Solve these using quadratic formula
Write out your work!

I need the vertex
Axis of symmetry
Intercepts -also include the function to find the intercepts-

(3) The linear equation is y = 1/3x + 1

(4) The solution to the system is (-1, -1/2)

(5) The factored expression of 45x² + 30x - 40 is 5(3x - 2)(3x + 4)

(6) The factored expression of 4x² - 36 is 4(x - 3)(x + 4)

(7) The surface area is 176

(8) The volume is 144

From the graph, we have the following points

(0, 1) and (-3, 0)

A linear relation is represented as

y = mx + c

So, we have

y = mx + 1

Using the other point, we have

-3m + 1 = 0

m = 1/3

So, we have

y = 1/3x + 1

Next, we solve graphically

To do this, we plot the graphs and write out the point of intersection

In this case, the intersection point is (-1, -1/2)

So, the solution is (-1, -1/2)

Here, we have

45x² + 30x - 40

When factored, we have

45x² + 30x - 40 = 5(3x - 2)(3x + 4)

Also, we have

4x² - 36

When factored, we have

4x² - 36 = 4(x - 3)(x + 4)

For figure (8), we have

Surface area = 2 *(9 * 4 + 4 * 4 + 9 * 4)

Evaluate

Surface area = 176

Also, we have

Volume = 9 * 4 * 4

Evaluate

Volume = 144

Hence, the surface area is 176 and the volume is 144

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The equation of line L2 in the form y = mx + c is y = (2/3)x + 23/3.  the x-intercept of L3 is -14/3. the y-intercept of L3 is -7. The point of intersection between L2 and L3 is (-88/13, 41/13).  The x-intercept of L3 is -14/3.

(a) To find the equation of line L2 perpendicular to line L1, we need to determine the slope of L1 and then use the negative reciprocal of that slope for L2.

Given the equation of line L1: 3x + 2y = 12

Rearrange the equation to isolate y:

2y = -3x + 12

y = (-3/2)x + 6

The slope of L1 is -3/2.

Since L2 is perpendicular to L1, the slope of L2 is the negative reciprocal of -3/2, which is 2/3.

Using the slope-intercept form y = mx + c, where m is the slope and c is the y-intercept, we can write the equation of L2:

y = (2/3)x + c

To find c, we substitute the coordinates of the point (2,9) into the equation:

9 = (2/3)(2) + c

9 = 4/3 + c

c = 9 - 4/3

c = 27/3 - 4/3

c = 23/3

Therefore, the equation of line L2 in the form y = mx + c is:

y = (2/3)x + 23/3

(b)(i) To find the equation of line L3 parallel to line L1, we can use the same slope as L1, which is -3/2, and substitute the given point (-4,-1) into the point-slope form y - y1 = m(x - x1):

y - (-1) = (-3/2)(x - (-4))

y + 1 = (-3/2)(x + 4)

y + 1 = (-3/2)x - 6

y = (-3/2)x - 7

Therefore, the equation of line L3 in the form ax + by = c is:

(3/2)x + y = -7

(ii) To find the x-intercept of L3, we set y = 0 and solve for x:

0 = (-3/2)x - 7

(3/2)x = -7

x = (-7)(2/3)

x = -14/3

Therefore, the x-intercept of L3 is -14/3.

To find the y-intercept of L3, we set x = 0 and solve for y:

y = (-3/2)(0) - 7

y = -7

Therefore, the y-intercept of L3 is -7.

(iii) To find the point of intersection between L2 and L3, we can set the equations of L2 and L3 equal to each other and solve for x and y:

(2/3)x + 23/3 = (-3/2)x - 7

To solve for x:

Multiply both sides by 6 to eliminate the fractions:

4x + 46 = -9x - 42

4x + 9x = -42 - 46

13x = -88

x = -88/13

Substitute the value of x back into either equation to solve for y:

y = (2/3)(-88/13) + 23/3

y = -176/39 + 299/39

y = 123/39

y = 41/13

Therefore, the point of intersection between L2 and L3 is (-88/13, 41/13).

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a) The possible pairs of mutually exclusive outcomes are given below:

A and B

A and C

A and D

B and C

B and D

C and D

b) The Three of the outcomes that are said to be all mutually exclusive are:

c) The probability of having B and D is written as P(B or D) = P(B) + P(D)

Mutually exclusive events cannot happen at the same time, like rolling a kick the bucket and getting a 1 and a 2. In this case, there are four conceivable results: A, B, C, and D, a few of which we commonly select.

