State the domain of f(a,b) = e^ab

Answer:

[tex]y = 10cos (7x) - \frac{4}{7}sin ( 7x )[/tex]

B.

B.

[tex]y = \frac{17}{8}e^4^x - \frac{1}{8}e^-^4^x[/tex]

Step-by-step explanation:

Question 1:

- We are given a homogeneous second order linear ODE as follows:

                                [tex]y'' + 49y = 0[/tex]

- A pair of independent functions are given as ( y1 ) and ( y2 ):

                               [tex]y_1 = cos ( 7x )\\\\y_2 = sin ( 7x )[/tex]

- The given ODE is subjected to following initial conditions as follows:

                               [tex]y ( 0 ) = 10\\\\y ' ( 0 ) = -4[/tex]

- We are to verify that the given independent functions ( y1 ) and ( y2 ) are indeed the solution to the given ODE. If the functions are solutions then find the complete solution of the homogeneous ODE of the form:

                               [tex]y = c_1y_1 + c_2y_2[/tex]

Solution:-

- To verify the functions are indeed the solution to the given ODE. We will plug the respective derivatives of each function [ y1 and y2 ] into the ODE and prove whether the equality holds true or not.

- Formulate the second derivatives of both functions y1 and y2 as follows:

                           [tex]y'_1 = -7sin(7x) , y''_1 = -49cos(7x)\\\\y'_2 = -7cos(7x) , y''_2 = -49sin(7x)\[/tex]

- Now plug the second derivatives of each function and the functions itself into the given ODE and verify whether the equality holds true or not.

                           [tex]y''_1 + 49y_1 = 0\\\\-49cos(7x) + 49cos ( 7x ) = 0\\0 = 0\\\\y''_2 + 49y_2 = 0\\\\-49sin(7x) + 49sin ( 7x ) = 0\\0 = 0\\\\[/tex]

- We see that both functions [ y1 and y2 ] holds true as the solution to the given homogeneous second order linear ODE. Hence, are the solution to given ODE.

- The complete solution to a homogeneous ODE is given in the form as follows:

                            [tex]y = c_1y_1 + c_2y_2\\\\y = c_1*cos(7x) + c_2*sin(7x)\\[/tex]

- To complete the above solution we need to determine the constants [ c1 and c2 ] using the initial conditions given. Therefore,

                           [tex]y (0) = c_1cos ( 0 ) + c_2sin ( 0 ) = 10\\\\y'(0) = -7c_1*sin(0) + 7c_2*cos(0) = -4\\\\c_1 ( 1 ) + c_2 ( 0 ) = 10, c_1 = 10\\\\-7c_1(0) + 7c_2( 1 ) = -4 , c_2 = -\frac{4}{7}[/tex]

- Now we can write the complete solution to the given homogeneous second order linear ODE as follows:

                           [tex]y = 10cos (7x) - \frac{4}{7}sin ( 7x )[/tex]    .... Answer

Question 2

- We are given a homogeneous second order linear ODE as follows:

                               [tex]y'' -16y =0[/tex]

- A pair of independent functions are given as ( y1 ) and ( y2 ):

                               [tex]y_1 = e^4^x\\\\y_2 = e^-^4^x[/tex]

- The given ODE is subjected to following initial conditions as follows:

                               [tex]y( 0 ) = 2\\y'( 0 ) = 9[/tex]

- We are to verify that the given independent functions ( y1 ) and ( y2 ) are indeed the solution to the given ODE. If the functions are solutions then find the complete solution of the homogeneous ODE of the form:

                               [tex]y = c_1y_1 + c_2y_2[/tex]

Solution:-

- To verify the functions are indeed the solution to the given ODE. We will plug the respective derivatives of each function [ y1 and y2 ] into the ODE and prove whether the equality holds true or not.

- Formulate the second derivatives of both functions y1 and y2 as follows:

                         [tex]y'_1 = 4e^4^x , y''_1 = 16e^4^x\\\\y'_2 = -4e^-^4^x , y''_2 = 16e^-^4^x[/tex]  

- Now substitute the second derivatives of each function and the functions itself into the given ODE and verify whether the equality holds true or not.

