Which statement can be used to prove that a given parallelogram is a rectangle? A) The opposite sides of the parallelogram are congruent. B) The diagonals of the parallelogram are congruent. C) The diagonals of the parallelogram bisect the angles. D) The opposite angles of the parallelogram are congruent.

Answer:

Their intercepts are unique.

Explanation:

[tex]\displaystyle x^3 - 24x^2 + 192x - 508 = (x - 8)^3 + 4[/tex]

This graph's x-intercept is located at approximately [6,41259894, 0], and the y-intercept located at [0, −508].

[tex]\displaystyle g(x) = x^3[/tex]

The parent graph here, has both an x-intercept and y-intercept located at the origin.

I am joyous to assist you anytime.

Answer:

The dimensions of the garden that minimize the cost is 9.018 feet(length) and 13.528 feet(width)

Step-by-step explanation:

Let the length of garden be x

Let the breadth of garden be y

Area of Rectangular garden = [tex]Length \times Breadth = xy[/tex]

We are given that the area of the garden is 122 square feet

So, [tex]xy=122[/tex] ---A

A landscape architect wished to enclose a rectangular garden on one side by a brick wall costing $20/ft

So, cost of brick along length x = 20 x

On the other three sides by a metal fence costing $10/ft.

So, Other three side s = x+2y

So, cost of brick along the other three sides= 10(x+2y)

So, Total cost = 20x+10(x+2y)=20x+10x+20y=30x+20y

Total cost = 30x+20y

Substitute the value of y from A

Total cost = [tex]30x+20(\frac{122}{x})[/tex]

Total cost = [tex]\frac{2440}{x}+30x[/tex]

Now take the derivative to minimize the cost

[tex]f(x)=\frac{2440}{x}+30x[/tex]

[tex]f'(x)=-\frac{2440}{x^2}+30[/tex]

Equate it equal to 0

[tex]0=-\frac{2440}{x^2}+30[/tex]

[tex]\frac{2440}{x^2}=30[/tex]

[tex]\sqrt{\frac{2440}{30}}=x[/tex]

[tex]9.018 =x[/tex]

Now check whether it is minimum or not

take second derivative

[tex]f'(x)=-\frac{2440}{x^2}+30[/tex]

[tex]f''(x)=-(-2)\frac{2440}{x^3}[/tex]

Substitute the value of x

[tex]f''(x)=-(-2)\frac{2440}{(9.018)^3}[/tex]

[tex]f''(x)=6.6540[/tex]

Since it is positive ,So the x is minimum

Now find y

Substitute the value of x in A

[tex](9.018)y=122[/tex]

[tex]y=\frac{122}{9.018}[/tex]

[tex]y=13.528[/tex]

Hence the dimensions of the garden that minimize the cost is 9.018 feet(length) and 13.528 feet(width)

Answer:

d b f

Step-by-step explanation:

Answer:

Both ratios will increase where the accounts payable balance is paid off.

Step-by-step explanation:

The current ratio is given as

Current ratio = Current asset / current liabilities

Where the current assets are asset that can be converted into cash easily ( including cash and cash equivalents) while the current liabilities are liabilities to be settled in a short term, say 1 year.

Acid test ratio is given as

Acid test ratio = (Current asset -  Inventories) / current liabilities

Here, the current assets excludes the assets that are not so easily converted to cash.

From the two formulas stated above, where the accounts payable balance which is an element of the current liabilities is paid off, the current liabilities balance reduces thus resulting in an increase in both ratio.

Hence, current and the acid-test ratios will increase where the accounts payable balance is paid off.

Paying off accounts payable increases both the current and acid-test ratios, assuming they are originally greater than 1, indicating positive financial stability to investors and creditors.

The effect of paying off an accounts payable balance on both the current and the acid-test ratios, if they are greater than 1, would be an increase. The current ratio is calculated as current assets divided by current liabilities. When accounts payable (a current liability) is paid off, the denominator of the ratio decreases, leading to an increase in the ratio. Similarly, for the acid-test (or quick) ratio, once again we see a decrease in the denominator after pay off, leading to an increase in the ratio.

Although the net change in an entity’s financial position may seem neutral (decrease in an asset offset by a decrease in liabilities), these ratio increases can be viewed positively by investors and creditors as they imply a greater degree of short-term solvency.

