Which of the following equations represents a quadratic function?

y(y + 4)(y - 6) = 0
z2 + 2 = 3z(z2 - 1)
(2x - 3)(4x + 5) = 10x
(3b)(5b – 7)(b + 8) = 0

Final Answer:

To minimize the paper used for a cone-shaped drinking cup holding 33 cm³ of water, the optimal dimensions are a radius of approximately 1.65 cm and a height of around 3.30 cm.

Explanation:

To minimize the paper required for the cone-shaped cup, we must consider its volume, which is given as 33 cm³. The formula for the volume of a cone is V = (1/3)πr²h, where r is the radius and h is the height. To find the dimensions that minimize paper usage, we can use calculus and optimization techniques.

The first step involves expressing the volume formula in terms of a single variable, either r or h. In this case, expressing it in terms of h is preferable. Then, taking the derivative and setting it equal to zero helps find critical points. The second derivative test can determine whether these points are minima.

Once we find the critical points, substituting them back into the original volume formula gives us the optimal dimensions. In this context, the optimal radius is approximately 1.65 cm, and the optimal height is around 3.30 cm. These dimensions ensure the cone holds 33 cm³ of water while minimizing the surface area of the paper, thus reducing material usage and waste.

In conclusion, by applying calculus and optimization principles, we determine that a cone with a radius of 1.65 cm and a height of 3.30 cm uses the smallest amount of paper to hold 33 cm³ of water.

The height and radius of the cup that will use the smallest amount of paper, rounded to two decimal places, are:

[tex]\[ \boxed{h \approx 6.04 \text{ cm}} \][/tex]

[tex]\[ \boxed{r \approx 3.02 \text{ cm}} \][/tex]

These are the dimensions of the cone-shaped cup that will minimize the amount of paper used while still holding [tex]33 cm^3[/tex] of water.

To find the height and radius of the cone-shaped paper drinking cup that will use the smallest amount of paper, we need to minimize the surface area of the cone. The surface area [tex]\( A \)[/tex] of a cone consists of the base area and the lateral surface area, which can be expressed as:

[tex]\[ A = \pi r^2 + \pi r l \][/tex]

where [tex]\( r \)[/tex] is the radius of the base of the cone, and [tex]\( l \)[/tex] is the slant height of the cone. The slant height can be found using the Pythagorean theorem:

[tex]\[ l = \sqrt{r^2 + h^2} \][/tex]

where [tex]\( h \)[/tex] is the height of the cone. The volume [tex]\( V \)[/tex] of the cone is given by:

[tex]\[ V = \frac{1}{3} \pi r^2 h \][/tex]

We are given that the volume [tex]\( V \)[/tex] is [tex]33 cm^3[/tex]. We can use this to express [tex]\( h \)[/tex] in terms of [tex]\( r \)[/tex]:

[tex]\[ h = \frac{3V}{\pi r^2} \][/tex]

Substituting the volume into the equation, we get:

[tex]\[ h = \frac{3 \times 33}{\pi r^2} \][/tex]

Now, we substitute [tex]\( h \)[/tex] into the expression for [tex]\( l \)[/tex]:

[tex]\[ l = \sqrt{r^2 + \left(\frac{3 \times 33}{\pi r^2}\right)^2} \][/tex]

Substituting [tex]\( l \)[/tex] back into the surface area equation, we have [tex]\( A \)[/tex] as a function of [tex]\( r \)[/tex] :

[tex]\[ A(r) = \pi r^2 + \pi r \sqrt{r^2 + \left(\frac{3 \times 33}{\pi r^2}\right)^2} \][/tex]

To find the minimum surface area, we need to take the derivative of [tex]\( A \)[/tex] with respect to [tex]\( r \)[/tex] and set it equal to zero:

[tex]\[ \frac{dA}{dr} = 0 \][/tex]

Solving this equation will give us the value of [tex]\( r \)[/tex] that minimizes the surface area. Once we have [tex]\( r \)[/tex], we can substitute it back into the equation for [tex]\( h \)[/tex] to find the height that corresponds to the minimum surface area.

After performing the differentiation and solving for [tex]\( r \)[/tex], we find that the radius that minimizes the surface area is approximately 3.02 cm. Substituting this value into the equation for [tex]\( h \)[/tex], we find that the corresponding height is approximately 6.04 cm.

The function f(x) = −2x3 − x2 + 3x + 1 is represented by the graph with 3 real zeros, down on the left and up on the right, due to the negative leading coefficient and odd degree.

The function you're looking at is f(x) = −2x3 − x2 + 3x + 1. To determine the correct graph, we consider the leading term, −2x3. Since the coefficient of the highest-degree term (which is -2) is negative, the graph will start down on the left and go up on the right. The degree of the function is the highest power of x, which is 3, a odd number. For polynomials, if the degree is odd and the leading coefficient is negative, the end behavior will be down on the left and up on the right. Therefore, the correct graph will have the description: Graph with 3 real zeros, down on the left, up on the right.

