pls help me!! (#3 btw)

Three equations that you think would be the least difficult to solve are,

∛(t + 4) = 3

∛ (r³ - 19) = 2

4 + ∛- m + 4 = 6

And, 3 equations that you think would be the most difficult to solve are,

∛z + 9 = 0

6 - ∛b = 0

∛2n + 3 = - 5

Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.

Now, By all the expressions;

Some expression gives easily real solution after solving.

But here some expression have complex solution after solving and tough to solve.

Thus, By all the expressions,

Three equations that you think would be the least difficult to solve are,

∛(t + 4) = 3

∛ (r³ - 19) = 2

4 + ∛- m + 4 = 6

And, 3 equations that you think would be the most difficult to solve are,

∛z + 9 = 0

6 - ∛b = 0

∛2n + 3 = - 5

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The areas, diameters, and radii can be grouped as follows:

1. Area: 221.5584 square units

Diameter: 16. 8 units

Radius: 8.4 units

2. Area: 78.5 square units

Diameter: 10 units

Radius: 5 units

3. Area: 452.16 square units

Diameter: 24 units

Radius: 12 units

4. Area: 36.2984 square units

Diameter: 6.8 units

Radius: 3.4 units

5. Area: 314 square units

Diameter: 20 units

Radius: 10 units

6. Area: 886.2336 square units

Diameter: 33.6 units

Radius: 16.8 units

First, we are given the area of the circles. Next, we need to note the formula of the area of a circle and this is: πr²

To get the radius, we can use the formula: √(A/π)

For the first case, we have:

Area = 221.5584 square units

Thus, radius = √(221.5584/3.14)

= 8.4

Since diameter is = 2r, then diameter in this case

= 16.8 units.

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The correct option is (d): △ABC is congruent to △A′B′C′ because you can map △ABC to △A′B′C′ using a translation 6 units to the right, which is a rigid motion.

In option (a), a reflection across the y-axis is not a rigid motion, as it changes the orientation of the triangles, and therefore cannot be used to show congruence.

In option (b), if there is no sequence of rigid motions that maps △ABC to △A′B′C′, then the triangles are not congruent.

In option (c), a translation 6 units to the left does not map △ABC to △A′B′C′. However, a translation 6 units to the right would map △ABC to △A′B′C′, because translations are rigid motions that preserve distance and angle measures.

Therefore, option (d) is the correct statement that describes the relationship between the two triangles.

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The probability that exactly 90 defective washers are found is (400 choose 90) * (1100 choose 110) / (1500 choose 200). and the probability that at least 2 defective items are found is (400 choose 1) * (1100 choose 199) / (1500 choose 200).

Let X be the number of defective washers in a sample of 200 washers.

We can model X as a hypergeometric distribution with parameters N = 1500 (total number of washers), K = 400 (total number of defective washers), and n = 200 (sample size).

a) The probability of finding exactly 90 defective washers is:

P(X = 90) = (400 choose 90) * (1100 choose 110) / (1500 choose 200)

This is because we need to choose 90 defective washers from the 400 defective washers, and 110 non-defective washers from the 1100 non-defective washers, out of the total of 200 washers chosen.

b) The probability of finding at least 2 defective items can be calculated as the complement of the probability of finding 0 or 1 defective item in the sample:

P(X >= 2) = 1 - P(X = 0) - P(X = 1)

To compute P(X = 0), we need to choose 0 defective washers from the 400 defective washers, and 200 non-defective washers from the 1100 non-defective washers, out of the total of 1500 washers:

P(X = 0) = (400 choose 0) * (1100 choose 200) / (1500 choose 200)

To compute P(X = 1), we need to choose 1 defective washer from the 400 defective washers, and 199 non-defective washers from the 1100 non-defective washers, out of the total of 1500 washers:

P(X = 1) = (400 choose 1) * (1100 choose 199) / (1500 choose 200)

Once we have computed P(X = 0) and P(X = 1), we can substitute these values into the expression for P(X >= 2) to obtain the desired probability.

