Chiara pays $21 for 3 DVDs that she finds in a one-price bin. Gunther wants to know how much he would spend for 1 DVD in this bin. Chiara wants to know how many dollars she will need to purchase 9 DVDs the next time she goes.

Answer:

Step-by-step explanation:

Given is a rational algebraic expression in x as

[tex]\frac{5x+2}{x^4-4x^2}[/tex]

First let us factorize the denominator.

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

Let the given term be

[tex]\frac{A}{x} +\frac{B}{x^2} +\frac{C}{x-2} +\frac{c}{x+2} \\\\=\frac{Ax(x^2-4)+B(x^2-4)+cx^2(x+2)+Dx^2(x-2)}{x^4-4x^2}[/tex]

Equate the numerator to get

[tex]Ax(x^2-4)+B(x^2-4)+cx^2(x+2)+Dx^2(x-2)[/tex]=[tex]5x+2[/tex]

Substitute x =-2

[tex]4D (-4) = -8\\D=\frac{1}{2}[/tex]

Now substitute x =2

We get

[tex]16C=12\\C=\frac{3}{4}[/tex]

Put x=0

[tex]-4B=2\\B=\frac{-1}{2}[/tex]

Next equate X cube terms to 0 on right side

[tex]A+C+D=0\\A=\frac{-5}{4}[/tex]

Hence partial fraction is

[tex]\frac{-1}{2x^2} -\frac{5}{4x} +\frac{1}{2(x+2)} +\frac{3}{4(x-2)}[/tex]

To decompose the rational expression startfraction 5x + 2 over x4 - 4x2 endfraction into partial fractions, we need to factor the denominator. The partial fraction decomposition of the given rational expression is startfraction A over x + B over x2 + C over (x - 2)2, where A, B, and C are the determined coefficients.

To decompose the rational expression startfraction 5x + 2 over x4 - 4x2 endfraction into partial fractions, we need to factor the denominator. The denominator can be factored as x2(x2 - 4) which gives us two distinct linear factors x and x + 2, and a repeated linear factor x - 2. Therefore, we can express the rational expression as follows: startfraction A over x + B over x2 + C over (x - 2)2.

To find the values of A, B, and C, we can use the method of equating coefficients. By multiplying both sides of the equation by the common denominator, we can eliminate the fractions and solve for the unknowns. Finally, we substitute the values of A, B, and C back into the partial fraction decomposition expression.

Therefore, the partial fraction decomposition of the given rational expression is startfraction A over x + B over x2 + C over (x - 2)2, where A, B, and C are the determined coefficients.

We have been given a system of nonlinear equations.

[tex]\\y=40x^{2},y=19x+3[/tex]

In order to solve this system we can first equate the two equations in order to get a quadratic in x.

[tex]\\40x^{2}=19x+3[/tex]

Our next step is to bring all the terms on one side.

[tex]\\40x^{2}-19x-3=0[/tex]

Now we have to solve this equation. We can solve it by factoring using the splitting middle term method.

[tex]\\40x^{2}-24x+5x-3=0[/tex]

[tex]\\8x(5x-3)+(5x-3)=0[/tex]

[tex]\\(8x+1)(5x-3)=0[/tex]

Upon setting each of these factors equal to zero using zero product property we get

[tex]\\8x+1=0 \text{ or } 5x-3 = 0[/tex]

Upon solving both these equations we get

[tex]\\x=-\frac{1}{8} \text{ or } x=\frac{3}{5}[/tex]

Now we can substitute these values of x in the equation

[tex]y=19x+3[/tex]

We get

[tex]\text{For }x=- \frac{1}{8} \Rightarrow y=19(- \frac{1}{8})+3= \frac{5}{8}\\[/tex]

[tex]\text{For }x= \frac{3}{5}\Rightarrow y=19( \frac{3}{5})+3= \frac{72}{5}\\[/tex]

Therefore, our final set of solutions are

[tex](x,y)=(- \frac{1}{8},\frac{5}{8}) \text{ and } ( \frac{3}{5}, \frac{72}{5})[/tex]

Answer:

c 42 degrees

Step-by-step explanation:

Answer:

8 times

Step-by-step explanation:

I found the volume of the sphere. Then increased it how many times he blew. That was the ratio.