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We are given the functions:

P (d) = 0.75 d                                      ---> 1

C (P) = 1.14 P                                      ---> 2

The problem asks us to find for the final price after discount and taxes applied; therefore we have to find the composite function of the two given functions 1 and 2. To solve for composite function of the final price of the dishwasher with the discount and taxes applied, all we have to do is to plug in the value of P (d) with variable d into the equation of C (P). That is:

C (P) = 1.14 (0.75 d)

C (P) = 0.855 d

or

C [P (d)] = 0.855 d

Answer:

0.855d

Step-by-step explanation:

I took the test. I also checked a bunch of other answers by completing the test. at least 2 other people had the same answer as me.

The number of ways he can select his attire for the prom to look differently will be twenty-four (24).

A permutation is an act of arranging the objects or elements in order. Combinations are the way of selecting objects or elements from a group of objects or collections, in such a way the order of the objects does not matter.

Roger is renting a tuxedo for prom.

Once he has chosen his jacket, he must choose from three types of pants, four colors of vests, and two different styles of shoes.

Then the number of the ways he can select his attire for the prom will be

[tex]\rm Number \ of \ ways = ^3C_1 \times ^4C_1 \times ^2C_1 \\\\ Number \ of \ ways = 3 \times 4 \times 2\\\\Number \ of \ ways = 24[/tex]

More about the permutation and the combination link is given below.

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Answer:

The sum of the roots of the equation [tex]x^{2} + x = 2[/tex] is -1

Step-by-step explanation:

You have two options to find the sum of the roots,

[tex]x_{1,2} = \frac{-b\±\sqrt{b^{2}-4ac}}{2a} [/tex]

[tex]x^{2} + x - 2= [/tex] where:

a = 1

b = 1

c = -2

[tex]x_{1} = \frac{-1-\sqrt{1^{2}-4*1*-2}}{2*1}[/tex] = -2

[tex]x_{2} = \frac{-1+\sqrt{1^{2}-4*1*-2}}{2*1}[/tex] = 1

The sum of the roots is -2 + 1 = -1

    2. The second option is use the fact that a general quadratic equation is in the form of:

[tex]ax^{2}+bx+c=0[/tex]

if you divided by [tex]a[/tex] you get:

[tex]x^{2}+\frac{b}{a} x+\frac{c}{a} =0[/tex]

and always the sum of roots will be given for this expression [tex]x_{1} + x_{2} = \frac{-b}{a}[/tex]

Why this is true?

Because if we use the Quadratic Formula as follows:

[tex]x_{1} + x_{2} = \frac{-b+\sqrt{b^{2}-4ac}}{2*a} + \frac{-b-\sqrt{b^{2}-4ac}}{2a}[/tex]

[tex]x_{1} + x_{2} = \frac{-2b+0}{2a}}[/tex]

[tex]x_{1} + x_{2} = \frac{-b}{a}[/tex]

In the case of this equation:

[tex]x_{1} + x_{2} = \frac{-1}{1} = -1[/tex]

b. StartFraction x y Over p m EndFraction = 8

The maximum allowable vertical distance from a fixture outlet to the trap weir in plumbing is 24 inches. This standard ensures proper drainage and the maintenance of a water seal, preventing sewer gases from entering a building.

The vertical distance from a fixture outlet to the trap weir, which is a critical aspect of plumbing design, should not be more than 24 inches. The fixture outlet is the point where water exits the fixture, and the trap weir is the peak point inside a P-trap, which maintains a water seal to prevent sewer gases from entering the building.

It's important to adhere to this standard to ensure proper drainage and maintain the water seal. If the distance is too great, it could lead to poor drainage and a loss of the trap seal due to siphoning, which would allow sewer gases to enter the home or building.

Answer:

Hi!

The correct answer is {56.35, 70.55, 81.2, 116.7} .

Step-by-step explanation:

The set {17, 21, 24, 34} represents the values of x.

If you replace each value in the equation:

Then you have the values {56.35, 70.55, 81.2, 116.7} .

1.     Multiply the coefficients and round to the number of significant figures in the coefficient with the smallest number of significant figures.

2.     Add the exponents.

3.     Convert the result to scientific notation.

Answer:

Let's say that I have to multiply 2.5 x 10^3 and 6.23 x 10^5.

First, Let's multiply 10^3 and 10^5.

It would be 10^8.

next, let's multiply 2.5 and 6.23.

It is 15.575.

So, my answer is 15.575 x 10^8.

The correct answer is b

F is a point which is greater than zero and F must be in the location of 1/11 and it can be determine by using arithmetic operations.

Given :

F partitions the directed line segment from D to E into a 5:6 ratio.

Given that F partitions the directed line segment from D to E into a 5:6 ratio therefore, total segments is (5 + 6 = 11).

From point D to E in the given line segment there are 9 units. To divide the line segment of 9 unit into 11 unit, first find the distance between two units, that is:

[tex]\dfrac{9}{11}=0.82[/tex]

[tex]0.82\times 5 = 4.1[/tex]

Now, it can be say that F is a point which is greater than zero and F must be in the location of 1/11.

For more information, refer the link given below:

https://brainly.com/question/12431044