Drag the tiles to the correct boxes to complete the pairs.

Match each expression to its equivalent form.

To find which expression must be an odd integer when m + 1 is even, we analyze each option using substitution. The only option that results in an odd integer is 2m + 1.

If m + 1 is an even integer, then m must be an odd integer. Let's analyze each option to see which one of them must be an odd integer:

Therefore, the answer is (C) 2m + 1 as it is the only option that must be an odd integer.

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Final answer:

Option C, which is 2m + 1, must be an odd integer if m + 1 is an even integer because adding 1 to any even integer results in an odd integer.

Explanation:

Given that m + 1 is an even integer, we can determine which of the other expressions would therefore be an odd integer. We know that any even integer can be written in the form 2k, where k is an integer, because even integers are multiples of 2. Similarly, odd integers can be written as 2k + 1, where k is also an integer.

Given the options:

Therefore, the answer is C: 2m + 1.

Answer:

Prime Numbers

Step-by-step explanation:

Two sets are being considered in the given statement:

A universal set is defined as such a set from which all the other subsets are taken for a given case. Or in other words, a universal set is such a set to which all the subsets belong and this universal set contains all the elements being studied/considered.

For the given case, set of Even Numbers is the subset which is being taken from the set of prime numbers. The set of prime numbers contains all the elements which we are considering for this very scenario.

Therefore, the set of prime numbers is the universal set.

Answer:

c

Step-by-step explanation:

Answer: z= .6

Step-by-step explanation:

Numerator: When you do 5•.6, you get 3, and then 3-1 equals 2.

Denominator: 8 - 3 = 5.

Therefore you get 2/5 which makes z = .6

Answer:

Option B is correct.

Step-by-step explanation:

we are given [tex]f(x) = \frac{x}{2}-3[/tex]

and [tex]g(x) = 3x^2+x-6[/tex]

We need to find (f+g)(x)

We just need to add f(x) and g(x)

(f+g)x = f(x) + g(x)

[tex](f+g)(x)=(\frac{x}{2}-3)+(3x^2+x-6)\\(f+g)(x)=\frac{x}{2}-3+3x^2+x-6\\(f+g)(x)=3x^2+\frac{x}{2}+x-3-6\\(f+g)(x)=3x^2+\frac{3x}{2}-9\\[/tex]

So, Option B is correct.

For this case we have the following functions:

[tex]f (x) = \frac {x} {2} -3\\g (x) = 3x ^ 2 + x-6[/tex]

We must find [tex](f + g) (x).[/tex] By definition, we have to:

[tex](f + g) (x) = f (x) + g (x)[/tex]

So:

[tex](f + g) (x) = \frac {x} {2} -3+ (3x ^ 2 + x-6)\\(f + g) (x) = \frac {x} {2} -3 + 3x ^ 2 + x-6\\(f + g) (x) = + 3x ^ 2 + x + \frac {x} {2} -3-6\\(f + g) (x) = + 3x ^ 2 + \frac {2x + x} {2} -9\\(f + g) (x) = + 3x ^ 2 + \frac {3x} {2} -9[/tex]

Answer:

Option B

Answer:

V = 75 pi m^3

or approximately

V =235.5 m^3

Step-by-step explanation:

The volume of a cylinder is given by

V = pi r^2 h

where r is the radius and h is the height

V = pi * 5^2  *3

V = 75 pi m^3

If we use 3.14 as an approximation for pi

V =235.5 m^3

Answer:

volume = 75(pi) m^3 (exactly)

volume = 235.6 m^3 (approximately)

Step-by-step explanation:

volume of cylinder = pi * radius^2 * height

volume = pi * (5 m)^2 * 3 m

volume = 75(pi) m^3 (exactly)

volume = 235.6 m^3 (approximately)

Answer:

Choice A. [tex]F(x) = -(x - 4)^{2} -3[/tex].

Step-by-step explanation:

Both F(x) and G(x) are quadratic equations. The graphs of the two functions are known as parabolas. All four choices for the equation of F(x) are written in their vertex form. That is:

[tex]y = a(x - h)^{2} + k[/tex], [tex]a \ne 0[/tex]

where

For the parabola G(x),

In other words,

Hence choice A. [tex]F(x) = -(x - 4)^{2} -3[/tex].

The only equation that correctly represents the graph of F(x) as a vertically stretched and reflected version of the graph of G(x), shifted to the right by 4 units.

Option A. [tex]F(x)=-(x-4)^{2}-3[/tex] is correct.  