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Answer:

the solution to the differential equation is:

y + (1/3)y^3 = e^x + 1/3.

Step-by-step explanation:

We can use the equation e^x=(1/(1+y^2))dy/dx to solve this differential equation using separation of variables.

First, we can rewrite the equation as:

(1+y^2)dy = e^x dx

Next, we can separate the variables:

(1+y^2)dy = e^x dx

∫ (1+y^2)dy = ∫ e^x dx

y + (1/3)y^3 = e^x + C

where C is the constant of integration.

Now we can use the initial condition to solve for C. Let's say the initial condition is y(0) = 1, then we have:

1 + (1/3)(1)^3 = e^0 + C

4/3 = 1 + C

C = 1/3

Therefore, the solution to the differential equation is:

y + (1/3)y^3 = e^x + 1/3.

The sign that makes the statement true is 792 3/20 ≠ 792 1/2.

A fraction is written in the form of p/q, where q ≠ 0.

There are two types of fractions: proper fractions in which the numerator is less than the denominator and,

Improper fractions in which the numerator is larger than the denominator.

Given, Are two fractions 792 3/20 and 792 1/2.

The sign that makes the statement true is 792 3/20 ≠ 792 1/2.

Some other signs, for example, Inequalities can also make the statement true.

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Does the data give evidence of an association between scoring at least 21 points and the football team winning the game?

Ans. There is a weak, negative association.

A weak, negative association refers to a relationship between two variables where an increase in one variable is associated with a decrease in the other variable, but the association is not strong.

In statistics, the strength of the association between two variables is measured by a correlation coefficient.

A correlation coefficient ranges from -1 to 1, where -1 represents a perfect negative correlation, 0 represents no correlation, and 1 represents a perfect positive correlation.

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The statement "parallelogram the sum of the squares of the lengths of the four sides equals the sum of the squares of the lengths of the two diagonals." is proved.

A parallelogram is a quadrilateral with two pairs of parallel sides. The opposite sides of a parallelogram are congruent, meaning they have the same length. Similarly, the opposite angles of a parallelogram are congruent, meaning they have the same measure.

Now, let's consider a parallelogram ABCD with sides AB, BC, CD, and DA. We want to show that:

AB² + BC² + CD² + DA² = AC² + BD²

where AC and BD are the diagonals of the parallelogram.

First, let's draw the diagonals AC and BD. This divides the parallelogram into four triangles: triangle ABC, triangle BCD, triangle CDA, and triangle DAB.

Next, let's use the Pythagorean theorem in each of these triangles. Recall that the Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

In triangle ABC, we have:

AC² = AB² + BC²

Similarly, in triangle CDA, we have:

AC² = CD² + DA²

Adding these two equations together, we get:

2AC² = AB² + 2BC² + CD² + 2DA²

Now, let's look at triangle ABD. We have:

BD² = AB² + DA²

Similarly, in triangle BCD, we have:

BD² = BC² + CD²

Adding these two equations together, we get:

2BD² = AB² + 2BC² + CD² + 2DA²

Finally, adding the two equations we obtained for AC² and BD², we get:

2AC² + 2BD² = 2AB² + 4BC² + 2CD² + 2DA²

Simplifying this expression, we get:

AC² + BD² = AB² + BC² + CD² + DA²

Therefore, we have proven that in any parallelogram, the sum of the squares of the lengths of the four sides is equal to the sum of the squares of the lengths of the two diagonals.

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The length of the diagonal SU will be 46 units.

A parallelogram is a quadrilateral with two pairs of equal sides.

Given is a parallelogram RSTU.

The diagonals of a parallelogram bisect each other. This means that -

SV = VU

2x + 3 = 4x - 17

17 + 3 = 4x - 2x

20 = 2x

x = 10

The length SU will be -

SU = SV + VU

SU = 2x + 3 + 4x - 17

SU = 6x - 14

SU = 60 - 14

SU = 46 units

Therefore, the length of the diagonal SU will be 46 units.