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Answer:

[tex]6^{14}[/tex]

Step-by-step explanation:

Use the "Power Law" of exponents that tells us that when you have a base to a power "n" and all that raised to a power "m", it is the same as writing the original base to the single exponent which is the product of n time m:

[tex](b^n)^m=b^{n*m}[/tex]

therefore in your case, the base "b" is 6, the exponent "n" is 2, and the exponent "m" is 7. Then:

[tex](6^2)^7=6^{2*7}=6^{14}[/tex]

Final answer:

To simplify the expression [tex](6^2)^7[/tex] using a single exponent, multiply the inner exponent 2 by the outer exponent 7, which yields 6¹⁴.

Explanation:

To write the expression [tex](6^2)^7[/tex] using a single exponent, you need to apply the rule for raising a power to a power. This rule states that you multiply the exponents together. So for our expression, we have the base number 6 raised to the power of 2, and this result is then raised to the power of 7. To combine them into a single exponent, you multiply 2 by 7, which gives you 14.

Therefore,[tex](6^2)^7[/tex] can be simplified to 6¹⁴. This is because when you raise a power to another power, the powers are multiplied: for example, (a^b)^c = a^(b*c).

Answer:

a) Frequency table:

Category       Frequency

 Dog                     4

 Cat                       3

 Fish                      2

 Hamster               1

b) Relative frequencies of each animal type

c) Popularity

Explanation:

a. Make a frequency table for the results.

There are four kind of pets: dog, cat, hamster, and fish.

A frequency table shows the number of items for each category (kind of pets).

Count the number of each kind of pet:

With that you build your frequency table.

Frequency table:

Category       Frequency

 Dog                     4

 Cat                       3

 Fish                      2

 Hamster               1

b. Relative frequencies of each animal type.

The relative frequency is how often an outcome appears divided by the total number of outmcomes.

Here the total number of outcomes is 10 (the ten pets).

So, calculate each relative frequency:

An important feature of the relative frequency is that they must add up 1. Check:

c. Using the relative frequencies explain which animals are most and least popular.

Popularity is determined by the frequency with each outcome is repeated. The most popular is the most repeated. The least popular is the least repeated.

What is the question?

Answer:

The answer is in the explanation.

Step-by-step explanation:

450x2=900 167x2=334 Then you add those together and you get 1,234 then you do 234x2=468 156x2=312 then you add those two, and you get 780 then you add 1,234+780= 2014. For his brother you do 254x2=508 and 56x2=112 then you add them and you get 620. Then you subtract 2014-620= 1394. That is the final answer. Hope it helps!

Answer:

Area of Δ ABC = 21.86 units square

Perimeter of Δ ABC = 24.59 units

Step-by-step explanation:

Given:

In Δ ABC

∠A=45°

∠C=30°

Height of triangle = 4 units.

To find area and perimeter of triangle we need to find the sides of the triangle.

Naming the end point of altitude as 'D'

Given [tex]BD\perp AC[/tex]

For Δ ABD

Since its a right triangle with one angle 45°, it means it is a special 45-45-90 triangle.

The sides of 45-45-90 triangle is given as:

Leg1 [tex]=x[/tex]

Leg2 [tex]=x[/tex]

Hypotenuse [tex]=x\sqrt2[/tex]

where [tex]x[/tex] is any positive number

We are given BD(Leg 1)=4

∴ AD(Leg2)=4

∴ AB (hypotenuse) [tex]=4\sqrt2=5.66 [/tex]  

For Δ CBD

Since its a right triangle with one angle 30°, it means it is a special 30-60-90 triangle.

The sides of 30-60-90 triangle is given as:

Leg1(side opposite 30° angle) [tex]=x[/tex]

Leg2(side opposite 60° angle) [tex]=x\sqrt3[/tex]

Hypotenuse [tex]=2x[/tex]

where [tex]x[/tex] is any positive number

We are given BD(Leg 1)=4

∴ CD(Leg2) [tex]=4\sqrt3=6.93[/tex]

∴ BC (hypotenuse) [tex]=2\times 4=8 [/tex]  

Length of side AC is given as sum of segments AD and CD

[tex]AC=AD+CD=4+6.93=10.93[/tex]

Perimeter of Δ ABC= Sum of sides of triangle

⇒ AB+BC+AC

⇒ [tex]5.66+8+10.93[/tex]

