Enter your answer in the box.
x=

Answer:

1

I've attached a screenshot from my graphing calculator of [tex]f(x)[/tex] in blue and [tex]f^{-1}(x)[/tex] in red. Notice how the red line (inverse function) has (3, 1)

Step-by-step explanation:

[tex]f(x) = 3^x\\f^{-1}(x) =\ ?\\[/tex]

We must first figure out the inverse function of [tex]f(x)[/tex] which is  [tex]f^{-1}(x)[/tex]

[tex]y = f(x) = 3^x\\y = 3^x\\log\ y = log\ 3^x\\log\ y = x \times log\ 3\\x = \frac{log\ y}{log\ 3}\\[/tex]

We say y = f(x) to begin with, but after find x = ...

we must 'swap' x and y

[tex]x = \frac{log\ y}{log\ 3}\\y = \frac{log\ x}{log\ 3}\\[/tex]

[tex]f^{-1}(x) = \frac{log\ x}{log\ 3}[/tex]

[tex]f^{-1}(3) = \frac{log\ 3}{log\ 3} = 1[/tex]

Answer:

using [tex]\pi[/tex]:   65.45 in³ (nearest hundredth)

using [tex]\pi =3.14[/tex]:   65.42 in³ (nearest hundredth)

Step-by-step explanation:

The radius of the sphere is half the side length of the cube (see attached diagram).  Therefore, the side length of the cube = 2r

Given:

[tex]\textsf{Volume of a cube}=x^3\quad \textsf{(where}\:x\:\textsf{is the side length)}[/tex]

[tex]\implies 125=(2r)^3[/tex]

[tex]\implies \sqrt[3]{125}=2r[/tex]

[tex]\implies 5=2r[/tex]

[tex]\implies r=\dfrac52[/tex]

Substitute the found value of r into the volume of a sphere equation:

[tex]\begin{aligned}\textsf{Volume of a sphere} & =\dfrac43 \pi r^3\\\\ & =\dfrac43 \pi \left(\dfrac52\right)^3\\\\ & =\dfrac43 \pi \left(\dfrac{125}{8}\right)\\\\ & =\dfrac{500}{24} \pi\\\\ & =\dfrac{125}{6} \pi\\\\ & =65.45\:\sf in^3\:(nearest\:hundredth) \end{aligned}[/tex]

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

Explanation:

Let C be the corner point.

x = distance from P to C

That makes segment AP to be 70-x feet long

Focus on the right triangle PBC. Use the pythagorean theorem to find the hypotenuse PB.

[tex]a^2 + b^2 = c^2\\\\c = \sqrt{a^2 + b^2}\\\\PB = \sqrt{(PC)^2 + (CB)^2}\\\\PB = \sqrt{x^2 + 90^2}\\\\PB = \sqrt{x^2 + 8100}\\\\[/tex]

If we knew what x was, then we could find a numeric value for PB.

---------------------

It costs $28 per foot to run cable along the ground. This is the portion from A to P. So it costs 28(70-x) dollars to run that portion of cable above ground. Simply multiply the cost per foot by the number of feet.

Similarly, the cost from P to B is [tex]53\sqrt{x^2+8100}[/tex] since it costs $53 per foot to have it run underground.

The total cost is therefore [tex]28(70-x) + 53\sqrt{x^2 + 8100}[/tex]

The derivative of this will help determine when the cost is minimized.

Type that function into GeoGebra and have it compute the derivative.

You should find that the x intercept of the derivative curve is exactly x = 56

If x = 56, then 70-x = 70-56 = 14

This means AP = 14 feet is the amount of cable to run along main street.

This also leads to [tex]PB = \sqrt{x^2 + 8100} = \sqrt{56^2 + 8100} = 106[/tex]

The total length of the wire is 14+106 = 120 feet

It costs $28 per foot along the 14 ft section, so 28*14 = 392 dollars is the cost for this section.

It costs $53 per foot along the 106 ft section, so 53*106 = 5618 dollars is the cost for this other section.

