A certain car travels at a constant speed of 40 miles per hour. At this speed, the car can travel a distance of 25 miles for each gallon of fuel used. How many gallon sof fuel ar used when the car travels at this speed for 75 mins?

Answer:

(2, 1).

Step-by-step explanation:

Dilated by factor 3:

C = (6, 3) ----> C' would be (6 * 1/3, 3 *1.3) = (2, 1).

Answer:

The probability that the sample will contain exactly three​ successes is 0.2668.

Step-by-step explanation:

Given information:

Sample size = 10

Probability of a success, p=0.30

Probability of a failure q=1-p = 1-0.30 = 0.70

The binomial formula to determine the probability is

[tex]P(X=r)=^nC_rp^rq^{n-r}[/tex]

where, n is the sample size, r is required number of success, p is probability of success and q is probability of failure.

We need to find the probability that the sample will contain exactly three​ successes.

[tex]P(X=3)=^{10}C_3(0.30)^3(0.70)^{10-3}[/tex]

[tex]P(X=3)=^{10}C_3(0.30)^3(0.70)^{7}[/tex]

[tex]P(X=3)=(120)(0.0022235661)[/tex]

[tex]P(X=3)=0.266827932[/tex]

[tex]P(X=3)\approx 0.2668[/tex]

Therefore the probability that the sample will contain exactly three​ successes is 0.2668.

Using the binomial distribution, it is found that there is a 0.2668 = 26.68% probability that the sample will contain exactly three​ successes.

The formula is:

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]

The parameters are:

In this problem:

The probability that the sample will contain exactly three​ successes is P(X = 3), hence:

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

[tex]P(X = x) = C_{10,3}.(0.3)^{3}.(0.7)^{7} = 0.2668[/tex]

More can be learned about the binomial distribution at https://brainly.com/question/24863377

Answer:

70 sentences

Step-by-step explanation:

7 x 5 x 2= 70

Answer:

Jeff mows 9 yards of lawn in 30 minutes

Step-by-step explanation:

The equation that models the amount of time (t) in minutes it takes Jeff to mow an average sized yard is [tex]t=2n+12[/tex], where n is the number of yards he mows.

To find the number of yards Jeff mows in 30 minutes, we set the equation to 30 and solve for n.

[tex]\implies 2n+12=30[/tex]

Add -12 to both sides:

[tex]\implies 2n=30-12[/tex]

[tex]\implies 2n=18[/tex]

Divide both sides by 2

[tex]\implies \frac{2n}{2}=\frac{18}{2}[/tex]

[tex]\implies n=9[/tex]

Hence Jeff mows 9 yards of lawn in 30 minutes.

Final answer:

Jeff mows 9 lawns if it takes him 30 minutes, based on the equation t = 2n + 12

Explanation:

The question asks us to find out how many lawns Jeff mows if it takes him 30 minutes. The relationship between the amount of time (t) in minutes and the number of yards he mows (n) is given by the equation t = 2n + 12. Since we are given that t = 30 minutes, we can substitute the value of t into the equation to solve for n, the number of lawns.

30 = 2n + 12

By subtracting 12 from both sides of the equation we get:

18 = 2n

Next, we divide both sides of the equation by 2 to solve for n:

n = 9

Therefore, Jeff mows 9 lawns if it takes him 30 minutes.

Answer:

84=2l+2w

w=21

Step-by-step explanation:

84=2(l+w)

42=l+w

l=42-w

Area=l×w

A=(42-w)×w

Differentiate A=42w-w×w

with respective to "w".

dA/dw= 42-2w

For a minimum or maximum area

dA/dw=0

then, 42-2w=0

w=21

proving "A" is maximum when "w=21"

dA/dw>0 when w<21

dA/dw<0 when w>21

Therefore Area is maximum when "w=21"

The most extensive rectangular garden this man can build with his 84 feet of fencing would be a square with length and width of 21 feet each, yielding a total area of 441 square feet.

A rectangular garden's perimeter can be expressed as P = 2l + 2w, where l is the length and w is the park's width. Given 84 feet of fencing (the total perimeter), this equation becomes 84 = 2l + 2w. Simplified, this gives l + w = 42. Expressing w in terms of l gives w = 42 - l.

Next, breathing, we substitute w into the area formula A = l * w, get A = l * (42 - l). This area expression of one variable forms a quadratic equation, A = -l² + 42l. This suggests that the site is maximized from the quadratic formula when l = -b/2a = 42/2 = 21. So, the length and width of the most significant garden area are both 21 feet, representing a square.