⇒ [tex]24.59[/tex] units

Area of Δ ABC = [tex]\frac{1}{2}\times base\times height[/tex]

⇒  [tex]\frac{1}{2}\times 10.93\times 4[/tex]

⇒ [tex]21.86[/tex] units square

Option D

Function used to determine the profit the theater group earns from selling "x" tickets is f(x) = 12.50x - 150

Given that a theater group charges $12.50 per ticket for its opening play of the season

Also given that Production costs for the play are $150

To find: function used to determine the profit the theater group earns from selling "x" tickets

So "x" represents the number of tickets sold

Cost per ticket = $ 12.50

[tex]\text { cost of "x" tickets }=" x " \times \text { cost per ticket }[/tex]

[tex]\text { cost of "x" tickets }= x \times 12.50=12.50x[/tex]

Production costs = 150

Then the profit the theater group earns from selling "x" tickets is:

Profit earned from selling "x" tickets = cost of "x" tickets - production cost

Let f(x) denotes profit earned from selling "x" tickets

f(x) = 12.50x - 150

Thus option D is correct

Answer:

The pipe C alone can fill the tank in 14 hours .

Step-by-step explanation:

Given as :

The three pipes a , b , c can fill the pipes in 6 hours

They work for 2 hours

After that c pipe is close and a , b finish remaining work

Now, According to question

In 1 hour pipes ( a + b + c ) fill [tex]\frac{1}{6}[/tex] of the tank

∴ In 2 hour pipes ( a + b + c ) fill [tex]\frac{2}{6}[/tex] =  [tex]\frac{1}{3}[/tex] of the tank

Remaining ( 1 - [tex]\frac{1}{3}[/tex]  ) = [tex]\frac{2}{3}[/tex]  part is filled by pipes a and b in 7 hours

∴ The whole tank is filled by a and b in 7 × [tex]\frac{3}{2}[/tex] =  [tex]\frac{21}{2}[/tex] hours

∴ In 1 hour pipes A and b fill the tank in [tex]\frac{2}{21}[/tex] hours

∴ In 1 hour pipes C alone can fill the tank in[tex]\frac{1}{6}[/tex] - [tex]\frac{2}{21}[/tex] hours

Or,  In 1 hour pipes C alone can fill the tank in [tex]\frac{9}{126}[/tex] =  [tex]\frac{1}{14}[/tex]

Or, In 1 hour pipes C alone can fill the tank in 14 hours

Hence The pipe C alone can fill the tank in 14 hours . Answer

Answer:

475 billion dollars

Step-by-step explanation:

Let P be the customer credit

At the end of year X, 36% of P is gotten from automobile installment credit

57 billion credit is 1/3 of the automobile installment credit. This means that the total automobile installment credit 57*3 = 171 billion dollars

36% * P = 171

36/100 * P = 171

36P = 171 *100

P = 17100/36

P= 475 billion dollars

Answer: [tex]S_n=5(1-\dfrac{1}{n+1})[/tex] ; 5

Step-by-step explanation:

Given series : [tex][\dfrac{5}{1\cdot2}]+[\dfrac{5}{2\cdot3}]+[\dfrac{5}{3\cdot4}]+....+[\dfrac{5}{n\cdot(n+1)}][/tex]

Sum of series = [tex]S_n=\sum^{\infty}_{1}\ [\dfrac{5}{n\cdot(n+1)}]=5[\sum^{\infty}_{1}\dfrac{1}{n\cdot(n+1)}][/tex]

Consider [tex]\dfrac{1}{n\cdot(n+1)}=\dfrac{n+1-n}{n(n+1)}[/tex]

[tex]=\dfrac{1}{n}-\dfrac{1}{n+1}[/tex]

⇒ [tex]S_n=5\sum^{\infty}_{1}\dfrac{1}{n\cdot(n+1)}=5\sum^{\infty}_{1}[\dfrac{1}{n}-\dfrac{1}{n+1}][/tex]

Put values of n= 1,2,3,4,5,.....n

⇒ [tex]S_n=5(\dfrac{1}{1}-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+......-\dfrac{1}{n}+\dfrac{1}{n}-\dfrac{1}{n+1})[/tex]

All terms get cancel but First and last terms left behind.

