carry out polynomial subtraction.

[tex]{x^4+x^3+x^2+x} - {x^4-x^3-x^2-x}[/tex]

[tex]= x^4 + x^three + x^2 + x - x^4 + x^3 + x^2 + x ([/tex] distribute the negative sign to every term inside the 2nd set of brackets)

[tex]= 2x^3 + 2x[/tex] (integrate like terms)

Therefore, the difference of the two polynomials is [tex]2x^3 + 2x.[/tex]

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Answer: A  7x2+3y2-4x

Step-by-step explanation:

Answer:

6/MN = 8/9

8MN = 54

MN = 6 3/4

The correct answer is B.