The total min cost is 392+5618 = 6010 dollars

------------------

Side note: You could do all this without a calculator to compute the derivative and use algebra to end up with x = 56. However, I think the use of technology is beneficial because it's fast/efficient. In real world settings, you won't be likely to pull out pencil/paper to get things done. The modern world relies on computers. It's refreshing to see that your teacher encourages the use of technology.

Let me know if you need me to go over the algebraic steps and I'll update my answer.

Answer:

ir

Step-by-step explanation:

irregular

regular

what do you notice different?

the both have r, e, g, u, l, a, r, but one has ir at the beginning.

Answer:

Smaller

Step-by-step explanation:

2 2/5 * 1/6 = 2/

5

= 0.4

Conversion a mixed number 2 2/

5

to a improper fraction: 2 2/5 = 2 2/

5

= 2 · 5 + 2/

5

= 10 + 2/

5

= 12/

5

To find a new numerator:

a) Multiply the whole number 2 by the denominator 5. Whole number 2 equally 2 * 5/

5

= 10/

5

b) Add the answer from previous step 10 to the numerator 2. New numerator is 10 + 2 = 12

c) Write a previous answer (new numerator 12) over the denominator 5.

Two and two fifths is twelve fifths

Multiple: 12/

5

* 1/

6

= 12 · 1/

5 · 6

= 12/

30

= 2 · 6/

5 · 6

= 2/

5

Multiply both numerators and denominators. Result fraction keep to lowest possible denominator GCD(12, 30) = 6. In the following intermediate step, cancel by a common factor of 6 gives 2/

5

.

In other words - twelve fifths multiplied by one sixth = two fifths.

Answer:

Step-by-step explanation:

Use long multiplication to evaluate.

576

Answer:

With Graph and Without Graph.

Step-by-step explanation:

Without graphing calculator, you plug in your x-values into the equation

y = 16 -x^2 to solve for y-values.

f(x) = 16 - x^2

f(-4) = 16 - (-4)^2 = 16 - 16 =0

...

f(4) = 16 - (4)^2 = 16 - 16 =0

With Graphing Calculator.

Answer:

Step-by-step explanation:

Length of Fence

Cost

[tex]~~~~1+ \cos \theta = 2 \cos^2 \theta \\\\\implies 2\cos^2 \theta -\cos \theta -1 = 0\\\\\implies 2 \cos^2 \theta -2\cos \theta + \cos \theta -1 = 0\\\\\implies 2 \cos \theta( \cos \theta -1) +(\cos \theta -1)=0\\\\\implies (\cos \theta -1)(2 \cos \theta +1)=0\\\\\implies \cos \theta = 1, ~~\cos \theta = -\dfrac 12\\\\\implies \theta = 2n\pi,~~~ \theta = 2n\pi \pm \dfrac{2\pi}3[/tex]

Answer:

Smily face=5 lightning=8

Step-by-step explanation:

5+5=10 8x3=24      24+10=34

8x5=40              5x8=40

Both f(x) and g(x) include domain values of [12, ∞), and both functions increase over the interval (12, ∞). Then the correct option is D.

The domain means all the possible values of the x and the range means all the possible values of the y.

The functions are given below.

[tex]\rm f(x) = 12x \\\\g(x) = \sqrt{x - 12}[/tex]

Then the domain of f(x) is (-∞, ∞) and the domain of g(x) is (12, ∞). And both the functions increases in the interval of (12, ∞).

More about the domain and range link is given below.

https://brainly.com/question/12208715

#SPJ1

Answer:

62cm

Step-by-step explanation:

You were told the perimeter of the triangle is 38

but two sides are added to make a Pentagon. and that Pentagon is made up of a triangle and a square.

so if we find the side of the square, we can equally get the other two sides of the triangle.

hint: (because it's a square, we would be getting an isosceles triangle, meaning two sides are equal)

because one side of the pentagon added to the other two sides of the triangles gives us the perimeter of the triangle.

we first find the sides of the square with all sides equal by diving by 2

[tex] \frac{48}{2} = 12[/tex]

the sides of the square making the perimeter of the square 12cm.

now we subtract 12 from the perimeter of the triangle 38

[tex]38 - 12 = 26[/tex]

now that we have gotten one side we can get the other two by dividing the result by 2

[tex] \frac{26}{2} = 13[/tex]

so the other two sides of the triangle are 13cm.

now since we have the sides of the square and the triangle, we can find the perimeter of the pentagon by adding.

all the sides of the triangle without the base which is also the top of the square.

so instead of adding 4 times of 12, we add 3 times of 12 and 2 times of 13.