⇒ [tex]S_n=5(1-\dfrac{1}{n+1})[/tex]

Formula for the nth partial sum of the series :

[tex]S_n=5(1-\dfrac{1}{n+1})[/tex]

Also, [tex]\lim_{n \to \infty} S_n = 5(1-\dfrac{1}{n+1})[/tex]

[tex]=5(1-\dfrac{1}{\infty})\\\\=5(1-0)=5[/tex]

Answer:

Open circle to the right of 4

x > 4

Step-by-step explanation:

Add 7 to both sides

2x > 8

x > 4

Open circle to the right of 4

Answer:

P = 68/552 = 0.123 or 12.3%

Step-by-step explanation:

First, let's calculate the probability of getting a lemon donut. We have only 4 lemon donut among 24 donuts, so probability is:

P(A) = 4/24

Next, as we already ate the lemon donut, we only have 23 donuts now, and among these 23, 17 are custard filled, so probability of choosing one of those is:

P(B) = 17/23

But we want to know the probability that the custard filled donut is choosen after you eat the lemon one so:

P(B|A) = P(A) * P(B)

Replacing:

P(B|A) = 4/24 * 17/23

P(B|A) = 68/552 = 0.123 or 12.3%

Answer:

$824

Step-by-step explanation:

3% (rate of interest) of 800= 24

800 + 24= 824

The maximum area of the given rectangle will be A = 4(C/3)√(C/3).

The quantity of space enclosing a three-dimensional shape's exterior is its surface area.

In other meaning, if we say side square then it is an area of the square but for a cuboid, there are 6 faces so the surface area will be external to all 6 surfaces area.

As per the given rectangle inscribed in the parabola has been drawn,

The area of rectangle A = (x + x)y

A = 2x(C - x²)

A = 2Cx - 2x³

To find the maximum area, take the first derivative with respect to x.

dA/dx = 2C - 6x² = 0

C - 3x² = 0

x = √(C/3)

Therefore, the area will be as,

A = 2C√(C/3) - 2(√(C/3))³

A = 2C√(C/3) - 2(C/3)√(C/3)

A = 2√(C/3) [C - C/3]

A = 2√(C/3)(2C/3)

A = 4(C/3)√(C/3)

Hence "The maximum area of the given rectangle will be A = 4(C/3)√(C/3)".

To learn more about surface area,

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Answer:

-12

Step-by-step explanation:

Answer:

The first One

Step-by-step explanation:

I just got 100% on My Quiz

Answer:it will take the two plants 6 weeks before the heights are the same

Step-by-step explanation:

Jill planted two flowers in her garden.

The first flower is 2 inches tall, and it is growing 2.25 inches each week. Since the growth rate is in an arithmetic progression, we will apply the formula for finding the nth term of the series

Tn = a + (n - 1)d

Tn = the nth height of the first flower

a = the initial height of the first flower

d = the common difference in height of the first flower weekly

n = number of weeks

From the information given,

For the first flower,

a = 2

d = 2.25

Tn ?

n ?

Tn = 2 + (n - 1)2.25

For the second flower,

a = 5.75

d = 1.5

Tn ?

n ?

Tn = 5.75 + (n - 1)1.5

To determine the number of weeks that it will take until the two plants are the same height, we would equate Tn for both flowers. It becomes

2 + (n - 1)2.25 = 5.75 + (n - 1)1.5

2 + 2.25n - 2.25 = 5.75 + 1.5n - 1.5

Collecting like terms

2.25n - 1.5n = 5.75 - 1.5 - 2 + 2.25

0.75n = 4.5

n = 4.5/0.75

n = 6 weeks

To find out how many weeks it will be until the two plants are the same height, set up an equation and solve for x. The plants will be the same height after 5 weeks.

To find out how many weeks it will be until the two plants are the same height, we need to set up an equation. Let the number of weeks be represented by x. The height of the first plant can be represented as 2 + 2.25x, and the height of the second plant can be represented as 5.75 + 1.5x. Set these two expressions equal to each other: 2 + 2.25x = 5.75 + 1.5x.

To solve for x, subtract 1.5x from both sides: 2 + 0.75x = 5.75.

Then, subtract 2 from both sides: 0.75x = 3.75.

Finally, divide both sides by 0.75 to solve for x: x = 5.