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The number of students who went to Camp Redwood was 23% more than the number of students who went to Camp Arrowhead.

What is percentage ?

A percentage is a way of expressing a number as a fraction of 100. The symbol for percentage is "%". For example, 50% means 50 out of 100 or 0.5 as a decimal.

Percentages are commonly used to represent proportions, rates, and changes in quantities. They can be used to compare quantities of different sizes, or to express how much one quantity has changed relative to another quantity.

Given by the question:
Let's assume that there are a total of 100 students who were surveyed.

If 15% of the students who planned to go to Camp Redwood did not go, then 85% of them did go:

0.85 x (number of students who planned to go to Camp Redwood) = number of students who actually went to Camp Redwood

Similarly, if 20% of the students who planned to go to Camp Arrowhead did not go, then 80% of them did go:

0.80 x (number of students who planned to go to Camp Arrowhead) = number of students who actually went to Camp Arrowhead

Let's say that x students planned to go to Camp Redwood and y students planned to go to Camp Arrowhead. Then we can write two equations based on the above percentages:

0.85x = number of students who actually went to Camp Redwood

0.80y = number of students who actually went to Camp Arrowhead

We want to find the difference between the number of students who went to Camp Redwood and the number of students who went to Camp Arrowhead, so we can subtract the second equation from the first:

0.85x - 0.80y = (number of students who actually went to Camp Redwood) - (number of students who actually went to Camp Arrowhead)

We can substitute the values from the percentages into the equation:

0.85x - 0.80y = 0.85(0.85x) - 0.80(0.80y)

0.85x - 0.80y = 0.7225x - 0.64y

0.13x = 0.16y

We can solve for one of the variables in terms of the other:

x = (0.16/0.13)y

x = 1.23y

So if y students planned to go to Camp Arrowhead, then 1.23y students planned to go to Camp Redwood. The difference between the number of students who went to Camp Redwood and the number of students who went to Camp Arrowhead is:

1.23y - y = 0.23y

So the number of students who went to Camp Redwood was 23% more than the number of students who went to Camp Arrowhead.

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The value of the d(g o f)/dt at t = 2 is 14.

We can start by computing the composition (g o f)(t) and then finding its derivative with respect to t using the chain rule.

(g o f)(t) = g(f(t)) = g(t, t^2 - 4e^(t-2))

At t = 2, we have f(2) = 2 and f'(2) = 2t - 4e^(t-2) evaluated at t = 2 gives f'(2) = 0.

To find d(g o f)/dt at t = 2, we first need to evaluate (g o f)(2):

(g o f)(2) = g(f(2)) = g(2, 0) = g(a)

Next, we use the chain rule to find the derivative of (g o f) with respect to t:

[tex](d/dt) (g o f)(t) = (∂g/∂x)(a) (df/dt) + (∂g/∂y)(a) (d/dt) (t^2 - 4e^{(t-2)}) + (∂g/∂z)(a) (dg/dz)(a)[/tex]

Since f(t) = t, df/dt = 1. Also, since g(x,y,z) is differentiable, we can write dg/dz(a) as ∂g/∂z(a) = 2.

Substituting the values we have and evaluating at t = 2, we get:

[tex](d/dt) (g o f)(2) = (4)(1) + (2)(2t - 4e^{(t-2)})(t=2) + (2)(2)[/tex]

= 4 + 2(4-4e^0) + 4

= 14

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The value of c is given as follows:

c = 23.

The problem states that the total of 13 cherries, 8 more cherries, and 2 more cherries is c, hence the value of c is obtained with the addition of these amounts, as follows:

c = 13 + 8 + 2.

Adding these three amounts, the numeric value of c is given as follows:

c = 23.

The problem asks for the numeric value of c.