                           [tex]y''_1 - 16y_1 = 0\\\\16e^4^x - 16e^4^x = 0\\\\0 = 0\\\\y''_2 - 16y_2 = 0\\\\16e^-^4^x - 16e^-^4^x = 0\\\\0 = 0[/tex]

- We see that both functions [ y1 and y2 ] holds true as the solution to the given homogeneous second order linear ODE. Hence, are the solution to given ODE.

- The complete solution to a homogeneous ODE is given in the form as follows:

                            [tex]y = c_1y_1 + c_2y_2\\\\y = c_1*e^4^x + c_2*e^-^4^x[/tex]

- To complete the above solution we need to determine the constants [ c1 and c2 ] using the initial conditions given. Therefore,

                           [tex]y ( 0 ) = c_1 * e^0 + c_2 * e^0 = 2\\\\y' ( 0 ) = 4 c_1 * e^0 - 4c_2 * e^0 = 9\\\\c_1 + c_2 = 2 , 4c_1 - 4c_2 = 9\\\\c_1 = \frac{17}{8} , c_2 = -\frac{1}{8}[/tex]

- Now we can write the complete solution to the given homogeneous second order linear ODE as follows:

                           [tex]y = \frac{17}{8} e^4^x - \frac{1}{8}e^-^4^x[/tex]   .... Answer

Answer:

[tex] = \frac{3}{2} [/tex]

Step-by-step explanation:

[tex]y = mx + c[/tex]

Here,

In this equation ,

[tex]2y - 3x = 8[/tex]

to find the value of m or the value of slope we have to solve for y

Let's solve,

[tex]2y - 3x = 8 \\ 2y = 8 + 3x \\ \frac{2y}{2} = \frac{8 + 3x}{2} \\ y = 4 + \frac{3x}{2} \\ y = \frac{3x}{2} + 4[/tex]

So, the slope is,

[tex] = \frac{3}{2}[/tex]

Answer:

13 students

Step-by-step explanation:

At least 30 and fewer than 67 makes it 30-66

So,

30-66 => 13 students

Answer:

16

Step-by-step explanation:

There are two columns in the diagram.

The column headed stem represents tens while the column headed leaf represents units. e.g. 2 3 = 23

So we just have to count how many of the numbers are less than 8 in the 6th Stem column and all the numbers below it, which are:

20, 23, 28, 31, 31, 34, 38, 40, 44, 50, 51, 53, 54, 65, 65, 66

Answer:

1 : 500,000

Step-by-step explanation:

The scale of a map scale refers to the relationship (or ratio) between the distance on a map and the corresponding distance on the ground.

In the given map:

1 mm​(map) = 500 m ​(actual)

1 meter = 1000 millimeter

Therefore:

500 meters = 1000 X 500 =500,000 millimeter

Therefore, the scale ratio of the map is:

1:500,000

Answer:

x=2√2 is the answer

Step-by-step explanation:

x²=8

TAKING SQUARE ROOT ON BOTH SIDES

√x²=√8

x=√2×2×2

x=√2²×√2

x=2√2

i hope this will help you

Answer:

The value of x is -2.828 or 2.828

Step-by-step explanation:

In order to eliminate of square of x, you have to square root both sides :

[tex] {x}^{2} = 8[/tex]

[tex] \sqrt{ {x}^{2} } = ± \sqrt{8} [/tex]

[tex]x = \sqrt{8} \\ x = 2 \sqrt{2} \: or \: 2.828[/tex]

[tex]x = - \sqrt{8} \\ x = - 2 \sqrt{2} \: or \: - 2.828[/tex]

Answer:

[tex]\boxed{\sf \ \ 9x^2-18x-7 \ \ divided \ by \ (3x+1) \ is \ (3x-7) \ }[/tex]

Step-by-step explanation:

Hello,

let's find a and b reals so that

[tex]9x^2-18x-7=(3x+1)(ax+b)[/tex]

[tex](3x+1)(ax+b)=3ax^2+(3b+a)x+b[/tex]

we identify the terms in [tex]x^2[/tex]

   9 = 3a

we identify the terms in x

   -18 = 3b + a

we identify the constant terms

   -7 = b

so ti goes with a = 9/3 = 3, b = -7

so we can write

[tex]9x^2-18x-7=(3x+1)(3x-7)[/tex]

so [tex]9x^2-18x-7 \ divided \ by \ (3x+1) \ is \ (3x-7)[/tex]

hope this helps

Answer:

D. No, because the differential equation does not have constant coefficients.

Step-by-step explanation:

The undetermined coefficient method cannot be applied to non homogeneous variables. The differential equation does not have constant variables therefore the method of undetermined superposition can not be applied. To complete a solution of non homogeneous equation the particular solution must be added to the homogeneous equation.