Answer:

[tex]\sin \theta = \dfrac{5}{13}[/tex] and [tex]\sec \theta = -\dfrac{13}{12}[/tex]

Step-by-step explanation:

Assume that the terminal side of thetaθ passes through the point (−12,5).

In ordered pair (-12,5), x-intercept is negative and y-intercept is positive. It means the point lies in 2nd quadrant.

Using Pythagoras theorem:

[tex]hypotenuse^2=perpendicular^2+base^2[/tex]

[tex]hypotenuse^2=(5)^2+(12)^2[/tex]

[tex]hypotenuse^2=25+144[/tex]

[tex]hypotenuse^2=169[/tex]

Taking square root on both sides.

[tex]hypotenuse=13[/tex]

In a right angled triangle

[tex]\sin \theta = \dfrac{opposite}{hypotenuse}[/tex]

[tex]\sin \theta = \dfrac{5}{13}[/tex]

[tex]\sec \theta = \dfrac{hypotenuse}{adjacent}[/tex]

[tex]\sec \theta = \dfrac{13}{12}[/tex]

In second quadrant only sine and cosecant are positive.

[tex]\sin \theta = \dfrac{5}{13}[/tex] and [tex]\sec \theta = -\dfrac{13}{12}[/tex]

Answer:

x4 + 12x^3 – 45x^2

Step-by-step explanation:

Answer:

Each container contains [tex]2.5 \ pounds \ \ OR \ \ \frac{11}{4} \ pounds \ \ OR \ \ 2 \frac{3}{4} \ pounds[/tex] of cat food.

Step-by-step explanation:

Given:

Amount of cat food = 8 1\4 pounds.

Also 8 1\4 can be rewritten as 8.25 pounds

Number of containers =3

We need to find the amount of cat food in each container.

Amount of cat food in each container can be calculated by Dividing Amount of cat food she has with number of Containers.

Amount of cat food in each container = [tex]\frac{8.25}{3}= 2.75 \ pounds[/tex]

2.75 pounds ca be rewritten as [tex]\frac{11}{4} \ pounds \ \ OR \ \ 2 \frac{3}{4} \ pounds[/tex]

Hence Each container contains [tex]2.5 \ pounds \ \ OR \ \ \frac{11}{4} \ pounds \ \ OR \ \ 2 \frac{3}{4} \ pounds[/tex] of cat food.

Answer:

  see below

Step-by-step explanation:

The signs on the bottom line alternate, so the value of k is, indeed, a lower bound.

_____

Comment on lower bound for this cubic

The signs of the coefficients alternate, so Descartes' rule of signs will tell you there are zero negative real roots. That is, 0 is a lower bound for real roots. No synthetic division is needed.

Answer: 5.1667 gallons of coffee had been dispensed

Step-by-step explanation:

Let x represent the number of gallons of coffee that had been dispensed.

By the end of each day, the coffee urn, which had started out with 7 1/4 gallons of coffee, was left with 2 1/12 gallons. This means that the initial number of gallons of coffee was 7 1/4 = 7.25 gallons

The amount left after dispense x gallons is 2 1/12 = 2.0833 gallons.

Therefore,

x +2.0833 = 7.25

x = 7.25 - 2.0833 = 5.1667 gallons

Answer:

Jim's current weight = 119 pounds

Step-by-step explanation:

1 Stone = 14 pounds

[tex]1\frac{1}{2}[/tex] stone = 1.5 stone

1.5 stone = 1.5 (14 pounds) = 21 pounds

Jim lost 21 pounds

Let X be Jim's Original Weight

Y be his present weight

As per given statement in the Question:

After losing 1.5 stones (21 pounds of weight) Jim now weighs Y

Present weight = original weight - 21

Y = X -21                                                        Equation 1

Also Current Weight = 85 % (Original weight)

Y = 85 % (X) =[tex]\frac{85X}{100}[/tex]

Y=[tex]\frac{85X}{100}[/tex]

put in Equation 1

[tex]\frac{85X}{100}[/tex] = X-21

85X = (X-21) 100

85 X = 100 X -2100

or

2100 = 100 X - 85X

2100 = 15X

or

15 X = 2100

[tex]X=\frac{2100}{15}[/tex]

X= 140 pounds ( Original Weight)

Current Weight = Y = Original weight - 21 (From Equation 1)

Y = X -21

Y = 140 -21

Y = 119 pounds (Current Weight)

Trigonometry can be used to model projectile motion, such as the flight of a baseball. Given the angle at which the ball leaves the bat and the initial velocity, you can determine the distance the ball will travel.  