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Answer: 45 degree angle

Answer : The total volume of figure is, [tex]162in^3[/tex]

Step-by-step explanation :

First we have to calculate the volume of cuboid A.

[tex]V_A=l\times b\times h[/tex]

where,

[tex]V_A[/tex] = volume of cuboid A

l = length of cuboid = 9 in

b = breadth of cuboid = 5 in

h = height of cuboid = 2 in

[tex]V_A=9in\times 5in\times 2in[/tex]

[tex]V_A=90in^3[/tex]

Now we have to calculate the volume of cuboid B.

[tex]V_B=l\times b\times h[/tex]

where,

[tex]V_B[/tex] = volume of cuboid B

l = length of cuboid = 9 in

b = breadth of cuboid = 1 in

h = height of cuboid = 2 in

[tex]V_B=9in\times 1in\times 2in[/tex]

[tex]V_B=18in^3[/tex]

Now we have to calculate the volume of cuboid C.

[tex]V_C=l\times b\times h[/tex]

where,

[tex]V_C[/tex] = volume of cuboid C

l = length of cuboid = 9 in

b = breadth of cuboid = 1 in

h = height of cuboid = (8-2=6) in

[tex]V_C=9in\times 1in\times 6in[/tex]

[tex]V_C=54in^3[/tex]

Now we have to calculate the total volume of figure.

Total volume of figure = [tex]V_A+V_B+V_C[/tex]

Total volume of figure = [tex]90in^3+18in^3+54in^3[/tex]

Total volume of figure = [tex]162in^3[/tex]

Thus, the total volume of figure is, [tex]162in^3[/tex]

For this case we have to graph, we can see in the figure that the graph of the Neperian logarithm grows faster than the graph of the logarithm.

The graph of ln (x) is the second attached

Answer:

Option A

The log (x) graph will grow slower than the ln (x) graph.

See attached image

Answer:

Length of arc AB= 2πr (angle) /360

= 2π×6× 30°/360

= π.

ANSWER

The length of arc AB is π units.

EXPLANATION

The length of an arc of a circle is calculated using the formula:

[tex]l = \frac{ \theta}{360} \times 2\pi \: r[/tex]

Where [tex] \theta = 30 \degree[/tex]

is the angle of sector and r=6 units is the radius of the circle.

We substitute these values into the formula to find the length of arc AB.

[tex]l = \frac{30}{360} \times 2 \times \pi \: \times 6[/tex]

[tex]l = \pi[/tex]

The length of the arc is π units.

So put 1 as the coefficient.

There were 26 fans at the game who were lucky enough to receive all three free items.

The Madd Batters minor league baseball team offered different incentives to its fans for the opening home game.

To determine how many fans were lucky enough to receive all three free items, we need to find the total number of fans that satisfy each condition.

The least common multiple of 75, 30, and 50 is 150.

So, we divide 4000 by 150 to find the number of fans that satisfy the conditions for all three items.

Therefore, 26 fans at the game were lucky enough to receive all three free items.

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Answer:

Step-by-step explanation:

Such questions generally arise in the context of measurement precision and/or accuracy. Apparently, we're to assume that these dimensions could be arrived at by rounding to the nearest km. In that case, they can be taken to have a possible error of ±0.5 km.

The minimum possible area is the product of the minimum possible dimensions: 1.5 km by 4.5 km = 6.75 km².

The maximum possible area is the product of the maximum possible dimensions: 2.5 km by 5.5 km = 13.75 km².

_____

Comment on combining measurement values

You will note that the nominal area is 2 km by 5 km = 10 km², and that the middle value between the minimum and maximum is slightly more than this, at 10.25 km².

It is typically the case that when measurements are combined by operations other than addition and subtraction, the nominal result is different from the middle result in the range of possibilities.

Answer:

The first choice is the one you want

Step-by-step explanation:

First thing you need to know about this greatest integer graph is that it is aptly called a step graph.  It literally looks like stair steps on your calculator: short horizontal lines that are not connected vertically.  Really cool graph.

Second thing you need to know is about transformations of functions.  ANY side-to-side movement in ANY function will be in a set of parenthesis (or absolute value symbols, or under a radical sign, or inside the greatest integer brackets, etc.) and ANY up or down movement will be either added or subtracted.  Added means you move the function up from its starting position, subtracting means you move the function down from its starting position.  Since we have no numbers inside the greatest integer brackets, there is no side-to-side movement.  Since there is a "-2" after the brackets, we are moving the whole function down.

