Applications of exponential functions

Answer:

The correct answer is option a.

a. 5( √3+ 1 )

Step-by-step explanation:

Given that the angle changes from 45° to 60° in 10 minutes.

This situation can be represented as right angled triangles [tex]\triangle[/tex]ABC (in the starting when angle is 45°)and [tex]\triangle[/tex]ABD (after 10 minutes when the angle is 60°).

AB is the tower (A be its top and B be its base).

Now, we need to find the time to be taken to cover the distance D to B.

First of all, let us consider [tex]\triangle[/tex]ABC.

Using tangent property:

[tex]tan\theta =\dfrac{Perpendicular}{Base}\\\Rightarrow tan 45=\dfrac{AB}{BC}\\\Rightarrow 1=\dfrac{h}{BC}\\\Rightarrow h = BC[/tex]

Using tangent property in [tex]\triangle[/tex]ABD:

[tex]\Rightarrow tan 60=\dfrac{AB}{BD}\\\Rightarrow \sqrt3=\dfrac{h}{BD}\\\Rightarrow BD = \dfrac{h}{ \sqrt3}\ units[/tex]

Now distance traveled in 10 minutes, CD  = BC - BD

[tex]\Rightarrow h - \dfrac{h}{\sqrt3}\\\Rightarrow \dfrac{(\sqrt3-1)h}{\sqrt3}[/tex]

[tex]Speed =\dfrac{Distance }{Time}[/tex]

[tex]\Rightarrow \dfrac{(\sqrt3-1)h}{10\sqrt3}[/tex]

Now, we can say that more distance to be traveled to reach the base of tower is BD i.e. '[tex]\bold{\dfrac{h}{\sqrt3}}[/tex]'

So, more time required = Distance left divided by Speed

[tex]\Rightarrow \dfrac{\dfrac{h}{\sqrt3}}{\dfrac{(\sqrt3-1)h}{10\sqrt3}}\\\Rightarrow \dfrac{h\times 10\sqrt3}{\sqrt3(\sqrt3-1)h}\\\Rightarrow \dfrac{10 (\sqrt3+1)}{(\sqrt3-1)(\sqrt3+1)} (\text{Rationalizing the denominator})\\\Rightarrow \dfrac{10 (\sqrt3+1)}{3-1}\\\Rightarrow \dfrac{10 (\sqrt3+1)}{2}\\\Rightarrow 5(\sqrt3+1)}[/tex]

So, The correct answer is option a.

a. 5( √3+ 1 )

Please, check the options of the question. The point-slope equation needs the slope, m, in the equation.

Answer:

The point-slope equation of the points (5,2) and (-1,-6) is

[tex] \\ y - 2 = \frac{4}{3}(x-5)[/tex]  or,

[tex] \\ y + 6 = \frac{4}{3}(x+1)[/tex]

which are the same as

[tex] \\ 3y-4x+14 = 0[/tex] , (which is not a point-slope equation, though)

Step-by-step explanation:

The point-slope equation is given by:

[tex] \\ y - y_{1} = m(x - x_{1})[/tex]

Where m is the slope of the line:

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

Having the points (5,2) and (-1,-6), then

[tex] \\ m = \frac{-6 - 2}{-1 -5}[/tex]

[tex] \\ m = \frac{-8}{-6}[/tex]

[tex] \\ m = \frac{4}{3}[/tex]

Then, the point-slope equation of the points (5,2) and (-1,-6) is

[tex] \\ y - 2 = \frac{4}{3}(x-5)[/tex] or

[tex] \\ y + 6 = \frac{4}{3}(x+1)[/tex]

The below graph represents both lines (they are the same line).

Answer:

Step-by-step explanation:

To find the height h we use tan

tan ∅ = opposite / adjacent

From the question

The adjacent is 2000 ft

The opposite is h

So we have

tan 15° = h / 2000

h = 2000 tan 15

h = 535.89 ft

h = 536 ft to the nearest foot

Hope this helps you

Answer:  6 feet from the pivot point

Step-by-step explanation:

Girl's weight x distance from center = Boy's weight x distance from center

65 (7) = 75x

[tex]\dfrac{65(7)}{75}=x[/tex]

6.066 = x

The boy needs to be placed 6 feet from the center (aka pivot point) which is the same as saying 1 foot from the end of the seesaw.