For example, If A is the outcome, C or D cannot be. To find mutually exclusive pairs, we check which outcomes cannot occur together. A and B are mutually exclusive because they cannot both be true.

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Answer:

-4

Step-by-step explanation:

5x=-13-7

5x=-20

x=-20/5

x=-4

Answer:

5x+7=-13

5x=-13-7

5x=-20

x=-20 divide by 5

x=-4

Answer:

130°

Step-by-step explanation:

<4=<8

180-50=130

Answer:

-5,0,1,2,3

Step-by-step explanation:

in a number line the negative numbers are always on your left hand side and as you head to the right they increase thus becoming positive integers

Check the picture below.

[tex]\textit{Law of Sines} \\\\ \cfrac{a}{\sin(\measuredangle A)}=\cfrac{b}{\sin(\measuredangle B)}=\cfrac{c}{\sin(\measuredangle C)} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{h}{\sin(157^o)}=\cfrac{240}{\sin(6^o)}\implies h\sin(6^o)=240\sin(157^o) \\\\\\ h=\cfrac{240\sin(157^o)}{\sin(6^o)}\implies h\approx 897.1~cm[/tex]

Make sure your calculator is in Degree mode.

The possible values of x in the circle using the theorem of intersecting secants are x = 7.38 and x = -11.38

From the question, we have the following parameters that can be used in our computation:

The intersecting secants

Using the theorem of intersecting secants, we have the following equation

(5 + 7) * 7 = (x + 4) * x

When the sum is evaluated, we have

12 * 7 = (x + 4) * x

When the product is evaluated, we have

84 = x² + 4x

So, we have

x² + 4x - 84 = 0

When solved for x, we get

x = 7.38 and x = -11.38

The value of x = -11.38 cannot be used

This is because x cannot be negative

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Answer:

2nd choice. -a^3 - 8ab^2

Step-by-step explanation:

(3a^3b - 4ab^2 + 5ab) - (4a^3b + 4ab^2 + 5ab)

Distribute the minus sign and take out the parenthesis.

3a^3b - 4ab^2 + 5ab - 4a^3b - 4ab^2 - 5ab

Combine like terms.

3a^3b - 4ab^2 -4ab^2 - 4ab^2 + 5ab - 5ab

-1a^3b - 8ab^2

Answer:

  [tex]f_{ave}=\dfrac{9\pi}{4}[/tex]

Step-by-step explanation:

You want the average value of f(x) = √(81 -x²) on the interval [0, 9].

The function f(x) defines a quarter circle of radius 9 in the first quadrant on the given interval. Its area is given by the formula in the problem statement:

  A = (1/4)πr² = (π/4)·81

The average value of the function is the area divided by the width of the interval:

  [tex]f_{ave}=\dfrac{\dfrac{81\pi}{4}}{9}\\\\\\\boxed{f_{ave}=\dfrac{9\pi}{4}}[/tex]

__

Additional comment

You will notice that the average value is π/4 times the radius. This is also true for a semicircle. The attachment shows the rectangle with area equal to that of the quarter circle.

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Answer: AE=10, AD=14, x=5, BC=36, DE=52

Step-by-step explanation: finding x

4x-32=2(3x-21) || 4x-32=6x-42 || 10=2x || 5=x

The equations has 8 a solution are

x/2 = 16

x/8 = 1

x/2 = 8

x/8 = 1

To check whether 8 is a solution for an equation, we substitute x = 8 into the equation and see if it satisfies the equation. If it does, then 8 is a solution; otherwise, it is not.

Substituting x = 8 into each equation, we get:

1. x + 6 = 8 + 6 = 14 (8 is not a solution)

2. 2x + 2 = 2(8) + 2 = 18 (8 is not a solution)

3. 4x - 2 = 4(8) - 2 = 30 (8 is not a solution)

4. 10 = 10 (8 is not a solution)

5. 2x = 4 (8 is not a solution)

6. 3x = 24 (8 is not a solution)

7. 8/2 = 4 (8 is a solution)

83 8/8 = 1 (8 is a solution)

Therefore, the correct answers are:

x/2 = 16

x/8 = 1

x/2 = 8

x/8 = 1

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Answer:  From the calculations, the pair of points with a slope of 4 is option c. (0, -3), (1, 1).