To answer this question, we can consider the following transformations of the graph of [tex]G(x)=x^2$:[/tex]

Vertical stretch: [tex]$F(x)=kx^2$[/tex] for some positive constant k.

This will stretch the graph of G(x) vertically by a factor of k.

Vertical shift: [tex]$F(x)=x^2+h$[/tex] for some constant $h$.

This will shift the graph of G(x) up or down by $h$ units.

* **Horizontal shift: [tex]F(x)=(x+h)^2[/tex] for some constant $h$. This will shift the graph of G(x) to the left or right by h units.

* **Reflection: [tex]F(x)=-x^2[/tex]. This will reflect the graph of G(x) across the x-axis.

The graph of F(x) in the image appears to be a vertically stretched and reflected version of the graph of G(x), shifted to the right by 4 units. This suggests that the equation of F(x) could be of the form:

F(x)=-k(x-h)^2 for some positive constant k and some constant h.

To find the values of k and h, we can use the fact that the graph of F(x) passes through the points (-2,3) and (4,-3).

Substituting x=-2 and y=3 into the equation above, we get:

[tex]3=-k(-2-h)^2[/tex]

and substituting x=4 and y=-3 into the equation above, we get:

[tex]-3=-k(4-h)^2[/tex]

Dividing the second equation by the first equation, we get:

[tex]\frac{1}{3}=\frac{(-2-h)^2}{(4-h)^2}[/tex]

Taking the square root of both sides, we get:

[tex]\pm \frac{1}{\sqrt{3}}=\frac{-2-h}{4-h}[/tex]

Solving for $h$, we get:

[tex]h=4\pm \frac{4}{\sqrt{3}}[/tex]

Substituting this value of h into either of the original equations, we can solve for k.

For example, substituting [tex]$h=4+\frac{4}{\sqrt{3}}$[/tex] into the first equation, we get:

[tex]3=-k\left(-2-4+\frac{4}{\sqrt{3}}\right)^2[/tex]

Solving for $k$, we get:

[tex]k=3\cdot \left(-6+\frac{4}{\sqrt{3}}\right)^2[/tex]

Therefore, one possible equation for $F(x)$ is:

[tex]F(x)=-3\left(x-4+\frac{4}{\sqrt{3}}\right)^2[/tex]

Another possible equation for $F(x)$ is:

[tex]F(x)=-3\left(x-4-\frac{4}{\sqrt{3}}\right)^2[/tex]

These two equations are equivalent, since they both represent the same reflected and vertically stretched version of the graph of G(x),  shifted to the right by 4 units.

Therefore, the answer to the question is:

[tex]A. F(x)=-(x-4)^{2}-3[/tex]

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Answer: D. 84 ft^2

Step-by-step explanation: The area of a trapezoid is (a+b)/2 x h. A and B are the bases, and h is the height. Plug in the numbers.

(8+16)/2 x 7 = 84

Your answer would be 84 ft^2. D.

Answer:

Yes.

Step-by-step explanation:

It can be noticed that 1/e can be written as e^(-1). Whenever a number or a variable is shifted from the numerator to the denominator or vice versa, the power of that number or that variable becomes negative. This means that 1/e = e^(-1). Since f(x) = (1/e)^x, f(x) can be written as:

f(x) = (e^(-1))^x.

Since there are two powers and the base is same, so both the powers will be multiplied. Therefore:

f(x) = e^(-x).

It can be seen that e is involved in the function and it has a negative power of x, so this means that f(x) is a decreasing exponential function!!!

Answer:

The correct option is A.

Step-by-step explanation:

The given expression is:

(a^-8b/a^-5b^3)^-3

We have to keep one thing in our mind that in a division, same base, the exponents are subtracted.

We will change the division into multiplication: The denominator will become numerator and the signs of the exponent become opposite.

(a^-8 *a^+5 * b *b^-3)^-3

=(a^-8+5 * b^1-3) ^-3

=(a^-3*b^-2)^-3

=(a)^-3*-3 (b)^-2*-3

= a^9 b^6

Thus the correct option is A....

Answer:

a^9b^6

Step-by-step explanation:

Answer: it’s B 0

Step-by-step explanation:

Answer:

  3 seconds

Step-by-step explanation:

The given equation describes a parabolic curve with zeros at x=-1 and x=7. The line of symmetry of the curve is halfway between the zeros, at ...

  x = (-1 +7)/2 = 6/2 = 3

The maximum of the curve is on the line of symmetry. The ball will reach its maximum height 3 seconds after being thrown.