Answer:

b. sqrt(120)

Step-by-step explanation:

a^2+b^2=c^2

a^2+7^2=13^2

13^2-7^2=a^2

120=a^2

sqrt(120)=a

This is using Pythagorean theorem

Answer:

  a) -900 L/min

  b) 63000 L

  c) v = -900t +63000

  d) 7200 L

Step-by-step explanation:

a) You are given two points on the curve of volume vs. time:

  (t, v) = (20, 45000) and (70, 0)

The rate of change is ...

  Δv/Δt = (0 -45000)/(70 -20) = -45000/50 = -900 . . . . liters per minute

__

b) In the first 20 minutes, the change in volume was ...

  (20 min)(-900 L/min) = -18000 L

So, the initial volume was ...

  initial volume -18000 = 45000

  initial volume = 63,000 . . . . liters

__

c) Since we have the slope and the intercept, we can write the equation in slope-intercept form:

  v = -900t +63000

__

d) Put the number in the equation and do the arithmetic.

When t=62, the amount remaining is ...

  v = -900(62) +63000 = -55800 +63000 = 7200

7200 L remain after 62 minutes.

Answer

[tex]P= 0.144[/tex] ways

the coin can land tails either exactly 8 times or exactly 5 times in

[tex]0.144[/tex] ways

Step by step explanation:

THis is a binomial distribution

Binomial distribution gives summary of the number of trials as well as observations as each trial has the same probability of attaining one particular value.

P(9)=(14,9).(0.5)⁹.(0.5)¹⁴⁻⁹

p(3)=(14,3).(0.5)⁹.(0.5)¹⁴⁻³

p=(9)+p(3)

p=C(14,9)(0.5)¹⁴ + C(14,3). (0.5)¹⁴

P= (0.5)¹⁴ [C(14,9) + C(14,3)]

P= (0.5)¹⁴ [2002 * 364]

P= 1/16384 * (2002 +364)

P= 91091/2048

P= 0.144

Hence,the coin can land tails either exactly 8 times or exactly 5 times in

[tex] 0.144[/tex] ways

Answer:

~0.3158

Step-by-step explanation:

Number of even numbers in the range of 1 - 38 is 38/2 = 19

=> P(E) = 19/38 = 1/2

Having: 38 = 3 x 12 + 2, then the number of numbers that is a multiple of 3 in the range of 1 - 38 is 12

=> P(M) = 12/38 = 6/19

Having: 38 = 6 x 6 + 2, then the number of numbers that is a multiple of 6 (or multiple of 2 and 3) is 6

=> P(E and M) = 6/38 = 3/19

Applying the conditional probability formula:

P(M|E) = P(E and M)/P(E) = (3/19)/(1/2) = 6/19 = ~0.3158

Answer:

x=3.9 or 39/10 and   y=3.13333 or 47/15

Step-by-step explanation:

Since both expressions (6x-3y) and (3-2x+6y) are equal to 14, separate the equations:

       6x-3y=14 and 3-2x+6y=14

Simplify the equations

       6x-3y=14 and -2x+6y=11

Now, line the equations up and pick a variable (either x or y) to eliminate

        6x-3y=14

        -2x+6y=11

In this case, let's eliminate y first. To do so make the y values in both equations the same but with opposite signs. Make both be 6y but one is +6y and the other -6y

Multiply (6x-3y=14) by 2 to get:

      12x-6y=28

Line the equations up and add or subtract the terms accordingly

      12x-6y=28

      -2x+6y=11

This becomes:

     10x+0y=39

Isolate for x

   x= 39/10 or x= 3.9

Now substitute the x value into either of the original equations

    6x-3y=14

    6(3.9)- 3y=14

Isolate for y

   23.4-14=3y

   3y= 9.4

  y= 3.1333 (repeating)  or y= 47/15

Answer: x = 39/10, y = 94/30

Step-by-step explanation:

