A group of friends wants to go to the amusement park. They have no more than $65 to spend on parking and admission. Parking is $9.75, and tickets cost $16.25 per person, including tax. Write and solve an inequality which can be used to determine pp, the number of people who can go to the amusement park.

The equation of the ellipse having foci  (±2,0) and vertices (±5,0) is [tex]x^{2} /4+y^{2} /21=1[/tex].

Given the foci of ellipse is (±2,0) and vertices (±5,0).

We are required to find the equation of ellipse having foci (±2,0) and vertices (±5,0).

Equation is the relationship between two or more variables that are expressed in equal to form. Equation of two variables looks like ax+by=c.

Equation of ellipse is as under:

[tex]x^{2} /a^{2} +y^{2} /b^{2} =1[/tex],where a is the semi major axis.

We have been given that a=5, c=2.

We know that [tex]c^{2} =a^{2} -b^{2}[/tex]

We have to find the value of b.

b=[tex]\sqrt{a^{2} -c^{2} }[/tex]

=[tex]\sqrt{5^{2} -2^{2} }[/tex]

=[tex]\sqrt{25-4}[/tex]

=[tex]\sqrt{21}[/tex]

Using the values of a and b in the equation [tex]x^{2} /a^{2} +y^{2} /b^{2} =1[/tex]

[tex]x^{2} /(2)^{2} +y^{2}/\sqrt{21} ^{2} =1[/tex]

[tex]x^{2} /4+y^{2} /21=1[/tex]

Hence the equation of the ellipse having foci  (±2,0) and vertices (±5,0) is [tex]x^{2} /4+y^{2} /21=1[/tex].

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Answer: 12x squared + 2x + 96

Step-by-step explanation: We first simplify the parentheses. 8x2 = 16 - 4x - 8 = 16-8-4x = 8-4x. 2x2 = 4 +3x+2 = 4+2+3x = 6+3x. So, we are left with (16-4x)(6+3x).  We rearrange the problem so it looks like this: (-4x + 16)(3x + 6). -4x(3x+6)  + 16(3x+6). Now we just multiply. And, we get 12 x squared + 2x + 96

A quantity of 84 shingles are required to cover the surface area of 336 square feet of the first roof.

A quantity of 110 shingles are required to cover the surface area of 220 square feet of the second root.

In this question we must determine both the surface area and the number of shingles to cover two roof configurations as well. In the first case, we calculate the surface area, which is the sum of the area of rectangles:

A = 2 · (8 ft) · (12 ft) + 2 · (6 ft) · (12 ft)

A = 336 ft²

And the number of shingles required to cover the root is:

n = (336 ft²) / (4 ft²)

n = 84

A quantity of 84 shingles are required to cover the surface area of 336 square feet of the first roof.

In the second, we only need to cover the very top surface. The area and the number of shingles are, respectively:

A = 2 · (5 ft) · (22 ft)

A = 220 ft²

n = (220 ft²) / (2 ft²)

n = 110

A quantity of 110 shingles are required to cover the surface area of 220 square feet of the second root.

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Assumed that sides are equal

CB

The number of servings of pasta were in the dish Lois made is 31.67.

Number of servings of pasta were in the dish Lois made = Number of cups of pasta Lois made / A serving of the pasta

= 6 1/3 ÷ 0.2

= 19/3 ÷ 0.2

= 19/3 × 1/0.2

= (19×1) / (3 × 0.2)

= 19/0.6

= 31.6666666666666

Approximately,

Number of servings of pasta were in the dish Lois made is 31.67 servings

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Answer:

x = -9

Step-by-step explanation:

-18+24 = -2(x+6) Distribute and simplify

6 = -2x-12 Add 12 to both sides

18 = -2x Divide by -2 on both sides

-9 = x Here's your answer

Hope this helps! :D

First, do -2 times x. that is -2x.  Next, do -2 times +6. That should be -12. The last steps are in order, add 18 on both sides. it should be -18+18 and -12+18. now you have 24 on the left and -2x and +6. -6 on both sides. 6-6 and 24-6. now you have 18 and -2x. so get rid of the -2x by doing -2x/-2 and on the left side 18/-2. So the answer should be -9=x.

Based on the given task content; the simplification of 7x - x to its simplest form is 6x.

Simplify 7x - x

7x and -x

So,

7x - x

= 6x

Therefore, 6x is the result of the simplification of 7x - x to its simplest form.

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2x4=8

8x2=16

the answer is 16

The two possible positions of point P are:

(3, -5) and (0.6, 5.8)

We know that point P lies on the line:

y = 4 - 3*x

And that the distance between P and the origin is √34, then if the coordinates of point P are (x, y), we have that:

[tex]\sqrt{34} = \sqrt{x^2 + y^2}[/tex]

Now, we can replace "y" in the distance equation by the linear equation, and also remove the square roots:

[tex]34 = x^2 + (4 - 3x)^2[/tex]

Now we can solve the quadratic equation for x:

[tex]34 = x^2 + 9x^2 + 16 - 24x\\\\10x^2 - 24x - 18 = 0\\\\[/tex]

The solutions are:

[tex]x = \frac{24 \pm \sqrt{(-24)^2 - 4*10*(-18)} }{2*10} \\\\x = \frac{24 \pm36}{20}[/tex]

So the two solutions are:

x = (24 + 36)/20 = 3

x = (24 - 36)/20 = -0.6

To get the points, we need to evaluate y on these values:

y = 4 - 3*3 = -5   So we have P = (3, -5)

y = 4 - 3*(-0.6) = 2.2 So we have the point (0.6, 5.8)

There are the two possible positions of point P.