_____

Additional comment
The maximum height will be 16 meters.

The ball will reach its maximum height 3 seconds after being thrown.

To find the time when the ball reaches its maximum height, we need to determine the vertex of the quadratic equation representing the ball's height. The equation h(x)=-(x+1)(x-7) can be rewritten in vertex form as h(x)=-x^2+6x-7. The x-coordinate of the vertex can be found using the formula x = -b/2a, where a, b, and c are the coefficients of the quadratic equation.

In this case, a = -1 and b = 6. Plugging these values into the formula, we get x = -6/-2 = 3. Therefore, the ball reaches its maximum height 3 seconds after being thrown.

Answer:

The correct option is A.

Step-by-step explanation:

Domain:

The expression in the denominator is x^2-2x-3

x² - 2x-3 ≠0

-3 = +1 -4

(x²-2x+1)-4 ≠0

(x²-2x+1)=(x-1)²

(x-1)² - (2)² ≠0

∴a²-b² =(a-b)(a+b)

(x-1-2)(x-1+2) ≠0

(x-3)(x+1) ≠0

x≠3 for all x≠ -1

So there is a hole at x=3 and an asymptote at x= -1, so Option B is wrong

Asymptote:

x-3/x^2-2x-3

We know that denominator is equal to (x-3)(x+1)

x-3/(x-3)(x+1)

x-3 will be cancelled out by x-3

1/x+1

We have asymptote at x=-1 and hole at x=3, therefore the correct option is A....

Answer:

Its A. I just passed the final exam with 5 minutes left of the SESSION.

Step-by-step explanation:

Answer:

2

Step-by-step explanation:

The average rate of change from x=1 to x=2 is the same as finding the slope of a line at x=1 and x=2.

So we are going to need to corresponding y coordinates.

What y corresponds to x=1? y=3

What y corresponds to x=2? y=5

So we have the ordered pairs (1,3) and (2,5).

Line the points up vertically and subtract vertically then put 2nd difference over 1st difference.

(2 , 5)

-(1  , 3)

-----------

1      2

The average rate of change is 2/1 or just 2.

Now since we were asked to find the average rate of change given the function was a line, it really didn't matter what two points you used on that line.

Answer: Fourth option

[tex]m=2[/tex]

Step-by-step explanation:

If we call m the average change rate of a function between [tex]x_1[/tex] and [tex]x_2[/tex], then, by definition:

[tex]m=\frac{f(x_2)-f(x_1)}{x_2-x_1}[/tex]

In this case the function is the line shown in the graph. Then we look for the values of [tex]y = f (x)[/tex] for [tex]x = 1[/tex] and [tex]x = 2[/tex]

When [tex]x=1[/tex] then [tex]f(x)=3[/tex]

When [tex]x=2[/tex] then [tex]f(x)=5[/tex]

Therefore

[tex]m=\frac{5-3}{2-1}[/tex]

[tex]m=\frac{2}{1}[/tex]

[tex]m=2[/tex]

Answer:

* sin Ф = -15/17     * cos Ф = 8/17      * tan Ф = -15/8

* csc Ф = -17/15    * sec Ф = 17/8      * cot Ф = -8/15

Step-by-step explanation:

Lets revise the trigonometric function of angle Ф

- Angle θ is in standard position

- Point  (8, -15) is on the terminal ray of angle θ

- That means the terminal is the hypotenuse of a right triangle x and y

  are its legs

∵ x-coordinate is positive and y-coordinate is negative

∴ angle Ф lies in the 4th quadrant

- The opposite side of angle Ф is the y-coordinate of the point on the

  terminal ray of angle Ф and the adjacent side to angle Ф is the

  x-coordinate of that point

∵ The length of the hypotenuse (h) = √(x² + y²)

∴ h = √[(8)² + (-15)²] = √[64 + 225] = √[289] = 17

∴ The length of the hypotenuse is 17

- Lets find sin Ф

∵ sin Ф = opposite/hypotenuse

∵ The opposite is y = -15

∵ The hypotenuse = 17

∴ sin Ф = -15/17

- Lets find cos Ф

∵ cos Ф = adjacent/hypotenuse

∵ The adjacent is x = 8

∵ The hypotenuse = 17

∴ cos Ф = 8/17

- Lets find tan Ф

∵ tan Ф = opposite/adjacent

∵ The opposite is y = -15

∵ The adjacent = 8

∴ tan Ф = -15/8

- Remember csc Ф is the reciprocal of sin Ф

∵ csc Ф = 1/sin Ф

∵ sin Ф = -15/17

∴ csc Ф = -17/15

- Remember sec Ф is the reciprocal of cos Ф

∵ sec Ф = 1/ cos Ф

∵ cos Ф = 8/17

∴ sec Ф = 17/8

- Remember cot Ф is the reciprocal of tan Ф

∵ cot Ф = 1/tan Ф

∵ tan Ф = -15/8

∴ cot Ф = -8/15

Answer:

-4

Step-by-step explanation:

So we are given f(t)=-2t+5.