3(12)+2(13) = 62

or

12 + 12 + 12 + 13 + 13 = 62cm

Answer:

Its B

Step-by-step explanation:

Answer:

20.52 cm²

area of semi-circle - area of triangle

[tex]\dashrightarrow \sf \dfrac{1}{2} \ \pi (radius)^2 \ - \ \sf \dfrac{1}{2} *base*height[/tex]

[tex]\rightarrow \sf \dfrac{1}{2} (3.14)(6)^2 - \dfrac{1}{2} *12*6[/tex]

[tex]\hookrightarrow \sf 56.52 \ - \ 36[/tex]

[tex]\hookrightarrow \sf 20.52 \ cm^2[/tex]

Answer:

37,600

Step-by-step explanation:

940 X 40 = 37,600.

simple multiplication

Answer:

, 37,000

Step-by-step explanation:

940 x 40= 37,000

Answer:

2 ND option

solve and write

Step-by-step explanation:

Tangent is equal to sine over cosine. Cotangent is equal to cosine over sine. Therefore:

[tex]cos(x)( \frac{sin(x)}{cos(x)} + \frac{cos(x)}{sin(x)} )[/tex]

Distribute the cos(x) into the sum to get:

[tex]sin(x) + \frac{ {cos(x)}^{2} }{sin(x)} [/tex]

Get a common denominator by multiplying the first term by sine over sine to get:

[tex] \frac{ {sin(x)}^{2} }{sin(x)} + \frac{ {cos(x)}^{2} }{sin(x)} [/tex]

The numerator adds to equal 1 due to a common trigonometric identity. Therefore the only remaining term is:

[tex] \frac{1}{sin(x)} [/tex]

Just find the vertex and then compare with graph

#a

Parabola opening upwards

Graph 3

#2

Vertex at (-3,0)

Graph IV

#3

Parabola opening downwards as a is -ve

Graph I

#4

One graph is left

Answer:

a)  graph iii)

b)  graph iv)

c)  graph i)

d)  graph ii)

Step-by-step explanation:

Vertex form of a quadratic equation:  [tex]y = a(x - h)^2 + k[/tex]

where [tex](h, k)[/tex] is the vertex (turning point)

First, determine the vertices of the parabolas by inspection of the graphs:

Next, write each given equation in vertex form and compare to the vertices above.

[tex]\textsf{a)}\quad y=2x^2-8[/tex]

[tex]\textsf{Vertex form}: \quad y=2(x-0)^2-8[/tex]

⇒ Vertex = (0, -8)

Therefore, graph iii)

[tex]\textsf{b)} \quad y=(x+3)^2[/tex]

[tex]\textsf{Vertex form}: \quad y=(x+3)^2+0[/tex]

⇒ Vertex = (-3, 0)

Therefore, graph iv)

[tex]\textsf{c)} \quad y=-2|x-4|^2+8[/tex]

[tex]\textsf{Vertex form}: \quad y=-2|x-4|^2+8[/tex]

⇒ Vertex = (4, 8)

Therefore, graph i)

[tex]\textsf{d)} \quad y=(x-3)^2-8[/tex]

[tex]\textsf{Vertex form}: \quad y=(x-3)^2-8[/tex]

⇒ Vertex = (3, -8)

Therefore, graph ii)

Jennifer idea of  roping off more total area if the tie two ropes together to make one giant BBQ  is a true statement.

The total surface area is known to be the addition of all the sum of any given area of surfaces of a solid together.

Conclusively, Note that by joining together the 2 parties, one can cut off some activities that will be be doubled if it were two separate event and as such, the use of space is been maximize and then the area of the space they will use for the event will be reduced.

Learn more about  total surface area from

https://brainly.com/question/26638955

Answer:

40m^2 :)

Step-by-step explanation:

Area of a triange = (h·b)/2

Now lets solve :)

4 + 6 = 10

(8·10)/2

80/2

40

Have an amazing day!!