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The second store offered the better buy at the price of $92.00.

What is better buy?

Better buy refers to the lowest price out of the prices offered by the three stores.

In order to determine the lowest price, the no discount price of the first store needs to be compared to the prices of  two other stores, bearing that after-discount price is the pre-discount price multiplied by 1 minus the discount rate.

First store price=$95.00

Second store after-discount price=pre-tax discount price*(1-discount rate)

pre-discount price=$115

discount rate=20%

Second store after-discount price=$115*(1-20%)

Second store after-discount price=$92.00

Third store after-discount price=pre-tax discount price*(1-discount rate)

pre-discount price=$105

discount rate=10%

Third store after-discount price=$105*(1-10%)

Third store after-discount price=$94.50

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By the use of the completing the square method, the maximum height is 3 ±√29/4 + 9 m.

The maximum height has been modeled by the use of the equation f(x)= -4x^2 +24x -29. Now we know that f(x)= h so we can write;

h = -4x^2 +24x -29

Now we have;

0 = -4x^2 +24x -29

-4x^2 +24x -29= 0

x^2 - 6x = -29/4

(x - 3)^2 = -29/4 + 3^2

(x - 3)^2 =  -29/4 + 9

x = 3 ±√29/4 + 9 m

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Answer:

[tex]\textsf{D.} \quad \begin{cases} y \leq 2x+2 \\ y\geq -\dfrac{1}{2}x \end{cases}[/tex]

Step-by-step explanation:

Both lines in the given system of inequalities are linear equations.

Slope-intercept form of a linear equation:  y = mx + b

(where m is the slope and b is the y-intercept)

Positive slope (m):  as x increases, y increases.

Negative slope (-m):  as x increases, y decreases.

When graphing inequalities:

The inequality represented by the blue line has a positive slope and shading below the line.

⇒ y ≤ mx + b

From inspection of the graph, its y-intercept is 2.

⇒ y ≤ mx + 2

The inequality represented by the yellow line has a negative slope and shading above the line.

⇒ y ≥ -mx + b

From inspection of the graph, it's y-intercept is zero.

⇒ y ≥ -mx

The only pair of inequalities that satisfies the above conditions is:

[tex]\textsf{D.} \quad \begin{cases} y \leq 2x+2 \\ y\geq -\dfrac{1}{2}x \end{cases}[/tex]

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¹³C₈ + ¹³C₉ as a single term from Pascal's Triangle is ¹⁴C₉

Pascal's triangle is a triangle written in such a way that it forms the coefficients of a binomial expansion. The coefficients of the terms are gotten through combination.

Combination is the number of ways r in which n objects can be selected. It is given by ⁿCₓ = n!/x!(n - x)!

Since we have ¹³C₈ + ¹³C₉ and we want to write it as a single term, we have that

So,  ¹³C₈ + ¹³C₉ =  13!/8!5! +  13!/9!4!

= 13!/(8! × 5 × 4!) +  13!/(9 × 8! × 4!)

= 13!/8!4![1/5 + 1/9]

= 13!/8!4! × [(9 + 5)/45]

= 13!/8!4! × 14/45]

= 13!/8!4! × 14/(9 × 5)]

= 14 × 13!/8! × 9 × 4! × 5)]

= 14!/9!5!

= 14!/9!(14 - 9)!

= ¹⁴C₉

So,  ¹³C₈ + ¹³C₉ as a single term from Pascal's Triangle is ¹⁴C₉

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The given set of functions are not linearly independent.

Given,

[tex]f_{1} (x) = x\\f_{2} (x) = x^{2} \\f_{3} (x) = 6x-2x^{2}[/tex]

We need,

[tex]c_{1} f_{1} (x)+c_{2} f_{2} (x)+c_{3} f_{3}(x)=0[/tex]

Substituting the values in equation we get,

[tex]c_{1} x+c_{2} x^{2} +c_{3} (6x-2x^{2} )=0\\[/tex]

Computing the equation we get,

[tex]c_{1} x+c_{2} x^{2} +c_{3} 6x-c_{3} 2x^{2}=0[/tex]

[tex](c_{1} +6c_{3} )x+(c_{2} -2c_{3} x^{2} =0[/tex]

This resolves to two equations

[tex](c_{1} +6c_{3})x =0\\(c_{2} -2c_{3} )x^{2} =0[/tex]

These will have an infinite set of solutions:

[tex]c_{1} =-6c_{3} \\c_{2} =2c_{3}[/tex]

Two functions are said to be linearly independent if neither function is a constant multiple of the other.