We are asked to find the value a such that f(a)=13.

If f(t)=-2t+5, then f(a)=-2a+5.  I just replaced old input,x, with new input, a.

So we want to solve f(a)=13 for a.

f(a)=13

Replace f(a) with -2a+5:

-2a+5=13  

Subtract 5 on both sides:

-2a    =8

Divide both sides by -2:

 a       =-4

Check:

Is -2t+5=13 for t=-4?

-2(-4)+5=8+5=13, so yep.

The value of t is -4 for which the function will give 13 as output.

A mathematical relationship from a set of inputs to a set of outputs is called a function.

The given function is F(t) = -2t+5

So,   -2t + 5 = 13

⇒  -2t = 8

⇒ t = -4

So, for the value of t = -4 the given function will give the output 13

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Answer:

Option D 5.1962

Step-by-step explanation:

we have

The number [tex]9[/tex] and the number [tex]\sqrt{3}[/tex]

I assume for the choices provided that the operation is calculate  [tex]9[/tex] divided by [tex]\sqrt{3}[/tex]

so

[tex]\frac{9}{\sqrt{3}} =5.1962[/tex] (rounded to the nearest ten thousandth)

Answer:

C. 44.9569

Step-by-step explanation:

Answer:

Ali-13

Ben-26

Joe-34

Step-by-step explanation:

Ali-x

Ben-2x

Joe-2x+8

x+2x+2x+8=73

3x+2x+8=73

5x+8=73

5x=73-8

5x=65

x=65/5

=13

Ali sells 13 tickets

Ben sells 13×2

=26 tickets

Joe sells (13×2)+8

=34 tickets

Ali sells 13 tickets. Ben sells 26 tickets. Joe sells 34ctickets

Let

Ali = x

Ben = 2x

Joe = 2x+8

⇒ x+2x+2x+8=73

⇒ 3x+2x+8=73

⇒ 5x+8=73

⇒ 5x=73-8

⇒ x=65/5

⇒ 13

Ali sells 13 tickets.

Ben sells 13×2

= 26 tickets.

Joe sells (13×2)+8

= 34 tickets.

An equation is a mathematical statement that is made up of two expressions connected by an equal sign.

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Answer:

y = 1

Step-by-step explanation:

Given function is:

[tex]\frac{x^2+8x+3}{x^2+3x+1}[/tex]

In order to find the horizontal asymptote, the coefficients of highest degree variable of numerator and denominator are divided.

In our case,

both the numerator and denominator have 1 as he co-efficient of x^2

So the horizontal asymptote is y = 1/1

Hence, third option y=1 s correct ..

Final answer:

The correct horizontal asymptote for the given rational expression is y = 1, found by comparing the degrees and leading coefficients of the numerator and denominator.

Explanation:

For the expression y = (x² - 8x + 3) / (x² + 3x + 1), the correct horizontal asymptote is determined by looking at the degrees of the polynomials in the numerator and the denominator. When the degrees are the same, the horizontal asymptote is found by dividing the leading coefficients. In this case, both polynomials are of degree 2, and the leading coefficients are both 1. Thus, the horizontal asymptote is y = 1.

The perimeter of a rectangle, is all of the sides added up.

In a rectangle, there are two widths (right and left), and two lengths (up and down)

So:

Perimeter = 2 × width + 2 × length.

To get the equation we just substitute in the values that we are given:

length = (x)

width = (x - 4)

perimeter = 87

So just put these values into:  perimeter = 2 × width + 2 × length:

87  =  2(x) + 2(x - 4)

_____________________________

Answer:

The equation is:       87 = 2(x) + 2(x - 4)

(note: this can be simplified down like so:

87 = 2x + 2x - 8

87 = 4x - 8

95 = 4x

23.75 = x    

)

[tex]\bf 8~~,~~\stackrel{8\cdot \frac{1}{2}}{4}~~,~~\stackrel{4\cdot \frac{1}{2}}{2}~~,~~\stackrel{2\cdot \frac{1}{2}}{1}~~,~~\stackrel{1\cdot \frac{1}{2}}{\cfrac{1}{2}} \\\\\\ a_n=\cfrac{1}{2}\cdot a_{n-1}\qquad \begin{cases} a_1=\textit{previous term}\\ a_n=\textit{current term}\\ a_1=\textit{first term}\\ \qquad 8 \end{cases}[/tex]

Answer:

26.50

Step-by-step explanation:

First we need to calculate the sales tax

25 * 6%

Changing to decimal from

25 * .06

1.50

Then we add the tax to the cost of the shirt

25+1.50

26.50

We owe 26.50

Answer:

2nd choice.

