A supplier to a car manufacturer produces a certain valve and seal used in their cars. The sizes of these seals and


valves are closely monitored to ensure the parts actually work. Here are summary statistics on the diameters for


these valves and seals (in millimeters).


Mean


Standard deviation


Valve


Hv = 50


OV = 0. 3


Seal


Ils = 51


Os = 0. 4


Both distributions are approximately normal. A seal properly fits a valve if the seal's diameter is larger than the


valve's diameter, but the difference can't be more than 2 mm. Suppose we choose a valve and seal at random


and calculate the difference between their diameters. We can assume that their diameters are independent.

Triangle FGH is a right triangle because (HG)²= (FG)²+ (FH)²

Similar triangles are triangles that have the same shape, but their sizes may vary. The ratio of corresponding sides of similar triangles are equal.

Therefore;

6/5 = 3.6/FH

represent FH by x

6/5 = 3.6/x

6x = 5 × 3.6

6x = 18

divide both sides by 6

x = 18/6 = 3

Since FH is 3, this means that the sides of triangle FGH are Pythagorean triple, hence FGH is a right triangle.

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Answer:

The perfect factored square trinomial that is equivalent to the expression x^2 + 8x + 16 is:

(x + 4)^2

To see why this is the case, you can expand the expression (x + 4)^2 using the FOIL method:

(x + 4)^2 = (x + 4) * (x + 4)

= x^2 + 4x + 4x + 16

= x^2 + 8x + 16

So, x^2 + 8x + 16 can be factored as (x + 4)^2, which is a perfect square trinomial.

At point (-43,0), f has a maximum. At point (0,4), f has a minimum. At point (463,0), f has a maximum. At point (0,0), f may have a critical point. none of the given points are critical points of the function f (x, y) = xy + 33 - 48y.

Hi! To analyze the critical points of the function f(x, y) = xy + 33 - 48y, we first need to find the partial derivatives with respect to x and y:

fx = ∂f/∂x = y
fy = ∂f/∂y = x - 48

Now, we can analyze the given points:

1. (-43, 0)
At this point, fx = 0 and fy = -48. Since both partial derivatives are not equal to 0, this point is not a critical point.

2. (0, 4)
At this point, fx = 4 and fy = 0. Again, both partial derivatives are not equal to 0, so this is not a critical point.

3. (463, 0)
At this point, fx = 0 and fy = 463 - 48 = 415. Since both partial derivatives are not equal to 0, this is not a critical point.

4. (0, 0)
At this point, fx = 0 and fy = -48. Since both partial derivatives are not equal to 0, this is not a critical point.

5. (0, -4)
At this point, fx = -4 and fy = -48. Again, both partial derivatives are not equal to 0, so this is not a critical point.

In summary, none of the given points are critical points of the function f(x, y) = xy + 33 - 48y.

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The relationship between the areas of the three squares is that square A plus square B equals the area of square C.

The Pythagorean theorem is a fundamental idea in geometry that states that for any right-angled triangle, the square of the length of the longest side (opposite the right angle) is equal to the sum of the square of the lengths of the two remaining sides. This equation can be expressed as:

[tex]a^2 + b^2 = c^2[/tex]

Thus, the relationship between the areas of the three squares is that square A plus square B equals the area of square C.

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The true statements are:

A valid conclusion is :

The range of a data set is the difference between the highest and lowest values in the set.

To calculate the range, you subtract the smallest value from the largest value.

The mean of a data set, also called the average, is the sum of all the values in the set divided by the total number of values in the set.

To calculate the mean, you add up all the values and divide by the total number of values.

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a) The mean monthly rainfall amount for City A is 334.17 inches and for City B is 5.83 inches.

b) The MAD for City A is 464.28 inches and for City B is 0.46 inches.

c) The median for City A is 5 inches and for City B is 6 inches.