Step-by-step explanation:  

To determine the slope between two points, we can use the formula:

slope = (change in y) / (change in x)

Let's calculate the slope for each pair of points:

a. (1, 4), (–1, 4)

The change in y is 4 - 4 = 0

The change in x is -1 - 1 = -2

slope = 0 / -2 = 0

b. (0, 4), (4, 0)

The change in y is 0 - 4 = -4

The change in x is 4 - 0 = 4

slope = -4 / 4 = -1

c. (0, –3), (1, 1)

The change in y is 1 - (-3) = 4

The change in x is 1 - 0 = 1

slope = 4 / 1 = 4

d. (3, 0), (–1, –1)

The change in y is -1 - 0 = -1

The change in x is -1 - 3 = -4

slope = -1 / -4 = 1/4

Answer:

c. (0, –3), (1, 1)

Step-by-step explanation:

To find the slope, we use the slope formula

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

a. (1, 4), (–1, 4)

m = (4-4)/(-1-1) = 0/-2 = 0

b. (0, 4), (4, 0)

m = ( 4-0)/(0-4) = 4/-4 = -1

c. (0, –3), (1, 1)

m = ( -3-1)/(0-1) -4/-1 = 4

d. (3, 0), (–1, –1)

m = (0--1)/(3 - -1) =1/4

After considering all the given options we conclude that the value of the given formula is y₁ = y₂ - m(x₂ - x₁) which is Option C.

To evaluate for y₁ in the equation m = (y₂ - y₁) / (x₂ - x₁), we can apply algebraic manipulation to isolate y₁.
In the first step, we  multiply both sides of the equation by (x₂ - x₁) to eliminate the denominator. Therefore,
m(x₂ - x₁) = y₂ - y₁
In the second step, we can subtract y₂ from both sides of the equation to isolate y₁.
Therefore,
y₁ = y₂ - m(x₂ - x₁)
So, the solution for y₁ is given by the formula:
y₁ = y₂ - m(x₂ - x₁)
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Answer:

Triangular Pyramid

Step-by-step explanation:

It is triangular and looks like a pyramid. It also has a square base and four triangular faces, which is usual for a triangular pyramid.

Answer:

C. 62.8 mi

Step-by-step explanation:

Formulas to know:

[tex]c = \pi d[/tex]

[tex]c=2\pi r[/tex]

For this problem use c = [tex]\pi[/tex]d

[tex]c =\pi d[/tex]

[tex]c = 3.14*20[/tex]

[tex]c = 62.8[/tex]

Answer:

  x = 106

Step-by-step explanation:

You want the measure of the unknown angle (x°) in the quadrilateral with angles 47°, 138°, and 69° given.

The sum of angles in a quadrilateral is 360°, so the measure of x is ...

  x = 360 -47 -138 -69 = 106

Angle x° is 106°.

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Answer:

Part A:

Let x be the amount the artist receives for her products this week. Then, the absolute value equation that represents the scenario is:

|250 - x| ≤ 25

Part B:

To solve the equation, we need to consider two cases:

Case 1: 250 - x ≤ 25

In this case, we have:

250 - x ≤ 25

x ≤ -225

x ≥ 225

Case 2: 250 - x ≥ -25

In this case, we have:

250 - x ≥ -25

x ≥ -275

x ≤ 275

Therefore, the solution to the absolute value equation is:

225 ≤ x ≤ 275

Part C:

The minimum amount the artist received for her products is $225, and the maximum amount is $275.

Step-by-step explanation:

The height of the building is approximately 45.44 ft when it casts a 68-ft shadow under an angle of elevation of 34 degrees.It's important to note that this calculation assumes the ground is flat and the sun's rays are parallel. Real-world conditions may introduce slight variations.

To determine the height of the building, we can use the           trigonometric relationship involving the angle of elevation, the height of the building, and the length of its shadow.

Let's denote the height of the building as h. The length of the shadow cast by the building is given as 68 ft, and the angle of elevation of the sun is 34 degrees.

Using the tangent function, we have the equation:

tan(34°) = h / 68To

solve for h, we can rearrange the equation:

h = 68 * tan(34°)

Calculating this expression, we find:

h ≈ 68 * 0.668178 = 45.439344 ft.

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Answer: 8.3

Step-by-step explanation: Add all the numbers together, 11.7+5.4+17.3+3.3+3.8+9.4+6.9 equals 57.8 BUT WAIT! We aren't done, we have to divide the answer by the sum of all answers, so since there are 7 numbers we are adding together, we divide by 7, so 57.8 divided by 7 equals 8.3 (which is rounded to the nearest tenth) and theres your asnwer. Hope this helps!