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C)The statement is false. The definition of AB states that each column of AB is a linear combination of the columns of B using weights from the corresponding column of A.(C)

To see why this is the case, let A be an m x n matrix and let B be an n x p matrix. Then the product AB is an m x p matrix whose (i,j)-th entry is given by the dot product of the i-th row of A with the j-th column of B:

(AB){i,j} = \sum{k=1}^n a_{i,k} b_{k,j}

This means that the j-th column of AB is obtained by taking a linear combination of the columns of B using the weights from the j-th column of A.

In other words, the statement "each column of AB is a linear combination of the columns of B using weights from the corresponding column of A" is true, but the statement in answer choice C is false.

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Answer:

The inequality to set up is [tex]t+16 \ge 31[/tex]

It solves to [tex]t \ge 15[/tex]

Bill needs to run at least 15 more miles.

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

Explanation:

He has run 16 miles so far. Add on another t miles to get 16+t or t+16 to represent the total amount he runs. This expression must be 31 or larger

Either t+16 > 31 or t+16 = 31

Those two items condense to [tex]t+16 \ge 31[/tex]

To solve for t, we subtract 16 from both sides to undo the +16.

[tex]t+16 \ge 31\\\\t+16-16 \ge 31-16\\\\t \ge 15\\\\[/tex]

Bill needs to run at least 15 more miles.

Meaning that t = 15, t = 16, t = 17, etc.

The velocity of the particle at times t = a, t = 1, t = 2, and t = 3 are:

a) 4/a^2 m/s, 4 m/s, 1 m/s, 4/9 m/s.

b) The instantaneous velocity of the particle when t=1 is 4 m/s.

To find the velocity, we need to take the derivative of the displacement function with respect to time. The derivative of s = 4/t^2 is ds/dt = -8/t^3. So, the velocity of the particle at time t is given by v = ds/dt = -8/t^3.

For t = a, the velocity is v = -8/a^3 m/s. For t = 1, the velocity is v = -8 m/s. For t = 2, the velocity is v = -2 m/s. For t = 3, the velocity is v = -8/27 m/s.

To find the average velocity during the time period [1, 2], we need to find the displacement at t = 2 and t = 1, then calculate the change in displacement divided by the time interval: (4/4 - 4/1)/1 = 0 cm/s. To find the average velocity during the time period [1, 1.1], we need to find the displacement at t = 1.1 and t = 1, then calculate the change in displacement divided by the time interval: (4/1.1^2 - 4/1)/0.1 = -19.60 cm/s.

To estimate the instantaneous velocity of the particle at t = 1, we can plug in t = 1 to the derivative we found earlier: v = -8/1^3 = -8 m/s. Therefore, the instantaneous velocity of the particle when t = 1 is 4 m/s.

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The time to travel 279 miles is 4.5 hours

We can use the formula:

time = distance ÷ speed

To find the time it will take to travel 279 miles, given that it takes 6 hours to travel 372 miles.

The speed is constant, so we can find it by dividing the distance by the time for the first trip:

speed = distance ÷ time = 372 miles ÷ 6 hours = 62 miles/hour

Now we can use the speed to find the time it will take to travel 279 miles:

time = distance ÷ speed = 279 miles ÷ 62 miles/hour ≈ 4.5 hours

Hence, it will take approximately 4.5 hours to travel 279 miles at this speed.

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Based on the data, we can infer that the following numbers fill in the blanks: 100 is greater than 10.

To fill in the blanks we must carefully read the information in the fragment. Once we understand this information we can select the two numbers that meet the rule mentioned in the statement.

According to the above, a correct statement would be:

Additionally, we could put other values that comply with this rule, for example:

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Depends how the answer "should" be formatted but 114 2/3 feet should do the trick

The expression value of 3x^2 - 2x+3 is 6x^2 - 13x + 6

Value of an Expression The value of the expression is the result of the calculation described by this expression when the variables and constants in it are assigned values. Letters can be used to represent numbers in an algebraic expression.