(a) To find the mean monthly rainfall amount for each city, we need to add up all the rainfall amounts and divide by the number of months:

For City A: (5 + 2.5 + 6 + 2008.5 + 5 + 3) / 6 = 334.17 inches

For City B: (7 + 6 + 5.5 + 6.5 + 5 + 6) / 6 = 5.83 inches

(b) To find the mean absolute deviation (MAD) for each city, we need to find the absolute deviations from the mean for each data point, then calculate the average of those absolute deviations:

For City A:

Mean = 334.17 inches

Absolute deviations from the mean: |5 - 334.17| = 329.17, |2.5 - 334.17| = 331.67, |6 - 334.17| = 328.17, |2008.5 - 334.17| = 1674.33, |5 - 334.17| = 328.17, |3 - 334.17| = 331.17

MAD = (329.17 + 331.67 + 328.17 + 1674.33 + 328.17 + 331.17) / 6 = 464.28 inches

For City B:

Mean = 5.83 inches

Absolute deviations from the mean: |7 - 5.83| = 1.17, |6 - 5.83| = 0.17, |5.5 - 5.83| = 0.33, |6.5 - 5.83| = 0.67, |5 - 5.83| = 0.83, |6 - 5.83| = 0.17

MAD = (1.17 + 0.17 + 0.33 + 0.67 + 0.83 + 0.17) / 6 = 0.46 inches

(c) To find the median for each city, we need to arrange the data points in order and find the middle value:

For City A: {2.5, 3, 5, 5, 6, 2008.5}

Median = 5 inches

For City B: {5, 5.5, 6, 6, 6.5, 7}

Median = 6 inches

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Note that the expected value of the retailers profit is - $114. This means he made a loss.

To find the expected value we must proceed as follows

Expected Value - E(F) is

Probability of F - P(F)

= 100 x ($100 - $200) + (P(F) = $50) x ($50 - $200) + (P(F) = $ -200) x ( $ - 200 - $200) +  (P(F)   = $- 350) x ($ -350  $ 200)

= (0.9) x (-100) + (0.07 ) x (-150) + (0.01)  x  (-550)  + (0.02) x (-400)

= - 90  - 10.5  - 5.5 -8

E(F) = $ -114

So it is right to state that the expected value of the retailer's profit per protection plan sold is -$114, which  is a loss.

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Full Question:

An electronics retailer sells mobile phones with an optional protection plan for $100. In case of an accident, the customer pays a $50 deductible and the retailer covers the rest of the repair cost, which is typically $200 per repair. The protection plan covers a maximum of 333 repairs.

Let X be the number of repairs a randomly chosen customer uses under the protection plan, and let F be the retailer's profit from one of these protection plans. The probability distributions of X and F are:

X = number of repairs: 0 1 2 3

Probability: 0.90 0.07 0.02 0.01

F = retailer profit: $100-$50-$200-$350

Probability: 0.90 0.07 0.02 0.01

The task is to find the expected value of the retailer's profit per protection plan sold.

The equation for revenue R in terms of the number of rides x is given by R = px, where p is the amount charged per ride (in dollars).

The equation for the revenue R in terms of the number of rides can be derived by multiplying the number of rides with the amount charged per ride.

Let the amount charged per ride be p (in dollars).

Then, the equation for revenue R can be written as R = px.

Note that the amount charged per ride is not given in the problem. It can be assumed that the amount charged is a fixed amount for all the rides.

However, the equation for revenue can still be written in terms of the variable p as R = px.

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If he put 40 red balls, 55 purple balls, 45 yellow balls, and 60 green balls into the ball pit. while his sister is playing, one ball rolls out of the pit. Therefore, the probability that the ball that rolled out of the pit is red is 0.2.

The probability of selecting a red ball from the ball pit can be found by dividing the number of red balls by the total number of balls in the pit.

Total number of balls = 40 + 55 + 45 + 60 = 200

Probability of selecting a red ball = Number of red balls / Total number of balls

Probability of selecting a red ball = 40 / 200

Probability of selecting a red ball = 0.2

Therefore, the probability that the ball that rolled out of the pit is red is 0.2.