6x - 3y = 3 - 2x + 6y,

Now solving this becomes

6x + 2x -3y - 6y = 3

8x - 9y = 3 ------------------- 1

3 - 2x + 6y. = 14

-2x + 6y = 14 - 3

-2x. + 6y = 11

Now multiply both side by -1

2x. - 6y = -11 ----------------- 2

Solve equations 1 & 2 together

8x - 9y. = 3

2x - 6y = -11

Using elimination method

Multiply equation 1 through by 2 ,and equation 2 be multiplied by 8

16x - 18y = 6

-16x - 48y = -88 ------------------------- n, now subtract

30y = 94

Therefore. y = 94/30.

Now substitute for y in equation 2

2x - 6y = -11

2x - 6(94/30) = -11

2x - 94/5 = -11

Now multiply through by 5

10x - 94 = -55

10x = -55 + 94

10x = 39

x = 39/10

Answer:

14.16 miles further.

Step-by-step explanation:

Campbell to Summerfield

9.52mi + 4.62 mi + 10.08 mi = 24.24 mi

Princeton to summerfield

is 10.08 mi

Therefore difference is 24.24mi - 10.08mi = 14.16 mi

Hope this helps.

Answer:

P = 60 cm

Step-by-step explanation:

To solve this question we will use the property of the tangents drawn from a point to a circle.

"Length of tangents drawn from a point to a circle are equal in measure."

By this property,

Therefore, BG ≅ FB ≅ 7.5 cm

AG ≅ AH ≅ 6.5 cm

CF ≅ EC ≅ 8.5 cm

Since m∠B ≅ m∠D

Therefore, length of tangents FB ≅ GB ≅ DE ≅ DH ≅ 7.5 cm

Since, Perimeter of ABCD = AB + BC + CD + DA

AB = 7.5 + 6.5 = 14 cm

BC = 7.5 + 8.5 = 16 cm

CD = 8.5 + 7.5 = 16 cm

DA = 7.5 + 6.5 = 14 cm

Now Perimeter = 16 + 16 + 14 + 14 = 60 cm

Therefore, P = 60 cm will be the answer.

Answer:

40 cm by 44 cm

Step-by-step explanation:

A square sheet of paper has area 1600 cm^2.

area = s^2

1600 cm^2 = s^2

s = sqrt(1600 cm^2)

s = 40 cm

The side of the square was 40 cm, so the square measured 40 cm by 40 cm.

One side of 4 cm short, so the paper should have been 40 cm by 44 cm.

Answer:

332.88 million

Step-by-step explanation:

you do the thing

Answer:

10 miles

Step-by-step explanation:

3 mi/1 hr  x (h hours + 2/3 hr) = 5 mi/1 hr  x  h hours

3h + 2 = 5h

2 = 2h

h = 1 hour

3mi/hr  x 1 2/3 hr = 5 miles

5 mi/hr x  1 hr = 5 miles

He hiked 10 miles. (

Answer:

The graph of g( x ) is the graph of f(x) stretched vertically by a factor of 2.

Option C is the correct option.

Step-by-step explanation:

Solution,

f ( x ) = 2ˣ

g ( x ) = 2 ( 2 ˣ )

2 is multiplied with f(x)

2 is greater than 1

so, Vertical stretch by a factor of 2.

Option C is correct.

Hope this helps...

Best regards!!

We want to compare the functions f(x) and g(x), given that we know that g(x) is a transformation of f(x).

The correct option is B, "the graph of g(x) is the graph of f(x) stretched vertically by a scale factor of 2."

Here we know that:

f(x)  =2^x

g(x) = 2*f(x) = 2*2^x

First, remember that a general vertical dilation/stretch of scale factor k is written as:

g(x) = k*f(x)

So only by that and knowing that g(x) = 2*f(x), we can conclude that the graph of g(x) is the graph of f(x) dilated/stretched vertically by a scale factor of 2.