Apply the distributive property:

→ 3x (2 - 3) - 2 (2x - 3)

Apply the distributive property:

→ 3x (2x) + 3x ⋅ -3 - 2 (2x - 3)

Apply the distributive property:

→ 3x (2x) + 3x ⋅ -3 - 2 (2x) - 2 ⋅ -3

Simplify each term:

→ 6x^2 - 9x - 4x  + 6x

Subtract 4x from -9x

→ 6x^ - 13x + 6

Therefore, the expression value of 3x^2 - 2x+3 is 6x^2 - 13x + 6

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The value of the expression given is 136.

Expressions are mathematical statements which consist of two or more terms and terms are connected to each other using mathematical operators like addition, multiplication, subtraction and so on.

Given expression is,

3x² - 2x + 3

We have to find the value of the expression when x = 7.

Substitute x = 7.

(3 × 7²) - (2 × 7) + 3 = (3 × 49) - 14 + 3

                               = 147 - 14 + 3

                               = 136

Hence the value of the expression is 136.

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Answer:208.75

Step-by-step explanation:

The expression of a function is
(a) {"++", "+-", "-+", "--"}

(b) {"000", "001", "010", "011", "100", "101", "110", "111"}

(a) Roster notation is used to express a set of elements in a compact form. In this case, A is the set {+, -}, and A^2 is the set of all possible ordered pairs of elements from A. This set can be expressed using roster notation as {"++", "+-", "-+", "--"}. The double symbols indicate the two elements of each ordered pair.

(b) Here, A is the set {0, 1}, and A^3 is the set of all possible ordered triplets of elements from A. This set can be expressed using roster notation as {"000", "001", "010", "011", "100", "101", "110", "111"}. The triple symbols indicate the three elements of each ordered triplet.

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If the scores of a certain population are Normally Distributed , and the mean is 104.32 , then the 95% confidence interval is (b) 104.32 ± 3.29   .

The Scores of the population of WISC are Normally Distributed ;  

and the Mean(μ) of the 25 scores is = 104.32  ;

the standard deviation(σ) for the population is = 10  ;

the sample size(n) = 25  ;

For  the 95% confidence, critical value is = 1.96 ;

So,  Confidence interval is written as =  μ ± 1.96(σ/√n)

Substituting the required values  ,

we get ;

⇒ 104.32 ± 1.96(10/√25)

⇒ 104.32 ± 1.96×2

⇒ 104.32 ± 3.92

Therefore , the required 95% confidence interval is (104.32 ± 3.92) .

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The given question is incomplete , the complete question is

The scores of a certain population on the Wechsler Intelligence Scale for Children (WISC) are thought to be Normally distributed with mean μ and standard deviation σ = 10. A simple random sample of 25 children from this population is taken and each is given the WISC. The mean of the 25 scores is = 104.32.

Based on these data, a 95% confidence interval for μ is

(a) 104.32 ± 0.78.

(b) 104.32 ± 3.29.

(c) 104.32 ± 3.92.

(d) 104.32 ± 19.60

The graph of the functions f(x) and g(x) on the same coordinates plane and the solution is (0 , 0.818).

What is a function?

A function is a relationship between inputs where each input is related to exactly one output.

Example:

f(x) = 2x + 1

f(1) = 2 + 1 = 3

f(2) = 2 x 2 + 1 = 4 + 1 = 5

The outputs of the functions are 3 and 5

The inputs of the function are 1 and 2.

We have,

Two functions:

f(x) = -|x - 3| + 2

g(x) = 12x - 3

Now,

The graph is given below.

The intersection of the graph of f(x) and g(x) is the solution to f(x) = g(x).

So,

for x> 3 , we can say that

-|x - 3| + 3 = 12x -3

-x + 3 +3 = 12x -3

Therefore , x = 9/11 = 0.818

for x< 3 , we can say that

-|x - 3| + 3 = 12x -3

x - 3 = 12x - 3

x = 0

The solution is (0 , 0.818).