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The expression represents the second partial sum for 2(0. 4) + 2(0. 4)2 2(0. 4)2 + 2(0. 4)3 2 + 2(0. 4) 0 + 2(0. 4)1 is 0.8.

The second partial sum of a sequence refers to the sum of the first two terms of the sequence.

The given sequence is: 2(0.4) + 2(0.4)^2 + 2(0.4)^2 + 2(0.4)^3 + 2(0.4)^0 + 2(0.4)^1

To find the second partial sum, we simply add the first two terms of the sequence:

2(0.4) + 2(0.4)^2 = 0.8

Therefore, the expression that represents the second partial sum for the given sequence 2(0. 4) + 2(0. 4)2 2(0. 4)2 + 2(0. 4)3 2 + 2(0. 4) 0 + 2(0. 4)1 is 0.8.

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The dealer bought 42 radios.

Let x be the number of radios the dealer bought.

Let y be the price the dealer paid for each radio.

We know that the dealer bought x radios for a total of $1,008, so:

x * y = 1008

We also know that the dealer gave away 6 radios and sold the rest for $14 more than she paid for each radio, breaking even. This means that the total revenue from selling the remaining radios is equal to the total cost of buying them:

(x - 6) * (y + 14) = x * y

Simplifying this equation, we get:

xy + 14x - 6y - 84 = xy

14x - 6y = 84

7x - 3y = 42 (dividing by 2 on both sides)

Now we have two equations:

x * y = 1008

7x - 3y = 42

We can use substitution or elimination to solve for x and y. Let's use elimination by multiplying the second equation by y/3 and adding it to the first equation:

x * y + (7x - 3y) * (y/3) = 1008 + 42 * (y/3)

xy + 7xy/3 - y²/3 = 1008 + 14y

10xy/3 - y²/3 - 14y - 1008 = 0

Multiplying both sides by 3, we get:

10xy - y² - 42y - 3024 = 0

Now we can use the quadratic formula to solve for y:

y = (-b ± sqrt(b² - 4ac)) / 2a

where a = -1, b = -42, and c = -3024:

y = (-(-42) ± sqrt((-42)² - 4(-1)(-3024))) / 2(-1)

y = (42 ± sqrt(42² - 4*3024)) / 2

y = (42 ± 126) / 2

y = 84 or y = -42

Since the price of a radio cannot be negative, we can discard the second solution and conclude that y = 84.

Now we can solve for x using the first equation:

x * y = 1008

x * 84 = 1008

x = 12

Therefore, the dealer bought 12 radios.

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Answer:

if the diagonal is 10 then the sides are 3*2 and 4*2 which is 6 and 8 respectively because the diagonal makes it a right angled triangle whereby the the 3,4,5 line steps in, so if the diagonal(hypotenuse) is 10 the 10/5 is 2 then you multiply both 3 and 4 by 2 and that gives you the length of two sides

The value of n which is the number of frogs Heather has is 6.

The number of frogs Heather has is calculated as follows;

let the number of frogs Heather has = n

So Ava has 2 fogs, which is equal to 1/3 n.

The value of n which is the number of frogs Heather has is calculated as follows;

(1/3) n = 2

multiply both sides by 3;

n = 3 x 2

n = 6

The division using a diagram, is determined as;

    0         0

    I           I

    I           I

    I            I

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Answer:

$15.92

Step-by-step explanation:

We Know

2 liters of Gatorade cost $3.98

How much do 8 liters cost?

We take

3.98 x 4 = $15.92

So, 8 liters cost $15.92

One possible set of parametric equations for the line is:

x = -4 + 4t
y = 8 - t
z = 7 - 4t

To see why these work, let's first consider the equation of the plane: -x + 4y + z = 8. This can also be written in vector form as:

[ -1, 4, 1 ] · [ x, y, z ] = 8

where · denotes the dot product. This equation says that the normal vector to the plane is [ -1, 4, 1 ], and that any point on the plane satisfies the equation.