Also, Khan :)

The ratio of the current volume of the balloon to the original volume of the balloon is 8.

The volume of a sphere is given by (4\3)πr³ where r is the radius of the sphere.

The initial volume of the balloon = [tex]\frac{4}{3} \pi 2^{3}[/tex]

The initial volume of the balloon =[tex]\frac{32}{3} \pi[/tex]

Radius of the balloon after first breath = 2+1 =3cm

The Radius of the balloon after the second breath = 2+2=4cm

So, the new volume of the balloon = [tex]\frac{4}{3} \pi 4^{3}[/tex]

The new volume of the balloon = [tex]\frac{256}{3} \pi[/tex]

The ratio of new volume to the initial volume = [tex]\frac{\frac{256}{3 } \pi }{\frac{32}{3}\pi }[/tex]

The ratio of new volume to the initial volume =8

Hence, the ratio of the current volume of the balloon to the original volume of the balloon is 8.

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1. Brett was right because her baseball reached a maximum height of 103.52 feet.

2. A table that justifies Brett's claim is shown below.

In Mathematics and Geometry, the maximum value of any function is a point on the graph where the function reaches its highest point. Additionally, the maximum value of Brett's quadratic function can be determined by finding the axis of symmetry;

Axis of symmetry, x = -b/2a

[tex]h(t) = -16t^2 + 79t + 6[/tex]

where:

a = -16,  b = 79, and c = 6.

By substituting the values of a and b into the formula, we have;

Axis of symmetry, x = -79/2(-16)

Axis of symmetry, x = 79/32 = 2.46875.

Now, we can determine the maximum value (vertex) as follows;

[tex]h(t) = -16t^2 + 79t + 6\\\\h(2.46875) = -16(2.46875)^2 + 79(2.46875) + 6[/tex]

h(2.46875) = 103.52 feet.

Based on the graph, we can logically deduce that the maximum height attained by Andre's baseball was below 80 feet. Hence, Brett was right.

Lastly, we would complete the table as follows;

t            Brett         Andre

0             6                 6

1              69              58

2             100             78

3              99              67

4              66               6

5               0                

In conlusion, we can logically deduce that Brett was right because her baseball reach a maximum height of 103.52 feet, which was greater than Andre's.

Answer: The possible source of error in this activity is " the accuracy of making and recording measurements"

The height of each and every student in the class has to be taken very accurately. Even if there was a single error in either making the calculation or even recording it, the average height will take a hit and will result in the entire exercise being a failure.

Hence, it’s very important to accurately measure the height and record and even have a second person do the same activity so that the numbers recorded are accurate.

Answer: 2 apples

Step-by-step explanation:

Logic

The volume of the coffee in the can if the can has a radius of 4 inches and height of 9 inches is 452.16 inches³.

The can must be in the shape of a cylinder.

The formula to find the volume of the cylinder is,

Volume of cylinder = π r² h

Here r is the radius of the base of the cylinder and h is the height of the cylinder.

Given, Radius, r = 4 inches and Height, h = 9 inches

Volume of the coffee = π (4)² (9)

                                    = 144 π

                                    = 144 × 3.14

                                    = 452.16 inches³

Thus, the volume of the coffee in the can is 452.16 inches³.

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Answer:

At 10 PM both candles will be same height.

Step-by-step explanation:

Let h represent the number of hours.

We have  twelve inch candle and an 18 inch candle are lit at 6pm and the 12-in. candle burns 0.5 inches every hour. The 18 inch candle burns 2 inch every hour.

Height 18 inch candle after h hours, = 18 - 2h

Height 12 inch candle after h hours, = 12 - 0.5h

At same height we have

                18 - 2h = 12 - 0.5h

                   1.5 h = 18 - 12

                         h = 4 hours.

Time = 6 PM + 4 hours = 10 PM

At 10 PM both candles will be same height.

Answer:

At 10 PM, both the candles will be of same height.

Step-by-step explanation:

Two candles are lit at 6 PM.

The length of candles is 12 inch and 18 inch.

The rate of burning of 12 inch candle is 0.5 inch per hour.

The rate of burning of 18 inch candle is 2 inch per hour.

Let [tex]h[/tex] represents the number of hours of burning.