Answer:

Trigonometry can be used to model projectile motion, such as the flight of a baseball. Given the angle at which the ball leaves the bat and the initial velocity, you can determine the distance the ball will travel.

Step-by-step explanation:

The inequality is: 5>x≥0

Step-by-step explanation:

We have to write the inequality one by one

The inequality symbols are used to write inequalities.

So,

5 is greater than x will be written as:

5>x

And

0 is less than or equals to x:

0≤x or x≥0

We have to combine the both inequalities so that the variable is not repeated. The inequality symbols have to be written carefully while writing the compound inequalities.

5>x≥0

Keywords: Inequality, Relationships

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Answer: option B is the correct answer

Step-by-step explanation:

The diagram of the tree is shown in the attached photo. The triangle ABC formed is not a right angle triangle. The last angle, angle C is gotten by subtracting the sum of angle A and angle B from 180(sum of angles) in a triangle is 180). It becomes

C = 180-(98+24)= 180 -122

C = 58 degrees

To find the height of the tree, we would apply the sine rule

a/sinA = b/sin B = c/ sinC

We would apply b/sin B = c/ sinC

b/sin24 = 27/sin58

b/0.4067 = 27/0.8480

Cross-multiplying,

27 ×0.4067 = b × 0.8480

10.9809 = 0.8480b

b = 10.9809/0.8480 = 12.949

Approximately 13 meters

The bug crawls 13 meters from the base to the top of the tree

Answer:

12.5 cm

Step-by-step explanation:

16 - 9 = 7

7 (1/2) = 3.5

9 + 3.5 = 12.5

15 - 3.5 = 12.5

Answer:

1740%.

Step-by-step explanation:

We have been given that the pawnbroker loans you ​$600. One month​ later, you get the guitar back by paying the pawnbroker ​$1470.

We will use simple interest formula to solve our given problem.

[tex]A=P(1+rt)[/tex], where,

A = Final amount after t years,

P = Principal amount,

r = Annual interest rate in decimal form,

t = Time in years.

1 month = 1/12 year

[tex]1470=600(1+r*\frac{1}{12})[/tex]

[tex]1470=600+\frac{600}{12}*r[/tex]

[tex]1470=600+50*r[/tex]

[tex]1470-600=600-600+50*r[/tex]

[tex]870=50*r[/tex]

[tex]50r=870=[/tex]

[tex]\frac{50r}{50}=\frac{870}{50}[/tex]

[tex]r=17.4[/tex]

Since our interest rate is in decimal form, so we will convert it into percentage by multiplying by 100 as:

[tex]17.4\times 100\%=1740\%[/tex]

Therefore, you paid an annual interest rate of 1740%.

Answer:

[tex]P(X>8)=e^{-1}[/tex]

Step-by-step explanation:

The exponential distribution is "the probability distribution of the time between events in a Poisson process (a process in which events occur continuously and independently at a constant average rate). It is a particular case of the gamma distribution". The probability density function is given by:

[tex]P(X=x)=\lambda e^{-\lambda x}, x>0[/tex]

And 0 for other case. Let X the random variable that represent "The number of years a radio functions" and we know that the distribution is given by:

[tex]X \sim Exp(\lambda=\frac{1}{8})[/tex]

We can assume that the random variable t represent the number of years that the radio is already here. So the interest is find this probability:

[tex]P(X>8|X>t)[/tex]

We have an important property on the exponential distribution called "Memoryless" property and says this:

[tex]P(X>a+t| X>t)=P(X>a)[/tex]

Where a represent a shift and t the time of interest.

On this case then [tex]P(X>8|X>t)=P(X>8+t|X>t)=P(X>8)[/tex]

We can use the definition of the density function and find this probability:

[tex]P(X>8)=\int_{8}^{\infty} \frac{1}{8}e^{-\frac{1}{8}x}dx[/tex]

[tex]=\frac{1}{8} \int_{8}^{\infty} e^{-\frac{1}{8}x}dx[/tex]

[tex]=[lim_{x\to\infty} (-e^{-\frac{1}{8}x})+e^{-1}]=0+e^{-1}=e^{-1}[/tex]