Now, since the line we want is perpendicular to the plane, its direction vector must be parallel to the normal vector to the plane. In other words, the direction vector of the line must be some multiple of [ -1, 4, 1 ]. Let's call this direction vector d.

To find d, we can use the fact that the dot product of two perpendicular vectors is zero. So we have:

d · [ -1, 4, 1 ] = 0

Expanding this out, we get:

-1d1 + 4d2 + 1d3 = 0

where d1, d2, d3 are the components of d. This equation tells us that d must be of the form:

d = [ 4k, k, -k ]

where k is any non-zero scalar (i.e. any non-zero real number).

Now we just need to find a point on the line. We're given that the line passes through (-4, 8, 7), so this will be our starting point. Let's call this point P.

We can now write the parametric equations of the line in vector form as:

P + td

where t is any scalar (i.e. any real number). Substituting in the expressions for P and d that we found above, we get:

[ -4, 8, 7 ] + t[ 4k, k, -k ]

Expanding this out, we get the set of parametric equations I gave at the beginning:

x = -4 + 4tk
y = 8 + tk
z = 7 - tk

where k is any non-zero scalar.
To find a set of parametric equations for the line, we first need to determine the direction vector of the line. Since the line is perpendicular to the plane given by -x + 4y + z = 8, we can use the plane's normal vector as the direction vector for the line. The normal vector for the plane can be determined by the coefficients of x, y, and z, which are (-1, 4, 1).

Now that we have the direction vector (-1, 4, 1) and the point the line passes through (-4, 8, 7), we can write the parametric equations as follows:

x(t) = -4 - t
y(t) = 8 + 4t
z(t) = 7 + t

So, the set of parametric equations for the line is {x(t) = -4 - t, y(t) = 8 + 4t, z(t) = 7 + t}.

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Answer:

unfortunately there's no questions to be answered

The area under the standard normal distribution curve between z = 0 and z = 0.98 is:

                         0.8365 - 0.5000 = 0.3365

To find the area under the standard normal distribution curve between z = 0 and z = 0.98, we can use a standard normal distribution table or a calculator that can compute normal probabilities.

Using a standard normal distribution table, we can look up the area corresponding to a z-score of 0 and a z-score of 0.98 separately and then subtract the two areas to find the area between them.

The area under the standard normal distribution curve to the left of z = 0 is 0.5000 (by definition). The area under the curve to the left of z = 0.98 is 0.8365 (from the standard normal distribution table).

So the area under the standard normal distribution curve between z=0 and z=0.98 is approximately 0.3365.

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The probability of the event BE falling on a random point AK is 4/11

A probability event can be defined as a set of outcomes of an experiment. In other words, an event in probability is the subset of the respective sample space.

In this problem, we need to determine our sample space;

The sample space = 11

The number of favorable outcomes = 4

The probability of a random point on AK to be on BE will be;

P = 4 / 11

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The most appropriate domain of the function is 0 ≤ x ≤ 22. This is because Alex can only walk from his home to the store within a maximum of 22 minutes, and the distance he walks can only be modeled within that time frame.

It is given that the function f(x) = 2.5x, which models the distance Alex walks in x minutes after leaving his house to go to the store. It takes him 22 minutes to walk from his home to the store. The most appropriate domain of the function is the range of x values that make sense in this context.

Step 1: Identify the minimum and maximum values for x.
In this case, the minimum value for x is when Alex starts walking, which is 0 minutes. The maximum value for x is when he reaches the store, which is 22 minutes.

Step 2: Express the domain as an interval.
The domain of the function can be written as an interval from the minimum to the maximum value, including both endpoints. Therefore, the domain is [0, 22].

Therefore, the most appropriate domain of the function f(x) = 2.5x, which models the distance Alex walks in x minutes after leaving his house to go to the store, is [0, 22].