Then the correct option is B, "the graph of g(x) is the graph of f(x) stretched vertically by a scale factor of 2."

If you want to learn more, you can read:

https://brainly.com/question/16670419

Answer:

x=6.28 inches

y=191.08 inches

Step-by-step explanation:

Let the dimensions of the rectangle be x and y

Area of the rectangular sheet

x*y=1200 in^2}

x = circumference of the cylinder

This means x=2πr

Volume of a cylinder=πr^2h

h=y

Volume of the cylind=πr^2(y)=600 in^3

From x=2πr

r=x/2π

Substitute r=x/2π into Volume=πr^2(y)=600 in^3

We have,

Volume of the cylinder=πr^2(y)=600 in^3

π*(x/2π)^2(y)=600

(x^2/4π)y=600

Recall, x*y=1200

y=1200/x

Substitute y=1200/x into (x^2/4π)y=600

(x^2/4π)y=600

(x^2/4π)(1200/x)=600

1200x/4π=600

Multiply both sides by 4π

(x^2/4π)(1200/x)(4π)=600*4π

1200x=2400π

Divide both sides by 1200

1200x/1200 = 2400π/1200

x=2π

Substitute x=2π into y=1200/x

We have,

y=1200/2π

y=600/π

The dimensions are x=2π and y=600/π

Let π=3.14

x=2π

=2(3.14)

=6.28 inches

y=600/π

=600/3.14

=191.08 inches

Answer:

Reject H0

Age and pressure are related

Step-by-step explanation:

The null hypothesis is rejected or accepted on the basis of level of significance. When the p-value is greater than level of significance we fail to reject the null hypothesis and null hypothesis is then accepted. It is not necessary that all null hypothesis will be rejected at 10% level of significance. To determine the criteria for accepting or rejecting a null hypothesis we should also consider p-value. In the given scenario we reject the null hypothesis because job pressure and age are related to each other.

Answer:

123 Square Feet. Of Paint. Probably gonna take a little more than a sample-size quart, let me know how it turns out.

Step-by-step explanation:

You would need 11*15=165 square feet of paint. BUT

You need 42 less square feet, because there are 2, 7*3=21 square-foot windows.

165-42

Answer

Step-by-step explanation:

Answer:

See the argument below

Step-by-step explanation:

I will give the argument in symbolic form, using rules of inference.

First, let's conclude c.

(1)⇒a  by simplification of conjunction

a⇒¬(¬a) by double negation

¬(¬a)∧(2)⇒¬(¬c) by Modus tollens

¬(¬c)⇒c by double negation

Now, the premise (5) is equivalent to ¬d∧¬h which is one of De Morgan's laws. From simplification, we conclude ¬h. We also concluded c before, then by adjunction, we conclude c∧¬h.

An alternative approach to De Morgan's law is the following:

By contradiction proof, assume h is true.

h⇒d∨h by addition

(5)∧(d∨h)⇒¬(d∨h)∧(d∨h), a contradiction. Hence we conclude ¬h.  

Answer:

Step-by-step explanation:

f(x) = 2x² + x - 3

g(x) = x + 2

To find (f • g) (x) substitute g(x) into every x in f (x)

That's

(f • g) (x) = 2(x + 2)² + x + 2 - 3

Expand and simplify

(f • g) (x) = 2( x² + 4x + 4) + x - 1

= 2x² + 8x + 8 + x - 1

Group like terms

= 2x² + 8x + x + 8 - 1

We have the final answer as

Hope this helps you

Answer:

a. p`± z₀.₀₂₅[tex]\sqrt{ \frac{p`q`}{n}[/tex]  

b.0.6 ±  1.96 [tex]\sqrt \frac{0.6* 0.4}{1000}[/tex]  

c. { -1.96 ≤  p`± z₀.₀₂₅[tex]\sqrt{ \frac{p`q`}{n}[/tex]     ≥ 1.96} = 0.95  

Step-by-step explanation:

Here the total number of trials is n= 1000

The number of successes is p` = 600/1000 = 0.6. The q` is 1 - p`= 1- 0.6 = 0.4

The degree of confidence is 95 %  therefore z₀.₀₂₅ = 1.96 ( α/2 = 0.025)

a.  The formula used will be

p`± z₀.₀₂₅[tex]\sqrt{ \frac{p`q`}{n}[/tex]       ( z with the base alpha by 2 (α/2 = 0.025))

b. Putting the values

0.6 ±  1.96 [tex]\sqrt \frac{0.6* 0.4}{1000}[/tex]  

c. Confidence Interval in Interval Notation.