In [tex]h[/tex] hours, the remaining length of the 12 inch candle will be,

[tex]l_1=12-0.5\times h[/tex]

In [tex]h[/tex] hours, the remaining length of the 18 inch candle will be,

[tex]l_2=18-2\times h[/tex]

Now, it is required to find the time at which the length of both the candles will be same.

So, the time [tex]h[/tex] in hours will be,

[tex]l_1=l_2\\12-0.5h=18-2h\\1.5h=6\\h=4\rm hour[/tex]

So, after 4 hours, both the candles will be of same length.

Therefore, the time will be [tex]6+4=10[/tex] PM. At 10 PM, both the candles will be of same height.

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[tex] |\Omega|=100\\
|A|=14\\\\
P(A)=\dfrac{14}{100}=\dfrac{7}{50}=14\% [/tex]

Answer: Yes, the 3 foot 3 foot square opening in your ceiling is big enough to insert a  4x8 piece of plywood.

In  a 3x3 foot square, we can form two right triangles with same diagonal side or hypotenuse. .

Using the Pythagorean theorem/formula, we can find the length of the hypotenuse or the longest side of the triangle ( c) given the two sides a and b

Substitute values a=3 and b=3 to formula

a^2^+b^2^=C^2^=

3^2^+3^2^=C^2^=

9+9= C^2^

C= square root of 18

C=4.24 

A 4x8 piece of plywood can be inserted in a 4.24 opening diagonally.

In addition to ...

 degenerate ellipse

 point

it is also a degenerate circle (special case of ellipse).

___

The vertex of a double-napped cone is a point. So, the intersection of only the vertex with a plane is a point, not a degenerate parabola (a line) or degenerate hyperbola (pair of crossing lines).

Essentially, the plane must intersect at an angle greater than the angle of the edge of the cone with the axis of the cone. If the plane is not at the vertex, such an intersection angle will result in a circle or ellipse. As the plane gets closer to the vertex, the ellipse gets smaller, degenerating to a single point.

Answer:

The answer is:

        Degenerate ellipse.

         Point.

Step-by-step explanation:

We know that when a plane intersects a double-napped cone only at the cone's vertex then the cross-section that is obtained is the point.

Also we know that a point is also considered as a circle with radius 0.

and a circle is also a kind of a ellipse when the length of both the transverse and conjugate axis are equal.

Hence, the term that best describe the degenerate conic section that is formed is:

degenerate ellipse

point

Answer with explanation:

We know that a polynomial is classified based on the degree as follows:

Constant-

If it has a degree zero.

and the equation of such a polynomial is:

   p(x)=a

Linear--

It has degree 1.

and the equation of such a polynomial is:   p(x)=ax+b

Quadratic--

If it has degree two and the equation of the polynomial is given by:

     [tex]p(x)=ax^2+bx+c[/tex]

Based on number of Terms--

Monomial--

If it has just one term.

Binomial--

If it has two terms.

Trinomial--

If it has three terms.

Answer:

D) Reflecting over the y-axis and then reflecting over the line y = -x

Step-by-step explanation:

When the figure is rotate clock-wise or anti clock-wise then the coordinate of figure is get changed and this new coordinate can be find on the basis of degree of rotation.

The rules of rotation is:

Thus, Option (D) is correct option.

Firstly, simplife the function [tex] -x^3-x^2+4x+4 [/tex] in the following way:

[tex] -x^3-x^2+4x+4=-(x^3+x^2)+4(x+1)=-x^2(x+1)+4(x+1)=(x+1)(4-x^2)=(x+1)(2-x)(2+x)=-(x+2)(x+1)(x-2) [/tex].

There are three x-intercepts: x=-1; x=2 and x=-2 (therefore, options A and D are incorrect).

Since before brackets is sign "-", you should state that graph starts down on the left (you can check this substituting smaller negative number than -2 into function expression. For example, if x=-10, then f(-10)=-(-10+2)(-10+1)(-10-2)=-(-8)(-9)(-12)>0) and goes up on the right (you can check this substituting greater positive number than 2 into function expression. For example, if x=10, then f(-10)=-(10+2)(10+1)(10-2)=-(12)(11)(8)<0).

Answer: correct choice is B.