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The rate at which salt enters the tank at time t is constant & equal to 8 lb/min,

The rate at which salt enters the tank at time t is equal to the product of the concentration of the incoming brine & the rate at which it enters the tank

At time t, the amount of salt in the tank is y(t), & the volume of the brine in the tank is V(t)-

Therefore, the concentration of salt in the tank at time t is:-

c(t) = y(t) / V(t)

The rate at which brine enters the tank at time t is 4 gal/min, & the concentration of salt in the incoming brine is 2 lb/gal

So the rate at which salt enters the tank at time t is:-

2 lb/gal x 4 gal/min = 8 lb/min

Therefore, the rate at which salt enters the tank at time t is constant & equal to 8 lb/min, regardless of how much salt is already in the tank

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The probability of rolling any number other than 4 on a number cube with sides numbered 1, 2, 3, 4, 5, and 6 is 5/6, which can be written as a fraction in simplest form.

The complement of rolling a 4 on a number cube with sides numbered 1, 2, 3, 4, 5, and 6 is rolling any number other than 4, which includes rolling a 1, 2, 3, 5, or 6.

To find the probability of the complement, we need to add up the probabilities of rolling each of these numbers.

Since each number has an equal chance of being rolled, we can find the probability of rolling each number by dividing 1 by the total number of possible outcomes (which is 6, since there are six sides on the cube).

Then, we can add up the probabilities of rolling each of the five numbers in the complement:

P(rolling a 1, 2, 3, 5, or 6) = P(rolling a 1) + P(rolling a 2) + P(rolling a 3) + P(rolling a 5) + P(rolling a 6)

P(rolling any number other than 4) = 1 - P(rolling a 4)

P(rolling any number other than 4) = 1 - 1/6 = 5/6

Therefore, the probability of rolling any number other than 4 on a number cube with sides numbered 1, 2, 3, 4, 5, and 6 is 5/6, which can be written as a fraction in simplest form.

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The statement that explains the negative and nonlinear relationship between group size and percent woodland is given by the percent of woodland increases, the number of deer observed in a group decreases quickly at first and then more slowly.

This statement suggests that as the amount of woodland id increases, when the number of deer in a group decreases.

However, the rate of decrease is not constant, but rather decreases more slowly as the percent of woodland increases.

This suggests that there may be some threshold or tipping point.

At which the relationship between group size and percent woodland becomes less pronounced.

This kind of relationship is not uncommon in ecological studies.

Where factors like habitat availability, food availability, and predation risk can all influence animal behavior and population dynamics.

Nonlinear relationships like this one can help researchers better understand complex interplay between these factors and behavior of animals they study.

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Answer:

1 hour to build a website

5 hours to update websites

Step-by-step explanation:

x is the hours to build a website

y is the hours to update websites

x + y = 6 ------> 25x + 25y = 150

25x + 15y = 100

10y = 50

y = 5

25x + 15 (5) = 100

25x + 75 = 100

25x = 25

x = 1

So, LaShawn needs 1 hour to build a website and 5 hours to update websites to allow him to reach his goals.

The rate of the boat in still water is 5 miles per hour and rate of the boat in current is 3 miles per hour.

Let us represent the rate of boat in still water hence and rate of boat in current be y. Also, we know that speed = distance/time. Hence, keep the values in formula -

Converting mixed fraction to fraction, time = 3/2 hour

Time = 1.5 hour

1.5 (x + y) = 12 : equation 1

Divide the equation 1 by 3

0.5 (x + y) = 4 : equation 2

6 (x - y) = 12 : equation 3

Divide the equation 3 by 6

(x - y) = 2

x = 2 + y : equation 4

Keep the value of x from equation 4 in equation 2

0.5 (2 + y + y) = 4

1 + y = 4

y = 4 - 1

y = 3 miles/ hour

Keep the value y in equation 4 to get x

x = 2 + 3

x = 5 miles per hour

The rate in still water and current is 5 and 3 miles per hour.