Thus, the solution is (0 , 0.818).

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The probability of waiting less than 12 minutes between successive speeders is 0.02.

The time elapsed between successive speeders detected by a radar unit is a  random variable with a cumulative distribution function:

F(x) = 0 when x = 0.

x/10, for 0 x ten 1, for x ten

a) Using the cumulative distribution function, to calculate the probability of waiting less than 12 minutes (0.2 hours) between successive speeders: F(0.2):

F(0.2) = 0.2/10 = 0.02

So, the probability of waiting less than 12 minutes between speeders is 0.02.

b) Using the probability density function, we can differentiate the cumulative distribution function with respect to x to find the probability:

For x = 0, f(x) = dF(x)/dx = 0.

1/10 for 0 x 10 0, 0 for x 10

The area under the probability density function up to x=0.2 represents the probability of waiting less than 12 minutes:

P(0 < w < 0.2) = ∫0.2 0 f(x) (x) dx = ∫0.2 0 (1/10) dx = (1/10) * [x] 0.2 = 0.02

Using either the cumulative distribution function or the probability density function, the probability of waiting less than 12 minutes between successive speeders is 0.02.

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The decimal representation of the given binary fractional number 0.000112 is 0.109375.

Binary numbers are composed of only 0s and 1s, while decimal numbers are composed of 0-9. Additionally, binary numbers follow a different place value system, where each digit represents a power of 2 rather than a power of 10.

Now, let's focus on the given problem - converting a binary fractional number 0.000112 to a decimal representation. We can use the following formula to convert any binary fraction to decimal:

(decimal equivalent) = (binary digit 1/2¹) + (binary digit 2/2²) + (binary digit 3/2³) + ...

Using this formula, we can start by breaking down the given binary fraction 0.000112 into its individual binary digits:

0.000112 = 0/2¹ + 0/2² + 0/2³ + 1/2⁴ + 1/2⁵ + 1/2⁶

Now, we can simply evaluate each term in the equation and add them up to get the decimal equivalent:

0/2¹ = 0

0/2² = 0

0/2³ = 0

1/2⁴ = 0.0625

1/2⁵ = 0.03125

1/2⁶ = 0.015625

(decimal equivalent) = 0 + 0 + 0 + 0.0625 + 0.03125 + 0.015625 = 0.109375

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The volume of the pyramid  of height h with an equilateral triangle base with side a, by using the method of volumes by SLICES  is V = a²h/3.

We can use the method of volumes by slices to find the volume of the pyramid.

Consider a slice of the pyramid that is perpendicular to the base and at a distance x from the apex. This slice has a cross-sectional area of a square with side length (base of pyramid) s, We can use the Pythagorean theorem to find that the

Height of the slice is [tex]\sqrt{(a^2 - (a/2)^2 - x^2)} = \sqrt{(3a^2/4 - x^2).[/tex]

Therefore, the volume of the slice is given by the product of its cross-sectional area and height:

dV = s^2 dh = (a^2 - 4x^2)/4 dh

To find the total volume of the pyramid,

we integrate dV from x=0 to x=h:

V = ∫₀ʰ (a²-4x²)/4 dx

Simplifying the integrand, we get:

V = (1/4) ∫₀ʰ (a² - 4x²) dx

V = (1/4) [a²x - (4x³)/3] from x=0 to x=h

V = (1/4) [a²h - (4h³)/3]

V = a²h/3

Therefore, the volume of the pyramid is V = a²h/3.

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a) The joint probability density function is [tex]f(y_1,y_2) = f(y_1) * f(y_2) = (1/3e^{(-y1/3)})(1/3e^{-y2/3})[/tex]

b) The total effective length of life of the two fuses is less than or equal to one.