Step-by-step explanation:

Let's compare the following:

[tex]f(x)=a\sin(b(x-c))+d[/tex] to

[tex]f(x)=-3\sin(4x-\pi))-5[/tex].

They are almost in the same form.

The amplitude is |a|, so it isn't going to be negative.

The period is [tex]\frac{2\pi}{|b|}[/tex].

The phase shift is [tex]c[/tex].

If c is positive it has been shifted right c units.

If c is negative it has been shifted left c units.

d is the vertical shift.

If d is negative, it has been moved down d units.

If d is positive, it has been moved up d units.

So we already know two things:

The amplitude is |a|=|-3|=3.

The vertical shift is d=-5 which means it was moved down 5 units from the parent function.

Now let's find the others.

I'm going to factor out 4 from [tex]4x-\pi[/tex].

Like this:

[tex]4(x-\frac{\pi}{4})[/tex]

Now if you compare this to [tex]b(x-c)[/tex]

then b=4 so the period is [tex]\frac{2\pi}{4}=\frac{\pi}{2}[/tex].

Also in place of c you see [tex]\frac{\pi}{4}[/tex] which means the phase shift is [tex]\frac{\pi}{4}[/tex].

The second choice is what we are looking for.

Answer: Second Option

Amplitude = 3; period = pi over two; phase shift: x equals pi over four

Step-by-step explanation:

By definition the sinusoidal function has the following form:

[tex]f(x) = asin(bx - c) +k[/tex]

Where

[tex]| a |[/tex] is the Amplitude of the function

[tex]\frac{2\pi}{b}[/tex] is the period of the function

[tex]-\frac{c}{b}[/tex] is the phase shift

In this case the function is:

[tex]f(x) = -3 sin(4x - \pi) - 5[/tex]

Therefore

[tex]Amplitude=|a|=3[/tex]

[tex]Period =\frac{2\pi}{b} = \frac{2\pi}{4}=\frac{\pi}{2}[/tex]

[tex]phase\ shift = -\frac{(-\pi)}{4}=\frac{\pi}{4}[/tex]

Answer:

C. [tex]432\text{ units}^2[/tex]

Step-by-step explanation:

We have been given an image of a right  prism. We are asked to find the surface area of our given prism.

The surface area of our given prism will be sum of area of two base right triangles and area of three rectangles.

[tex]SA=2B+Ph[/tex], where,

SA = Surface area,

B = Area of each base,

P = Perimeter of base,

h = Height of prism.

Let us find area of right base of prism.

[tex]B=\frac{1}{2}\times 6\times 8[/tex]

[tex]B=3\times 8[/tex]

[tex]B=24[/tex]

Perimeter of base will be [tex]6+8+10=24[/tex].

Upon substituting these values in surface area formula, we will get:

[tex]SA=2*24+24*16[/tex]

[tex]SA=24(2+16)[/tex]

[tex]SA=24(18)[/tex]

[tex]SA=432[/tex]

Therefore, the surface area of our given prism is 432 square units and option C is the correct choice.

For this case we have that by definition, the area of a circle is given by:

[tex]A = \pi * r ^ 2[/tex]

Where:

r: It is the radius of the circle

As data we have that[tex]r = 6 \ ft[/tex]

Then, replacing we have:

[tex]A = \pi * 6 ^ 2\\A = 36 \pi[/tex]

Taking [tex]\pi = 3.14[/tex]

A = 113.04

So, the pool area is [tex]113.04 \ ft ^ 2[/tex]

Answer:

[tex]113.04 \ ft ^ 2[/tex]

Answer:

9π ft²

Step-by-step explanation:

Answer:

The answer is the last option, D.

Step-by-step explanation:

A function with one solution can be added and have a variable and a constant leftover. The variables in the other three options cancel each other out, so none of them have one solution. The first and third equations have no solutions, and the second has infinitely many solutions. The fourth option has only one variable cancel out and still has a constant on the other side of the equation. The fourth option is correct.