If you do not know how to graph these without a calculator and you have no idea what this graph looks like, I recommend going to your calculator to see it.  First, call up your "y = " window.  Next, hit 2nd-->0 (catalog), then hit the x^2 button (this will take you to the letter I in the catalog).  Scroll down til you see "int( " and hit that button.  It will take you back to the "y = " window.  Enter an x after that set of parenthesis and then close it, then hit " - 2 " and then "graph".  Your steps should begin to appear.  Notice that the horizontal line between x = 0 and x = 1 is at y = -2.  The parent graph has this line between x = 0 and x = 1 on y = 0.  The -2 in ours moved the graph down from y = 0 to y = -2

Summing up, the first choice is the one you want as your answer.

Answer:

A:

The steps are at y=-2

Step-by-step explanation:

edge 2021

Answer:

AB= AC times AE/AD

Step-by-step explanation:

we know that

If two figures are similar, then the ratio of its corresponding sides is proportional

so

In this problem

AB/AC=AE/AD

solve for AB

AB=(AC)(AE)/AD

Answer:

AB= AC times AE/AD

Step-by-step explanation:

AB is on the segment AC, therefore it is proportional to AC .

In order to fulfill the requirement that triangle ABE is a smaller scaled replica of triangle ACD, a scale factor must be applied to the sides of the original figure; in this case, the factor is AE/AD.

Answer:

Answer is in the attachment.

Step-by-step explanation:

To graph x>2 consider first x=2. x=2 is a vertical line and if you want to graph x>2 you need to shade to the right of the vertical line.

To graph x+y<2, I will solve for y first.

x+y<2

Subtract x on both sides:

  y<-x+2

Consider the equation y=-x+2.  This is an equation with y-intercept 2 and slope -1 or -1/1.  So the line you have in that picture looks good for y=-x+2. Now going back to consider y<-x+2 means we want to shade below the line because we had y<.

Now where you see both shadings will be intersection of the shadings and will actually by your answer to system of inequalities you have.  In my picture it is where you have both blue and pink.

I have a graph in the picture that shows the solution.

Also both of your lines will be solid because your question in the picture shows they both have equal signs along with those inequality signs.

Just in case my one graph was confusing, I put a second attachment with just the solution to the system.

Answer:

B) There is a gap in Plot A, but not in Plot B.

Step-by-step explanation:

Plot A shows the number of hours ten girls watched television over a one-week period. Plot B shows the number of hours ten boys watched television over the same period of time. There is a gap in Plot A, but not in Plot B compares the shape of the dot plots.

The correct answer is B) There is a gap in Plot A, but not in Plot B

Explanation:

A dot plot is a type of graphic used in statistics to show information collected about a group of subjects or elements. In this, each dot or circle represents a subject related to a specific scale for example, in the case presented, the scale is the number of hours subjects watched television.

According to this, the statement that is true about the dot plots presented is that there is a gap in plot A because most subjects reported watching television for 3 to 7 hours; however, one of the subjects watched television 10 hours. This implies, a gap or space in the graphic because there is an interval in the scale with no subjects as most are grouped in a cluster except by one. This does not occur in plot B because all subjects form one cluster or group.

Answer:

3.58

Step-by-step explanation:

Given :

y = 2x + 4 -------- eq1

y = 2x - 4 -------- eq2

sanity check : both equations have same slope, so we can conclude that they are both parallel to one another.

Step 1: consider equation 1, pick any random x-value and find they corresponding y-value. we pick x = -2

This gives us y = 2(-2) + 4 = 0

Hence we get a point (x,y) = (-2,0)

Step 2: express equation 2 in general form (i.e Ax + By + C = 0)

y = 2x-4 -------rearrange---> 2x - y -4 = 0

Comparing with the general form, we get A = 2, B = -1, C = -4

Recall that the distance between 2 parallel lines is given by the attached formula (see attached picture).

substituting the values for A, B, C and (x, y) from the previous step:

d = | (2)(-2) + (-1)(0) + (-4) |  / √(2² + (-1)²)

d = | -4 + 0 - 4 |  / √(4 + 1)

d = | -8 |  / √5

d = 8  / √5

d = 3.5777

d = 3.58 (rounded 2 dec. pl)

Answer:

Explanation:

Correlation is a numerical value that tells how two variables or factors change together.

The correlation between two factors may be positive, negative or nonexistent.

A strong association is shown when the graph of the points representing the factors are reasonably well represented by a line.