Please rate and mark brainliest!!

Answer:

Step-by-step explanation:

4a - (4a-3)

= 4a -1(4a - 3)

= 4a - 4a + 3    <--- Distributing the -1 over the parenthes

= 3       <---- ( as 4a - 4a = 0)

Answer:

7290in ^2

Step-by-step explanation:

covert 30yd to feet which equals too 90ft.

Then multiple 90 x 81 = 7290

Inequalities are expressions not separated by an equal sign. The solution of the compound inequality will be -6 < x < 24

Inequalities are expressions not separated by an equal sign.

Given the compound inequalities

X - 7 < 17 or -6x >36

For the inequality

x - 7 < 17

x < 17 + 7

x < 24

Similarly for -6x >36

x < -36/6
x < -6

The solution of the compound inequality will be -6 < x < 24

Learn more on inequality here: https://brainly.com/question/11613554

Answer:

I have graphed them on desmos and given you three options to choose from:

Step-by-step explanation:

1. with lines (complete shape)

2. with lines (incomplete shape)

3. without lines

Hope this helps!

Answer:

here

Step-by-step explanation:

Answer:

If each tile has a number from 0 thru 9 then the possible outcomes for each tile are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

Answer:

[tex]\displaystyle \frac{1}{4}[/tex]

Step-by-step explanation:

Given:

[tex]\displaystyle\frac{4^{6} }{4^{7}}[/tex]

Expand:

[tex]\displaystyle\frac{4*4*4*4*4*4 }{4*4*4*4*4*4*4}[/tex]

Simplify ones, in this case [tex]\frac{4}{4}[/tex]:

[tex]\displaystyle \frac{1}{4}[/tex]

     If you raise 4 to the power of 6, to the power of 7, and then simplify the fraction, you will get the same result. This method is much quicker when you have numbers with the same bases (here the base is 4) so I used this way.

FGHIJ ≅ STUQR

Solve for HI

[tex]\sf \dfrac{HI}{GH} = \dfrac{UQ}{UT}[/tex]

[tex]\rightarrow \sf \dfrac{HI}{56} = \dfrac{25}{35}[/tex]

[tex]\rightarrow \sf HI = 40[/tex]

Solve for QR

[tex]\sf \dfrac{IJ}{GH} = \dfrac{QR}{UT}[/tex]

[tex]\rightarrow \sf \dfrac{72}{56} = \dfrac{QR}{35}[/tex]

[tex]\rightarrow \sf QR = 45[/tex]

Solve for ST

[tex]\sf \dfrac{FG}{GH} = \dfrac{ST}{UT}[/tex]

[tex]\rightarrow \sf \dfrac{48}{56} = \dfrac{ST}{35}[/tex]

[tex]\rightarrow \sf ST =30[/tex]

[tex]\\ \rm\Rrightarrow \dfrac{GH}{HI}=\dfrac{UT}{UQ}[/tex]

[tex]\\ \rm\Rrightarrow \dfrac{56}{HI}=\dfrac{35}{25}=\dfrac{7}{5}[/tex]

[tex]\\ \rm\Rrightarrow HI=56(5)/7=8(5)=40[/tex]

#QR

[tex]\\ \rm\Rrightarrow \dfrac{GH}{JI}=\dfrac{UT}{RQ}[/tex]

[tex]\\ \rm\Rrightarrow \dfrac{56}{72}=\dfrac{35}{RQ}[/tex]

[tex]\\ \rm\Rrightarrow \dfrac{7}{9}=\dfrac{35}{RQ}[/tex]

[tex]\\ \rm\Rrightarrow RQ=35(9)/7=5(9)=45[/tex]

#ST

[tex]\\ \rm\Rrightarrow \dfrac{GH}{FG}=\dfrac{UT}{ST}[/tex]

[tex]\\ \rm\Rrightarrow \dfrac{56}{48}=\dfrac{7}{6}=\dfrac{35}{ST}[/tex]

[tex]\\ \rm\Rrightarrow ST=6(35)/7=6(5)=30[/tex]

Answer:

y = 3x

Step-by-step explanation:

i cant see any equations

but the equaiton would be..