Answer:

(x-1) ( x -i) (x+i)

Step-by-step explanation:

x^3 -2x^2 +x-2

Factor by grouping

x^3 -2x^2      +x-2

x^2(x-2)      +1(x-2)

Factor out (x-2)

(x-2) (x^2+1)

Rewriting

(x-1) ( x^2 - (-1)^2)

(x-1) ( x -i) (x+i)

Answer:

Should be b

Step-by-step explanation:

Since it's a multiple choice question you know that -2 or 2 has to be a root for the cubic.

You can test both -2 and 2 and see that replacing x for 2 has the expression evaluate to 0.

Then, since you know the imaginary roots have to be conjugates, you get B.

Answer: 150

Step-by-step explanation:

How many months are in a year? 12.

The average rate per month is therefore 1800/12 = 150.

Hope that helped,

-sirswagger21

Answer:

The diagram of the question is missing, I found a matching diagram, and it is attached to this answer

The 3D object produced is a cone with height 8 and diameter 8 (radius 4)

Step-by-step explanation:

A 3 dimensional solid figure can be formed when a 2 dimensional object is rotated about a line without displacing the object.

when the object in the diagram is rotated about line m, the rotation forms an object with a circular base of diameter 8 units (radius 4) from the base of the triangle and height 8 units, and the 3D object formed is called a cone.

Answer:

50,000

Step-by-step explanation:

ten thousand  thousand  hundreds   tens ones

4                            7                 5            9       2

When rounding to the ten thousands, we look at the thousands place

If it is 5 or higher we round the ten thousands place up

7 is five or higher so we round the 4 up one place  4 becomes 5 and the rest becomes 0

5 0 0 0 0

Answer:

$50,000

Step-by-step explanation:

=> $47,592

While rounding off to the nearest thousand, we check the thousands place. If the digit in the thousands place is greater than 5, 1 will be added to the T. Th. place while if its less than 5, there will be no change and The digits except the ten thousands place will all become zero.

So,

=> $50,000

Answer:

Read below

Step-by-step explanation:

To copy a segment, you have to open your compass to the length of the given segment. The instructions say to have an endpoint at R, so, with the compass open to the length of the given line segment, place one end of the compass at R and draw an arc that intersects the line that R lies on. This new segment is congruent to the given segment.

I hope this helps!

Answer:

3.784

Step-by-step explanation:

Answer:

The probability that the sample mean is less than 50 points = 0.002    

Step-by-step explanation:

Step(i):-

Given mean of the normal distribution = 56 points

Given standard deviation of the normal distribution = 12 points

Random sample size 'n' = 36 games

Step(ii):-

Let x⁻ be the random variable of normal distribution

Let x⁻ = 50

[tex]Z = \frac{x^{-}-mean }{\frac{S.D}{\sqrt{n} } }[/tex]

[tex]Z = \frac{50-56 }{\frac{12}{\sqrt{36} } }= -3[/tex]

The probability that the sample mean is less than 50 points

P( x⁻≤ 50) = P( Z≤-3)

                = 0.5 - P(-3 <z<0)

               = 0.5 -P(0<z<3)

               =  0.5 - 0.498

               = 0.002

Final answer:-

The probability that the sample mean is less than 50 points = 0.002

Answer:

56

2

.001

Step-by-step explanation:

The Central Limit Theorem for Means states that the mean of any sampling distribution of the means is equal to the mean of the population distribution. The standard deviation is equal to the standard deviation of the population divided by the square root of the sample size. So, the mean of this sampling distribution of the means with sample size 36 is 56 points and the standard deviation is 1236√=2 points. The z-score for 50 using the formula z=x¯¯¯−μσ is −3.

z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

-3.0 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001

-2.9 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.001 0.001 0.001

-2.8 0.003 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002

-2.7 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003

-2.6 0.005 0.005 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004

-2.5 0.006 0.006 0.006 0.006 0.006 0.005 0.005 0.005 0.005 0.005

Using the Standard Normal Table, the area to the left of −3 is approximately 0.001. Therefore, the probability that the sample mean will be less than 50 points is approximately 0.001.