A perfect positive correlation is when the two variables are related by a linear function with positive slope (the two factors grow together).

A perfect negative correlation exists when the two factors are related by a linear function with negative slope (one factor grows when the other factor decreases).

A nonexistent correlation is when the two factors are not related in any way to each other and so none function can be obtained.

Final answer:

Correlation measures how two variables are related, with the correlation coefficient indicating the strength and direction of this relationship. The coefficient ranges from -1 to +1, where +1 is a perfect positive correlation and -1 is a perfect negative correlation.

Explanation:

Correlation is a measure of the extent to which two factors are related. Specifically, it refers to how one variable changes in relation to another. The statistical measure used to describe this is called a correlation coefficient, represented by the letter r.

The value of r ranges from -1 to +1, where +1 signifies a perfect positive correlation, -1 signifies a perfect negative correlation, and 0 indicates no correlation at all.

Positive correlation happens when both variables change in the same direction, either increasing or decreasing together. Conversely, a negative correlation indicates that as one variable increases, the other decreases.

While correlation is a useful statistical tool to identify a relationship between two variables, it is crucial to understand that correlation does not imply causation.

This means that just because two variables are correlated, it does not necessarily mean that one variable causes the other to change.

Answer:

y-7 = 1/2(x-1) point slope form

y = 1/2x+13/2  slope intercept form

Step-by-step explanation:

y=-2x+9

This equation is in the form y= mx +b so the slope is -2

We want a line perpendicular

Take the negative reciprocal

m perpendicular is - (-1/2)

m perpendicular = 1/2

We have a slope of 1/2 and a point.  We can use point slope form

y-y1 = m(x-x1)

y-7 = 1/2(x-1)  point slope form

y-7 = 1/2x-1/2

Adding 7 to each side

y-7+7 =1/2x -1/2+7

y = 1/2x-1/2 +14/2

y = 1/2x +13/2  slope intercept form

The distance is,

[tex]d(A,B)=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]

Where [tex]A(x_1,y_1),B(x_2,y_2)\longrightarrow A(-3,-2),B(-1,-2)[/tex]

[tex]d(A,B)=\sqrt{(-1-(-3))^2+(-2-(-2))^2}=\sqrt{4}=2[/tex]

The distance between points A, B is 2 units.

Hope this helps.

r3t40

The formula for distance between two points is:

[tex]\sqrt{(x_{2} -x_{1})^{2} + (y_{2} -y_{1})^{2}}[/tex]

In this case:

[tex]x_{2} =-1\\x_{1} =-3\\y_{2} =-2\\y_{1} =-2[/tex]

^^^Plug these numbers into the formula for distance like so...

[tex]\sqrt{(-1 - (-3))^{2} + (-2 - (-2))^{2}}[/tex]

To solve this you must use the rules of PEMDAS (Parentheses, Exponent, Multiplication, Division, Addition, Subtraction)

First we have parentheses. Remember that when solving you must go from left to right

[tex]\sqrt{(-1 - (-3))^{2} + (-2 - (-2))^{2}}[/tex]

-1 - (-3) = 2

[tex]\sqrt{(2)^{2} + (-2 - (-2))^{2}}[/tex]

-2 - (-2) = 0

[tex]\sqrt{(2)^{2} + (0)^{2}}[/tex]

Next solve the exponent. Again, you must do this from left to right

[tex]\sqrt{(2)^{2} + (0)^{2}}[/tex]

2² = 4

[tex]\sqrt{4 + (0)^{2}}[/tex]

0² = 0

[tex]\sqrt{(4+0)}[/tex]

Now for the addition

[tex]\sqrt{(4 + 0)}[/tex]

4 + 0 = 4

√4 <<<This can be further simplified to...

2

***Remember that the above answers are in terms of units

Hope this helped!

~Just a girl in love with Shawn Mendes

Answer:

Conjugate radicals.

Step-by-step explanation:

The two roots a+sqrb and a-sqr b are called conjugate radicals.

Answer:

There's one important property which states that when a number is multiplied by its reciprocal, equals 1.

The commutative property of multiplication states that two or more numbers can be multiplied in any order, therefore, the expression:  (7)×(13/29)×(1/7) can be changed to  (7)×(1/7)×(13/29) and the result won't be altered.