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The complete question is-

It takes a boat 1 (1/2) hr to go 12 mi downstream, and 6 hr to return. Find the rate of the boat in still water and the rate of the current.

The solution to the given equation which is log₃(1/9) = 2x - 1 is equal to x = -1/2.

To solve the equation log₃(1/9) = 2x - 1, we need to isolate the variable x on one side of the equation. We can start by using the logarithm property that states that the logarithm of a number to a base is equal to the exponent to which the base must be raised to obtain that number. In other words, log₃(1/9) = x if and only if [tex]3^x[/tex] = 1/9.

So, let's rewrite the given equation using this property as follows:

[tex]3^{(log(1/9))[/tex] = [tex]3^{2x-1[/tex]

Simplifying the left-hand side using the logarithm property, we get:

1/9 = [tex]3^{(2x - 1)[/tex]

Now, we can solve for x by taking the logarithm of both sides to base 3:

log₃(1/9) = log₃([tex]3^{(2x - 1)[/tex])

-2 = (2x - 1) * log₃(3)

-2 = 2x - 1

2x = -1

x = -1/2

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Answer: -10.57

Step-by-step explanation:

Answer:

0.25 years

Step-by-step explanation:

Penelope invested $89,000 in an account paying an interest rate of 6⅜% compounded continuously.

To calculate the time it would take Penelope's money to double, use the continuous compounding interest formula.

[tex]\boxed{\begin{minipage}{8.5 cm}\underline{Continuous Compounding Interest Formula}\\\\$ A=Pe^{rt}$\\\\where:\\\\ \phantom{ww}$\bullet$ $A =$ final amount \\\phantom{ww}$\bullet$ $P =$ principal amount \\\phantom{ww}$\bullet$ $e =$ Euler's number (constant) \\\phantom{ww}$\bullet$ $r =$ annual interest rate (in decimal form) \\\phantom{ww}$\bullet$ $t =$ time (in years) \\\end{minipage}}[/tex]

As the principal amount is doubled, then A = 2P.

Given interest rate:

Substitute A = 2P and r = 0.06375 into the continuous compounding interest formula and solve for t:

[tex]\implies 2P=Pe^{0.06375t}[/tex]

[tex]\implies 2=e^{0.06375t}[/tex]

[tex]\implies \ln 2=\ln e^{0.06375t}[/tex]

[tex]\implies \ln 2=0.06375t\ln e[/tex]

[tex]\implies \ln 2=0.06375t(1)[/tex]

[tex]\implies \ln 2=0.06375t[/tex]

[tex]\implies t=\dfrac{\ln 2}{0.06375}[/tex]

[tex]\implies t=10.872896949...[/tex]

Therefore, it will take 10.87 years for Penelope's investment to double.

[tex]\hrulefill[/tex]

Samir invested $89,000 in an account paying an interest rate of 6¹/₄% compounded monthly.

To calculate the time it would take Samir's money to double, use the compound interest formula.

[tex]\boxed{\begin{minipage}{8.5 cm}\underline{Compound Interest Formula}\\\\$ A=P\left(1+\frac{r}{n}\right)^{nt}$\\\\where:\\\\ \phantom{ww}$\bullet$ $A =$ final amount \\ \phantom{ww}$\bullet$ $P =$ principal amount \\ \phantom{ww}$\bullet$ $r =$ interest rate (in decimal form) \\ \phantom{ww}$\bullet$ $n =$ number of times interest is applied per year \\ \phantom{ww}$\bullet$ $t =$ time (in years) \\ \end{minipage}}[/tex]

As the principal amount is doubled, then A = 2P.