Question Completion

PIE CHART NUMBERS:

Answer:

0.000063

Step-by-step explanation:

Number of Respondents, n=1400

Probability that they would rate their financial shape as​ excellent = 0.09

Number of Those who would rate their financial shape as excellent

=0.09 X 1400

=126

Therefore:

The probability that 4 people chosen at random would rate their financial shape as​ excellent

[tex]=\dfrac{^{126}C_4 \times ^{1400-126}C_0}{^{1400}C_4} \\=\dfrac{^{126}C_4 \times ^{1274}C_0}{^{1400}C_4}\\=0.000063 $(correct to 6 decimal places)[/tex]

Answer:

12,000

Step-by-step explanation:

The machine fills the containers at a rate of 50+5t milliliters (mL) per second.

Therefore, the rate of change of the number of containers, N is:

[tex]\dfrac{dN}{dt}=50+5t, 0\leq t\leq 60[/tex]

[tex]dN=(50+5t)dt\\$Taking integrals of both sides\\\int dN=\int (50+5t)dt\\N(t)=50t+\frac{5t^2}{2}+C $(C a constant of integration)\\\\When t=0, , No containers are filled, therefore:$ N(t)=0\\0=50(0)+\frac{5(0)^2}{2}+C\\C=0\\$Therefore, N(t)=50t+2.5t^2[/tex]

When t=60 seconds

[tex]N(60)=50(60)+2.5(60)^2\\N(60)=12000$ mL[/tex]

Therefore, 12,000 milliliters of acid solution are put into a container in 60 seconds.

Answer:

1/2

Step-by-step explanation:

There is a 50/50 chance for the coin to land either heads or tails. Convert that to probability and it is 1/2.

The only time a coin would not be 50/50 chance is if the coin is weighted.

Answer:

1/2 is probability

becoz one side is head or one is tail

Answer:

1:6, 7:36, 2:9

Step-by-step explanation:

2 : 9 → 8 : 36

1 : 6 → 6 : 36

7 : 36

Least → Greatest

1:6, 7:36, 2:9

Answer:

[tex]77.7-1.66\frac{3.6}{\sqrt{100}} =77.102[/tex]    

[tex]77.7+1.66\frac{3.6}{\sqrt{100}} =78.30[/tex]    

And the best option would be:

C. (77.1, 78.3)

Step-by-step explanation:

Information given

[tex]\bar X=77.7[/tex] represent the sample mean

[tex]\mu[/tex] population mean (variable of interest)

s=3.6 represent the sample standard deviation

n=100 represent the sample size  

Confidence interval

The confidence interval for the mean is given by the following formula:

[tex]\bar X \pm t_{\alpha/2}\frac{s}{\sqrt{n}}[/tex]   (1)

The degrees of freedom are given by:

[tex]df=n-1=100-1=99[/tex]

Since the Confidence is 0.90 or 90%, the significance would be [tex]\alpha=0.1[/tex] and [tex]\alpha/2 =0.05[/tex], and the critical value for this case would be [tex]t_{\alpha/2}=1.66[/tex]

And replacing we got:

[tex]77.7-1.66\frac{3.6}{\sqrt{100}} =77.10[/tex]    

[tex]77.7+1.66\frac{3.6}{\sqrt{100}} =78.30[/tex]    

And the best option would be:

C. (77.1, 78.3)

Answer:

a) Percentage of standardized test takers that scored 550 or less = 76.4%

b) Percentage of standardized test takers that scored 524 = 0.782%

Step-by-step explanation:

This is a normal distribution problem with

Mean = μ = 514

Standard deviation = σ = 50

a) Percentage of standardized test takers scored 550 or less = P(x ≤ 550)

We first normalize or standardize 550

The standardized score for any is the value minus the mean then divided by the standard deviation.

z = (x - μ)/σ = (550 - 514)/50 = 0.72

To determine the required probability

P(x ≤ 550) = P(z ≤ 0.72)

We'll use data from the normal distribution table for these probabilities

P(x ≤ 550) = P(z ≤ 0.72) = 0.76424 = 76.424%

The normal curve for this question and the b part are sketched in the first attached image to this solution.

b) Percentage of standardized test takers that scored 524 = P(x = 524)

On standardizing,

z = (x - μ)/σ = (524 - 514)/50 = 0.20

For this part, since it's an exact probability, we will use the normal distribution formula

P(z = Z) = [1/(σ√2π)] × e^(-z²/2)

Since z = (x - μ)/σ

It can be written properly as presented in the second attached image to this question.