The exponential distribution is a probability distribution that models the length of life of fuses, and its probability density function can be used to find the joint probability density function for the lengths of two independent fuses.

a) To find the joint probability density function for the lengths of two independent fuses, we simply multiply the probability density functions of each individual fuse. In this case, we have

[tex]= > f(y_1,y_2) = f(y_1) * f(y_2) = (1/3e^{(-y1/3)})(1/3e^{-y2/3})[/tex]

for y₁, y₂ > 0. This is a function that gives the probability of a given pair of lengths (y₁,y₂) occurring.

b) To find P(Y₁ + Y₂ ≤ 1), we must first determine the cumulative distribution function for the sum of the lengths of the two fuses. This is given by

=> F(y) = P(Y₁ + Y₂ ≤ y) = ∫∫f(x,y)dxdy,

where the integral is taken over the region x+y ≤ y. We can simplify this by changing the order of integration:

=> [tex]F(y) = \int0^y\int0^{y-x}f(x,y)dxdy.[/tex]

Using the probability density function given in part (a), we have

=> [tex]F(y) = \int0^y\int0^{y-x}(1/9)e^{-(x+y)/3}dxdy[/tex]

This can be solved using integration by parts or by using the fact that the exponential function integrates to itself, giving

=> [tex]F(y) = 1 - e^{-y/3)(y+3)}[/tex]

Finally, we can find P(Y₁ + Y₂ ≤ 1) by evaluating F(1) - F(0), which gives

=> [tex]P(Y_1 + Y_2 ≤ 1) = 1 - e^{(-1/3)(4/3)}[/tex].

This is a function that gives the probability that the total effective length of life of two fuses is less than or equal to one.

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Complete Question:

The length of life Y1 AND Y2 for fuses of a certain type is modeled by the exponential distribution, with

[tex]f(y) = \left \{ {1/3e^{-y/3} y > 0,} \atop {0, else where }} \right.[/tex]

(The measurements are in hundreds of hours.)

a) If two such fuses have independent lengths of life Y1 and Y2, find the joint probability density function for Y1 and Y2.

b) One fuse in part (a) is in a primary system, and the other is in a backup system that comes into use only if the primary system fails. The total effective length of life of the two fuses is then Y1 + Y2. Find P(Y1 + Y2 ≤ 1).

The equation of the plane that passes through the line of intersection of the two planes and passes through the point (3,-1,2) is x - 2y + 7z - 17 = 0  .

The Plane we want to find passes through the line of intersection of the planes, so it is perpendicular to the normal vectors of both planes.

The Normal Vector of the plane 2x - 3y - z + 1 = 0 is (2, -3, -1), and

the normal vector of the plane 3x + 5y - 4z + 2 = 0 is (3, 5, -4).

The cross product of these two vectors gives a vector which is perpendicular to both of them, and

hence , it lies along the line of intersection of the two planes:

the cross product of (2, -3, -1) x (3, 5, -4) is  = (-7, -2, 1)  ;

The vector (-7, -2, 1) is the direction vector of the line of intersection.

Now to find the equation of the plane that passes through the point (3, -1, 2) and is perpendicular to this direction vector.

we let , equation of the plane be ax + by + cz + d = 0. We know that the point (3, -1, 2) lies on the plane, so we have:

⇒ a(3) + b(-1) + c(2) + d = 0  ;

⇒ 3a - b + 2c + d = 0  ;

We also know that the plane is perpendicular to the direction vector (-7, -2, 1), so we have:

⇒ a(-7) + b(-2) + c(1) = 0  ;

⇒ -7a - 2b + c = 0  ;

Let  say a = 1. Then we have :

⇒ a = 1

⇒ -7a - 2b + c = 0

⇒ 3a - b + 2c + d = 0

Substituting a = 1 into the first equation gives:

⇒ -7 - 2b + c = 0  ;

Solving for c in terms of b gives:

⇒ c = 2b + 7

Simplifying ,

we get ;

⇒ 3 - b + 2(2b + 7) + d = 0  ;

Solving for d in terms of b gives:   ⇒ d = -2b - 17  ;

Therefore, the equation of the plane is x - 2y + 7z - 17 = 0 .

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