Now, given that 7 and 1/7 are reciprocals, the result is 1. Therefore, the result of the multiplication equals (13/29)

Answer:

communitive property

Step-by-step explanation:

Answer:

  √35

Step-by-step explanation:

The right triangles are all similar, so the ratios of long leg to hypotenuse are the same for all:

  z/7 = 5/z

  z² = 35 . . . . multiply by 7z

  z = √35

Answer:

Step-by-step explanation:

The right triangles are all similar, so the ratios of long leg to hypotenuse are the same for all:

 z/7 = 5/z

 z² = 35 . . . . multiply by 7z

 z = √35

Answer:

A 1/2

Step-by-step explanation:

x - 2y = 3

-2y = -x + 3

y = 1/2 x - 3/2

please mark brainliest :)

Answer:

A.  1/2

Step-by-step explanation:

Solve the equation for y.

x - 2y = 3

Subtract x from both sides.

-2y = -x + 3

Divide both sides by -2.

[tex] y = \dfrac{1}{2}x - \dfrac{3}{2} [/tex]

The slope-intercept equation of a line is y = mx + b, where m is the slope.

In the case, m = 1/2.

Answer: slope = 1/2

we know that θ is in the III Quadrant, and let's recall that on the III Quadrant sine and cosine are both negative, and since tangent = sine/cosine, that means that tangent is positive.  Let's also keep in mind that tan(π) = sin(π)/cos(π) = 0/-1 = 0.

well, the hypotenuse is just a radius unit, so is never negative, since we know sin(θ) = -(5/13), well, the negative number must be the 5, so is really (-5)/13.

[tex]\bf sin(\theta )=\cfrac{\stackrel{opposite}{-5}}{\stackrel{hypotenuse}{13}}\impliedby \textit{let's find the \underline{adjacent side}} \\\\\\ \textit{using the pythagorean theorem} \\\\ c^2=a^2+b^2\implies \pm\sqrt{c^2-b^2}=a \qquad \begin{cases} c=hypotenuse\\ a=adjacent\\ b=opposite\\ \end{cases} \\\\\\ \pm\sqrt{13^2-(-5)^2}=a\implies \pm\sqrt{144}=a\implies \pm 12 =a\implies \stackrel{III~Quadrant}{-12=a} \\\\[-0.35em] ~\dotfill[/tex]

[tex]\bf tan(\theta )\implies \cfrac{sin(\theta )}{cos(\theta )}\implies \cfrac{~~-\frac{5}{13}~~}{-\frac{12}{13}}\implies -\cfrac{5}{~~\begin{matrix} 13 \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~}\times -\cfrac{~~\begin{matrix} 13 \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~}{12}\implies \cfrac{5}{12} \\\\[-0.35em] ~\dotfill[/tex]

[tex]\bf tan(\pi +\theta )=\cfrac{tan(\pi )+tan(\theta )}{1-tan(\pi )tan(\theta )}\implies tan(\pi +\theta )=\cfrac{0+\frac{5}{12}}{1-0\left( \frac{5}{12} \right)} \\\\\\ tan(\pi +\theta )=\cfrac{~~\frac{5}{12}~~}{1}\implies tan(\pi +\theta )=\cfrac{5}{12}[/tex]

The value is 5/12.

To find the value of tan(π + θ) given that θ terminates in Quadrant III and sinθ = -5/13, we need to first note that in the third quadrant, both sine and tangent are negative. Since we already have the value for sine, we can use the Pythagorean identity to find the value for cosine. The identity is sin^2θ + cos^2θ = 1.

Starting with sinθ = -5/13, we square both sides to get sin^2θ = 25/169. Then, we use the Pythagorean identity to solve for cos^2θ which gives us cos^2θ = 1 - 25/169 = 144/169. Taking the positive and negative square root, since cosine is also negative in the third quadrant, we choose the negative root, cosθ = -12/13.

Now, tanθ is the ratio of sine to cosine, which is tanθ = sinθ/cosθ = (-5/13) / (-12/13). This simplifies to tanθ = 5/12. However, the question asks for tan(π + θ), not tanθ. The tangent function has a period of π, so tan(π + θ) = tanθ. Therefore, the answer is 5/12.

Find the common ratio between each number given:

8/4 = 2

16/8 = 2

32/16 = 2

The common ratio is constant at 2, which means the next number would be 32 x 2.

A geometric series equation is written as an = a1r^n-1

Where a1 is the first term, r is the ratio and n is the term you want to find.

r was determined to be 2. The first term is given as 4 ( the first number in the series).

This means the equation becomes an = 4(2)^n-1

The answer would be B.