Putting x = 524 or z = 0.20 in this expression, we get

P(x = 524) = P(z = 0.20) = 0.0078208539 = 0.782%

Hope this Helps!!!

Answer:

Step-by-step explanation:

Let the number be x.

The reciprocal of the number will be 1/x

If the sum of the number and its reciprocal is 41/20, this can be represented as;

[tex]x+\frac{1}{x} = 41/20\\\frac{x^{2}+1}{x} = \frac{41}{20} \\20x^{2} +20 = 41x\\20x^{2} -41x+20 = 0\\[/tex]

Uisng the general formula to get x

x = -b±√b²+4ac/2a

x = 41±√41²-4(20)(20)/2(20)

x = 41±√1681-1600/40

x = 41±√81/40

x = 41±9/40

x = 50/40 or 32/40

x = 5/4 or 4/5

if the value is 5/4, the other value will be 4/5

The numbers are 5/4 and 4/5

The smaller value is 4/5

The larger value is 5/4

Answer:

a=5/4 or 4/5

Therefore, the smaller value = 4/5

The larger value = 5/4

Step-by-step explanation:

Let the number be represented by a

And it's reciprocal be represented by 1/a

So we have

a + 1/a = 41/20

Cross Multiply

20( a +1/a) = 41

20a +20/a =41

Find the LCM which is a

20a² + 20 = 41a

20a² + 20 - 41a =0

20a² - 41a +20 = 0

20a²-25a - 16a + 20 =0

5a(4a - 5) -4( 4a - 5) = 0

(5a - 4)(4a - 5) = 0

5a - 4 = 0

5a = 4

a = 4/5

or

4a - 5 = 0

4a =5

a = 5/4

Therefore, the number which is represented by a is

1) a = 4/5 while it's reciprocal which is 1/a is 5/4

or

2) a = 5/4 which it's reciprocal which is 1/a = 4/5

Therefore, the smaller value = 4/5

The larger value = 5/4

Answer:

[tex]157.3-1.96\frac{15.6}{\sqrt{234}}=155.301[/tex]    

[tex]157.3+1.96\frac{15.6}{\sqrt{234}}=159.299[/tex]    

So on this case the 95% confidence interval would be given by (155.301;159.299)    

And the best option would be:

B. (155.3, 159,3)

Step-by-step explanation:

Information given

[tex]\bar X=157.3[/tex] represent the sample mean

[tex]\mu[/tex] population mean (variable of interest)

[tex]\sigma =15.6[/tex] represent the population standard deviation

n=234 represent the sample size  

Confidence interval

The confidence interval for the mean is given by the following formula:

[tex]\bar X \pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}[/tex]   (1)

The Confidence level is is 0.95 or 95%, the significance is [tex]\alpha=0.05[/tex] and [tex]\alpha/2 =0.025[/tex], the critical value for this case would be [tex]z_{\alpha/2}=1.96[/tex]

And replacing we got:

[tex]157.3-1.96\frac{15.6}{\sqrt{234}}=155.301[/tex]    

[tex]157.3+1.96\frac{15.6}{\sqrt{234}}=159.299[/tex]    

So on this case the 95% confidence interval would be given by (155.301;159.299)    

And the best option would be:

B. (155.3, 159,3)

Answer:

x2 = 60

Step-by-step explanation:

If the variables x and y are inversely proportional, the product x * y is a constant.

So using x1 and y1 we can find the value of this constant:

[tex]x1 * y1 = k[/tex]

[tex]12 * 15 = k[/tex]

[tex]k = 180[/tex]

Now, we can use the same constant to find x2:

[tex]x2 * y2 = k[/tex]

[tex]x2 * 3 = 180[/tex]

[tex]x2 = 180 / 3 = 60[/tex]

So the value of x2 is 60.