The area of the triangle is 18.6 units²

Given data ,

Let the triangle be represented as ΔABC

Now , the sides of the triangle are

We can use the law of cosines to find the length of the third side of the triangle:

c² = a² + b² - 2ab cos(C)

c² = 5² + 8² - 2(5)(8)cos(79°)

c ≈ 9.05

Now, we can use Heron's formula to find the area of the triangle:

s = (5 + 8 + 9.05)/2 = 11.525

A = √(s(s-a)(s-b)(s-c))

A = √(11.525(11.525-5)(11.525-8)(11.525-9.05))

A ≈ 18.6

Hence , the area of the triangle is approximately 18.6 square units

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Answer:

  276.460 mi²

Step-by-step explanation:

You want the surface area of a cylinder with radius 4 mi and height 7 mi.

The area of a cylinder is given by the formula ...

  A = 2πr(r +h)

  A = 2π(4 mi)(4 mi +7 mi) = 88π mi² ≈ 276.460 mi²

The area of the cylinder is about 276.460 square miles.

__

Additional comment

The surface area is the sum of the areas of the two circular bases and the lateral area. The lateral area is the product of the cylinder height and circumference. These are combined in the formula given.

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the volume of the cone to be filled up if it has the same radius and height with is  376. 6 cm³

To determine the volume, we need to know the formula for the volume of a cone.

The formula for the volume of a cone is expressed as;

V = πr²h/3

Such that the parameters are expressed as;

Substitute the values, we have;

Volume = 3.14 × 6² × 10/3

Find the square value

Multiply the value, we have;

Volume = 1130. 4/3

Divide the values

Volume = 376. 6 cm³

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The scale factor from triangle BAC to triangle GHF is 3/4.

We have,

The two triangles are:

ΔBAC and ΔGHF

Now,

The scale factor can be used as:

Taking the corresponding sides.

BA x M = GH

36 x M = 27

M = 27/36

M = 3/4

We can also see that,

EC x 3/4 = CF

28 x 3/4 = 21

7 x 3 = 21

21 = 21

Thus,

The scale factor from triangle BAC to triangle GHF is 3/4.

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The expected value of the winnings in this game is $1.23.

We have,

The expected value of a random variable is the sum of the products of each possible outcome and its probability.

In this case, the random variable is the winnings from the game and the payout probability distribution specifies the probabilities associated with each possible payout.

The expected value of the winnings is found by multiplying each payout by its probability and summing the products.

Expected value = (0)(0.50) + (1)(0.25) + (2)(0.13) + (4)(0.06) + (8)(0.06)

= 0 + 0.25 + 0.26 + 0.24 + 0.48

= 1.23

Therefore,

The expected value of the winnings in this game is $1.23.

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The volume of the prism is 1368 cubic feet.

We have,

The three steps that should be taken to calculate the volume of the prism are:

- Use the formula A = b h to find the area of the base, where b is the length of the base and h is the width of the base.

For a rectangular prism, the base is a rectangle, so the formula is:

A = length x width.

- Use the formula V = B h to find the volume of the prism, where B is the area of the base and h is the height of the prism.

For a rectangular prism, the formula.

V = length x width x height.

- Substitute the given values into the formulas and solve for the volume. For the given rectangular prism with a length of 9 and one-half feet, a width of 24 feet, and a height of 6 feet, we can find the volume by using the formula V = B h.

First, we find the area of the base by using the formula:

A = length x width:

A = (9.5 ft) x (24 ft)

= 228 sq ft

Next, we use the formula V = B h to find the volume:

V = (228 sq ft) x (6 ft)

= 1368 cubic feet

Therefore,

The volume of the prism is 1368 cubic feet.

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The area of the image of the rectangle under the transformation is 10 square units

From the question, we have the following parameters that can be used in our computation:

Matrix transformation: [tex]\left[\begin{array}{cc}-7&-2\\6&12\end{array}\right][/tex]

And, we have rectangle plotted on the graph

The transformation is a rigid transformation

This means that the area of the rectangle would be the same before and after the transformation

From the graph, the side lengths of the rectangle are

Length = 2

Width = 5

So, we have

Area = 2 * 5

Evaluate

Area = 10

Hence, the area of the image of the rectangle under the transformation is 10 square units

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