Answer:

The answer is B

Step-by-step explanation:

Answer:

Step-by-step explanation:

Given that an equation in variable x as follows:

[tex]e^x =2x+3[/tex]

Let us start with x=0

We have left side = 1 and right side = 3.

Error = 2

ii) x=1: left side = e =2.718 and right side = 6.

Error = 3.282

iii) x =2: Left side = 7.388 and right side = 7

Error = 0.388

iv) x=3

Left side = 20.079

Right side = 9

Error = 11.079

Since error is the least when x=0, we can say 0 is the best approximation.

Answer:

[tex]1.\ \ p(x)\geq0 & \text{ for all values of x.}\\\ 2.\ \sum\ p(x)=1[/tex]

Step-by-step explanation:

There are two requirements for a discrete probability​ distribution that must be satisfied as :-

1. Each probability must be greater than equals to zero.

2. Sum of all probabilities should be equals to 1.

The above conditions are also can be written as :

[tex]1.\ \ p(x)\geq0 & \text{ for all values of x.}\\\ 2.\ \sum\ p(x)=1[/tex]

A discrete probability distribution must satisfy two conditions: each individual outcome probability must be between 0 and 1, and the total of all outcome probabilities must equal to 1.

There are two key requirements that a set of data must meet to be considered a discrete probability distribution:

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Answer:

  60 inches

Step-by-step explanation:

Put the given numbers in the formula and solve for the remaining unknown.

  P = 2L +2W

  18 ft = 2L +2×(4 ft)

  10 ft = 2L . . . . . . . . . subtract 8 ft

  5 ft = L . . . . . . . . . . . divide by 2

You want this in inches, so you make use of the fact that each foot is 12 inches.

  5 ft = 5×(12 in) = 60 in

The length of Sarah's fenced yard is 60 inches.

Answer:

60 IN

Step-by-step explanation:

DID THE QUIZ AND GOT IT CORRECT

Answer:

Step-by-step explanation:

For a. we start by dividing both sides by 200:

[tex](1.05)^x=1.885[/tex]

In order to solve for x, we have to get it out from its position of an exponent.  Do that by taking the natural log of both sides:

[tex]ln(1.05)^x=ln(1.885)[/tex]

Applying the power rule for logs lets us now bring down the x in front of the ln:

x * ln(1.05) = ln(1.885)

Now we can divide both sides by ln(1.05) to solve for x:

[tex]x=\frac{ln(1.885)}{ln(1.05)}[/tex]

Do this on your calculator to find that

x = 12.99294297

For b. we will first apply the rule for "undoing" the addition of logs by multipllying:

[tex]ln(x*x^2)=5[/tex]

Simplifying gives you

[tex]ln(x^3)=5[/tex]

Applying the power rule allows us to bring down the 3 in front of the ln:

3 * ln(x) = 5

Now we can divide both sides by 3 to get

[tex]ln(x)=\frac{5}{3}[/tex]

Take the inverse ln by raising each side to e:

[tex]e^{ln(x)}=e^{\frac{5}{3}}[/tex]

The "e" and the ln on the left undo each other, leaving you with just x; and raising e to the power or 5/3 gives you that

x = 5.29449005

For c. begin by dividing both sides by 20 to get:

[tex]\frac{1}{2}=e^{.1x}[/tex]

"Undo" that e by taking the ln of both sides:

[tex]ln(.5)=ln(e^{.1x})[/tex]

When the ln and the e undo each other on the right you're left with just .1x; on the left we have, from our calculators:

-.6931471806 = .1x

x = -6.931471806

Question d. is a bit more complicated than the others.  Begin by turning the base of 4 into a base of 2 so they are "like" in a sense:

[tex](2^2)^x-6(2)^x=-8[/tex]

Now we will bring over the -8 by adding:

[tex](2^2)^x-6(2)^x+8=0[/tex]

We can turn this into a quadratic of sorts and factor it, but we have to use a u substitution.  Let's let [tex]u=2^x[/tex]

When we do that, we can rewrite the polynomial as

[tex]u^2-6u+8=0[/tex]

This factors very nicely into u = 4 and u = 2

But don't forget the substitution that we made earlier to make this easy to factor.  Now we have to put it back in:

[tex]2^x=4,2^x=2[/tex]

For the first solution, we will change the base of 4 into a 2 again like we did in the beginning:

[tex]2^2=2^x[/tex]

Now that the bases are the same, we can say that

x = 2

For the second solution, we will raise the 2 on the right to a power of 1 to get:

[tex]2^x=2^1[/tex]

Now that the bases are the same, we can say that

x = 1