{ -1.96 ≤  p`± z₀.₀₂₅[tex]\sqrt{ \frac{p`q`}{n}[/tex]     ≥ 1.96} = 0.95  

{ -z( base alpha by 2) ≤  p`± z₀.₀₂₅[tex]\sqrt{ \frac{p`q`}{n}[/tex]     ≥ z( base alpha by 2)  } = 1- α

Answer:

24.5 mpg

Step-by-step explanation:

(23,695 - 23,252) / 14 = 24.5mpg

Answer:

The car averaged a total of 24.5 miles per gallon between the two fillings

Step-by-step explanation:

Firstly, we calculate the difference between the mileages.

This will give us the total distance traveled.

That would be 23,695 - 23,352 = 343 miles

The tank capacity obviously is 14 gallons

So mathematically, miles per gallon averaged between the two fillings = distance traveled by the car/gallon of fuel used = 343/14 = 24.5 miles per gallon

Answer:

A(t) = 41,47 in²

Step-by-step explanation:

Let´s call "x" the cut of point to get to pieces of wire, we make a square from x and the regular octagon will be shaped with 24-x

Then Area of the square  A(s)  =  x²

Area of the octagon is  A(o)  = 1/2*p*length of apothem (d)

p = ( 24 - x )

length of apothem (d) :

The side of the octagon is equal to ( 24 - x ) / 8 half the side is  

( 24  -  x ) / 16

tan α  = ( 24- x ) 16 / d      since ∡s in octagon are 360 / 8  = 45°

α ( ∡ between apothem and one of the interiors ∡ of the octagon )half of 45 is  α = 22,5°  

tanα = 0,41

d = (24 - x ) / 16*0,41        d = ( 24 - x ) / 6,56

Then

A(t) = A(s) + A(o)

A(t) =  x²  + (1/2)* ( 24 - x ) ( 24 - x ) / 6,56

Note  A(t) = A(x)

A(x) =  x²  + (1/2) * (24 - x )²/ 6,56

A(x) = x² + ( 1/ 2*6,56) * ( (24)² -48*x + x² )

Taking derivatives on both sides of the equation

A´(x) = 2*x + ( 1/13,12)* ( - 48 + 2x )

A´(x) = 2*x - 48/ 13,12 + 2*x

A´(x) = 4*x - 3,66

A´(x) = 0     4x = 3,66        x  = 0,91 in    and  d =( 24 - x ) / 6,56

d = ( 24 - 0,91 ) / 6,56        d = 3,52

Then  A(s) = (0,91)²      A(s) = 0,83 in²

A(o) = 1/2 * ( 24 - 0,91 )* 3,52

A(o) = 40,63 in²

A(t) = 40,63 + 0,83

A(t) = 41,47 in²

Answer:

[tex]\frac{109}{122}[/tex]

Step-by-step explanation:

Well first we need to find the total amount of Winter Olympic medals won.

550 + 540 + 130

= 1220

Now we need to find the amount won from the Western and Northern Europe.

550 + 540

= 1090

Now we can make the following fraction,

1090/1220

Simplify

= 109/122

Thus,

the answer is [tex]\frac{109}{122}[/tex].

Hope this helps :)

Hi there!! (✿◕‿◕)

⭐⭐⭐⭐⭐⭐⭐⭐⭐⭐⭐⭐⭐⭐⭐⭐⭐⭐⭐⭐⭐⭐

Northern Europe: 550 medals

Western Europe: 540 medals

550 + 540 = 1,090

Northern Europe and Western Europe: 1,090

Other: 130

1,090 + 130 = 1,220

European Regions: 1,220 medals

1,090/1,220 = 109/122

Hope this helped!!  ٩(◕‿◕｡)۶

Answer:

B. -3x

Step-by-step explanation:

A term is defined as either a constant or a variable with a coefficient.