Answer:

x = 40

Step-by-step explanation:

The three angles in a triangle add to 180 degrees

x + 30 + (3x-10) = 180

Combine like terms

4x +20 = 180

Subtract 20 from each side

4x+20-20 = 180-20

4x= 160

Divide each side by 4

4x/4 =160/4

x = 40

Answer:

I measures 60° ⇒ 3rd answer

Step-by-step explanation:

* Look to the attached figure

- To prove that the two triangles are similar by SAS we must to

  find two proportional pairs of corresponding side and the measure

  of the including angles between them are equal

- The given is:

  m∠ F = 60°

  EF = 40 , FG = 20

  HI = 20 , IJ = 10

- To prove that the Δ EFG is similar to Δ HIJ we must to prove

# EF/ HI = FG/IJ ⇒ two pairs of sides proportion

# m∠ F = m∠ I ⇒ including angles equal

∵ EF = 40 and HI = 20

∴ EF/HI = 40/20 = 2

∵ FG = 20 and IJ = 10

∴ FG/IJ = 20/10 = 2

∵ EF/HI = FG/IJ = 2

∴ The two pairs of sides are proportion

∵ ∠ F is the including angle between EF and FG

∵ ∠ I is the including angle between HI and IJ

∴ m∠ F must equal m∠ I

∵ m∠ F = 60° ⇒ given

∴ m∠ I = 60°

* I measures 60°

To prove triangles are similar by the SAS similarity theorem when dealing with equilateral triangles, the angle between the proportional sides needs to be congruent. In this context, showing that the angle measures 30° would be necessary if one angle is already known to be 60°. Thus, either angle I or J should be proven to measure 30°, depending on which is between the proportional sides.

To prove that triangles are similar by the SAS similarity theorem, two conditions must be met: corresponding sides are in proportion, and the angles between those sides are congruent. In the context of proving triangle similarity with equilateral triangles, by using a bisector, we can form a smaller triangle whose hypotenuse is twice as long as one of its sides. Consequently, as the sides preserve this ratio, the angles approach 30° and 60°. Therefore, if we must prove that a triangle in a given figure is similar through the SAS similarity criterion, and we have a side-angle-side situation where one angle is already known to be 60°, the angle we would need to prove congruent would reasonably be 30°.

This conclusion is supported by the nature of equilateral triangles, where bisecting the angle will give us angles of 30° and 60°. Additionally, the similarity of triangles is further backed by the congruency of angles and the proportionality of sides, as mentioned in other contexts within the provided information. Therefore, based on the given context and the properties of equilateral triangles, we would aim to prove that angle I or angle J (whichever is between the proportional sides) measures 30°.

Answer:

24

Step-by-step explanation:

Changing the word problem into an equation you get that you are trying to find n^3+16. This is because they are looking for the cube of a number n plus 16. Plugging in n as 2 you get 2^3+16 which is 8+16 which is 24.

Answer:

24

Step-by-step explanation:

Answer:

[tex]x^2+y^2=25[/tex]

Step-by-step explanation:

Recall the following Pythagorean Identity:

[tex]\sin^2(\theta)+\cos^2(\theta)=1[/tex]

Let's solve the x equation for cos(t) and the y equation for sin(t).

After the solve we will plug into our above identity.

x=5cos(t)

Divide both sides by 5:

(x/5)=cos(t)

y=5sin(t)

Divide both sides by 5:

(y/5)=sin(t)

Now we are ready to plug into the identity:

[tex]\sin^2(t)+\cos^2(t)=1[/tex]

[tex](\frac{y}{5})^2+(\frac{x}{5})^2=1[/tex]

[tex]\frac{x^2}{5^2}+\frac{y^2}{5^2}=1[/tex]

Multiply both sides by 5^2:

[tex]x^2+y^2=5^2[/tex]

This is a circle with center (0,0) and radius 5.

All I did to get that was compare our rectangular equation we found to

[tex](x-h)^2+(y-k)^2=r^2[/tex]

where (h,k) is the center and r is the radius of a circle.

Answer:

  14.8 units³

Step-by-step explanation:

The volume of a "pointed" solid, such as a cone or pyramid, is one-third the product of its base area and height:

  V = (1/3)Bh

Filling in the given numbers and doing the arithmetic, we find the volume to be ...

  V = (1/3)(7.4 units²)(6 units) = 14.8 units³

The volume of the pyramid is 14.8 cubic units.