Here, it is clear that the given functions are not linearly independent.

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......................

The inverse function of f(x) = 1/2x + 1/4 is f-1(x) = 2x - 1/2

The function is given as:

f(x) = 1/2x + 1/4

Rewrite the function as

y = 1/2x + 1/4

Swap the positions of x and y

x = 1/2y + 1/4

Multiply through by 4

4x = 2y + 1

Subtract 1 from both sides

2y = 4x - 1

Divide by 2

y = 2x - 1/2

Rewrite as an inverse function

f-1(x) = 2x - 1/2

Hence, the inverse function of f(x) = 1/2x + 1/4 is f-1(x) = 2x - 1/2

See attachment for the graph of the inverse function

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Answer:

Step-by-step explanation:

e graph opens upward, and it has a vertex of (0,0)

What is vertex?

The vertex of an equation is the minimum or the maximum point on the graph of the equation

The equation of the graph is given as:

The above equation represents an absolute value function

An absolute value function is represented as:

So, by comparison, we have:

This means that the graph opens upward, and it has a vertex of (0,0)

See attachment for the graph of

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Answer:

=========================

We need to make a stem and leaf plot

Rearrange the data in the ascending order:

Group the data as follows:

The first two digits form a stem and the last digits form a leaf.

Make a plot:

       Stem   |  Leaf

Answer: 2 and 3, or 3 and 2

Step-by-step explanation:

a + b = 5 so a = 5 - b

Substitute a = 5 - b into ab = 6; (5 - b)b =  6

5b - [tex]b^{2}[/tex] = 6

[tex]b^{2}[/tex] - 5b + 6 = 0

(b - 2)(b - 3) = 0

So b is either 2 or 3

So a is either 3 or 2 depending on what b is

Answer:

60

Step-by-step explanation:

Answer: 60

Step-by-step explanation:

Parabolas could Lauren be describing are:

c) y2 = x

d) y2 = 10x

F)  y2 = 5x

The intersection of a cone with a plane that is parallel to the side of the cone results in the formation of a parabola, which is a symmetrical open plane curve. Under the influence of gravity, the route taken by a projectile takes the form of a curve similar to this one.

In the question that is presented to her, Lauren gives an example of a parabola that has a positive non-zero coordinate value on focus

Generally, the equation for focus is  mathematically given as

y=a(x-h)^2+k

In conclusion

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Answer:

Step-by-step explanation:

Refer to attached diagram with added details.

Circle O with:

Since r = OC + CD, the radius is 4 cm.

Consider right triangles OAC or OBC:

Recall the property of 30°x60°x90° triangle:

It means OC: OA = 1 : 2, so angles AOC and BOC are both 60° as adjacent to short legs.

In order to find the shaded area we need to find the area of sector OADB and subtract the area of triangle OAB.

Area of sector:

Area of triangle AOB:

The shaded area is:

Answer:

8.57.

Step-by-step explanation:

that would be 360/42

= 8.5714

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

i really hate how we actually have to type something.

Volume of prism = [tex]\frac{1}{2x^{3} } + x^{2}[/tex]

Given that:

The oblique prism below has an isosceles right triangle base and the length of the base is x

=> the area of the base: [tex]\frac{1}{2}[/tex] × x × x = [tex]\frac{1}{2}[/tex]

The vertical height of the prism is (x + 2)

=> The volume of the oblique prism is:

                    V = the base area * the vertical height

<=> V = [tex]\frac{1}{2}[/tex] [tex]x^{2}[/tex] (x + 2)

<=> V =  [tex]\frac{1}{2x^{3} } + x^{2}[/tex]

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Answer: y ≥ [tex]\frac{3}{2}[/tex]x + 3

Step-by-step explanation:

       Since the shaded area is above the line, we will use either > or ≥ for our inequality.

       Next, since this is a solid line we will use the ≥ symbol for our inequality. This means the answer is the first option.

the expression that represents the volume of the prism is  14x² + 7x cubic centimeters. Option 2

The  volume of a pentagonal prism is determined using the formula;

V = 5/2abh

where

Now, let's substitute the values given

Let the base be 'x'

The apothem is 2. 8 centimeters

height is 2x + 1

Volume = 5/ 2 × 2. 8 × x × (2x + 1)

Volume = 14/ 2 × x(2x + 1)

Volume = 7x(2x + 1)

Volume = 14x² + 7x

Thus, the expression that represents the volume of the prism is  14x² + 7x cubic centimeters. Option 2

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The largest integers of the numbers is 406.

The sum of five consecutive positive integers is 2020.

Therefore,

let

the first positive integer = x

Hence,

x + x + 1 + x + 2 + x + 3 + x + 4 = 2020

5x + 10 = 2020

5x = 2020 - 10

5x = 2010

x = 2010 / 5

x = 402

largest integer = 402 + 4 = 406

Therefore, the largest integers of the numbers is 406.

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