-3 is incorrect because there is no constant -3 in the expression.

-3x is correct because there is a -3x in the expression

(x + 4) is incorrect because that is a linear binomial and has yet to be distributed.

-7 is incorrect because it has to be distributed.

Answer:

-4x + 4

Step-by-step explanation:

4x - 2( 4x - 2 )

→ Expand out 2 ( 4x - 2 )

2 ( 4x - 2 ) = 8x - 4

→ Substitute the expanded bracket back into the expression

4x - (8x - 4)

→ Collect the 'x' values

-4x + 4

Answer:

Hey there!

Circumference of a circle=0.5[tex]\pi[/tex]r

3.14=0.5[tex]\pi[/tex]r

1=0.5r

r=2

Hope this helps :)

Answer:

1/2 meter

Step-by-step explanation:

The circumference of a circle can be found using the following formula:

c= pi* 2r

We know that the circumference of the circle is 3.14 meters. Therefore, we can substitute 3.14 in for c.

3.14= pi* 2r

We want to find out what r, or the radius is. To do this, we must get r by itself.

First, divide both sides of the equation by pi, or 3.14. We divide because 2r is being multiplied by pi, and division is the inverse of multiplication.

3.14= pi* 2r

3.14/3.14=3.14* 2r/3.14

3.14/3.14=2r

1=2r

Next, divide both sides by 2. We divide because 2 and r are being multiplied, and the inverse of division is multiplication.

1/2=2r/2

1/2=r

0.5=r

The radius of the circle is 1/2 or 0.5 meters.

Answer:

s = 1/t - 1/r

Step-by-step explanation:

1/r+s=1/t

Subtract 1/r from each side

1/r - 1/r+s=1/t- 1/r

s = 1/t - 1/r

Answer:

[tex]\huge{\boxed{s=t-r}}[/tex]

Step-by-step explanation:

if

[tex]\dfrac{1}{r+s}=\dfrac{1}{t}\qquad\text{cross multiply}\\\\(1)(t)=(1)(r+s)\\\\t=r+s\qquad\text{subtract}\ r\ \text{from both sides}\\\\t-r=r-r+s\\\\t-r=s\to\boxed{s=t-r}[/tex]

Answer:

If I getting the question correctly, your doubt is the order of the values you get after doing the math. I mean, you can do the calculations, but in the end, you don't form the coordinate pairs correctly, that's why I understand from your question. So, here is what you need to do.

One way to graph is by using the definitions of those kinds of functions. For example, let's say we want to find the points to draw the function: [tex]y=sin(x)[/tex]

Remember that trigonometric functions have a specific period, that means, their drawing repeats over and over again after a certain number. That period is [tex]2 \pi[/tex], that means this number represents a cycle.

So, the main thing you need to do is to pick a starting point to then, draw the curve according to the [tex]2 \pi[/tex] period.

Now, we know that sine functions intercept the origin of the coordinate system, as you can observe in the image attached, there you can see that from the origin you draw those waves making sure you intercept the x-axis at every [tex]\pi[/tex] number. In the end, you will have a sine function.

On the other hand, if you want to have a chart with all x-values and y-values. First, you need to set x-values: -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5. Then, you need to find each y-values for each of them.

Now, you have to draw a chart value, to keep in other the coordinate pairs, that way, you'll have the correct pairs at the end.

For example, for [tex]x=-5[/tex], we get [tex]y=-0.09[/tex], that means in the chart value, you are gonna form the pair [tex](-5, -0.09)[/tex], and now you have your first point of the drawing. Then, you keep repeating the process until you complete all the chart values.

As you can imagine, you're going to get really small decimal numbers, that's why I explained to you the first method, it's faster and easier.

Answer:

The number stays the same

Step-by-step explanation:

The power of 1 equals the number itself