Answer:

the step-by-step explanation for height first :

[tex]h=\sqrt{h^{2} } +r^{2} =26[/tex]

[tex]h=\sqrt{h^{2} } +10^{2} =676[/tex]

[tex]h=\sqrt{h^{2} } + 100 = 676[/tex]

[tex]100-100 = 0[/tex]

[tex]676-100=576[/tex]

[tex]\sqrt{576}[/tex]

[tex]height =[/tex] 24 m

________________

step-by-step explanation for the problem :

FORMULA :  [tex]v = \frac{1}{3}[/tex] · [tex]\pi[/tex] · [tex]r^{2}[/tex] · [tex]h[/tex]

v = [tex]\frac{1}{3}[/tex] · [tex]\pi[/tex] · [tex]10^{2}[/tex] · [tex]24[/tex] = [tex]800\pi[/tex] = [tex]2513.27412[/tex] = 2513

Answer:

196x^2y

Step-by-step explanation: The least common multiple (LCM) of two or more non-zero whole numbers is the smallest whole number that is divisible by each of those numbers. In other words, the LCM is the smallest number that all of the numbers divide into evenly.

Answer:

The polynomial is [tex]x^{3} - 1.46x^{2} - 3.93x + 6[/tex]

Step-by-step explanation:

A nth order polynomial f(x) has roots [tex]x_{1}, x_{2}, ..., x_{n}[/tex] such that [tex]f(x) = (x - x_{1})*(x - x_{2})*...*(x - x_{n}}[/tex],

Which of the following is a polynomial with roots: − square root of 3 , square root of 3, and −2?

So

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

[tex]x_{3} = -2[/tex]

Then

[tex](x - \sqrt{3})^{2}*(x - (-2)) = (x - \sqrt{3})^{2}*(x + 2) = (x^{2} -2x\sqrt{3} + 3)*(x + 2) = x^{3} + 2x^{2} - 2x^{2}\sqrt{3} - 4x\sqrt{3} + 3x + 6[/tex]

Since [tex]\sqrt{3} = 1.73[/tex]

[tex]x^{3} + 2x^{2} - 3.46x^{2} - 6.93x + 3x + 6 = x^{3} - 1.46x^{2} - 3.93x + 6[/tex]

The polynomial is [tex]x^{3} - 1.46x^{2} - 3.93x + 6[/tex]

Answer:

-0.28

Step-by-step explanation:

(1/5) : (-5/7)=(1*5)/(5*(-5))=-(7/25)=-0.28

Answer:

[tex]-7/25[/tex]

Step-by-step explanation:

[tex]1/5 \div -5/7[/tex]

Do the reciprocal of the second fraction.

[tex]1/5 \times 7/-5[/tex]

Multiply the first fraction by the reciprocal of the second fraction.

[tex]7/-25=-0.28[/tex]

The answer in decimal form is -0.28.

Answer:

To find it's percentage divide $5.60 by

$17.50 and multiply it by 100%

That is

5.60/ 17.50 × 100%

= 32%

Hope this helps you

Answer:

x=-4

Step-by-step explanation:

[tex]2x^2+8x=x^2-16[/tex]

Move everything to one side:

[tex]x^2+8x+16=0[/tex]

Factor:

[tex](x+4)^2=0[/tex]

By the zero product rule, x=-4. Hope this helps!

Answer:

x=-4

Step-by-step explanation:

Move everything to one side and combine like-terms

x²+8x+16

Factor

(x+4)²

x=-4

Answer: the probability that the tires will fail within three years of the date of purchase is 0.12

Step-by-step explanation:

The average lifetime of a set of tires is 3.4 years. It means that μ = 3.4

Decay parameter, m = 1/3.4 = 0.294

The probability density function is

f(x) = me^-mx

Where x is a continuous random variable representing the time interval of interest(the reliability period that we are testing)

Since x = 3 years,

Therefore, the probability that the tires will fail within three years of the date of purchase is

f(3) = 0.294e^-(0.294 × 3)

f(3) = 0.294e^- 0.882

f(3) = 0.12