Given values:

Substitute the values into the formula and solve for t:

[tex]\implies 2P=P\left(1+\dfrac{0.0625}{12}\right)^{12t}[/tex]

[tex]\implies 2=\left(1+\dfrac{0.0625}{12}\right)^{12t}[/tex]

[tex]\implies 2=\left(1+0.005208333...\right)^{12t}[/tex]

[tex]\implies 2=\left(1.005208333...\right)^{12t}[/tex]

[tex]\implies \ln 2=\ln \left(1.005208333...\right)^{12t}[/tex]

[tex]\implies \ln 2=12t \ln \left(1.005208333...\right)[/tex]

[tex]\implies t=\dfrac{\ln 2}{12 \ln \left(1.005208333...\right)}[/tex]

[tex]\implies t=11.1192110...[/tex]

Therefore, it will take 11.12 years for Samir's investment to double.

[tex]\hrulefill[/tex]

To calculate how much longer it would take for Samir's money to double than for Penelope's money to double, subtract the value of t for Penelope from the value of t for Samir:

[tex]\begin{aligned}\implies t_{\sf Samir}-t_{\sf Penelope}&=11.1192110......-10.872896949...\\&= 0.246314066...\\&=0.25\; \sf years\;(nearest\;hundredth)\end{aligned}[/tex]

Therefore, it would take 0.25 years longer for Samir's money to double than for Penelope's money to double.

a) The probability that, for at least 15 packs, the average weight of the erasers in the pack is at least 31.95 g is approximately 0.0384.

b) The probability that, on average, the unit finds at least 37.2 "good" erasers per day is approximately 0.3133.

a) To solve this problem, we need to use the central limit theorem. According to this theorem, the distribution of sample means becomes approximately normal, regardless of the shape of the population distribution, when the sample size is sufficiently large (usually, n >= 30). In this case, since the sample size is 45, we can assume that the distribution of sample means will be approximately normal.

Now, we need to find the probability that the average weight of at least 15 packs is at least 31.95 g. We can use the normal distribution to calculate this probability. We first calculate the z-score for this value as follows:

z = (31.95 - 31.9) / (0.163 / √(45)) = 1.77

Using a standard normal table or calculator, we can find the probability that a z-score is greater than or equal to 1.77. This probability is approximately 0.0384.

b) To solve this problem, we need to use the normal approximation to the binomial distribution. Since each eraser is either "good" or "bad", the number of "good" erasers that the unit finds each day follows a binomial distribution with parameters n = 50 and p = probability of finding a "good" eraser = (32.3 - 31.7)/(32.3 - 31.5) = 0.5.

Now, we need to find the probability that, on average, the unit finds at least 37.2 "good" erasers per day. We can use the normal distribution to calculate this probability. We first calculate the z-score for this value as follows:

z = (37.2 - 25) / 25 = 0.488

Using a standard normal table or calculator, we can find the probability that a z-score is greater than or equal to 0.488. This probability is approximately 0.3133.

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The value of k is 11 at which a and 6 will be orthogonal (form a 90 degree angle).

To find the value of k that makes vectors a and 6 orthogonal, we need to use the dot product formula:

a · 6 = 2(-1) + 3(5) + (-1)k = 0

Simplifying the above equation, we get:

-2 + 15 - k = 0

Combining like terms, we get:

13 - k = 0

Therefore, k = 13.

However, we need to check if this value of k makes vectors a and 6 orthogonal.

a · 6 = 2(-1) + 3(5) + (-1)(13) = 0

The dot product is zero, which means vectors a and 6 are orthogonal.

Thus, the final answer is k = 11.

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The possible dimensions of the rectangular prism having volume  =  12 in³,  are  Length = 2 in, width = 3 in, height = 2/3 in, and Length = 1 in, width = 12 in, height = 1/12 in.

To find the possible dimensions of the prism, we need to consider that the volume of a rectangular prism is given by the formula V = lwh, where l, w, and h are the length, width, and height of the prism, respectively.

Since the volume of the prism is given as 12 in³, we can write: 12 = lwh
Now, we need to find two sets of dimensions that satisfy this equation, where one of the dimensions is a fraction.

Let's try the first set of dimensions:
l = 2 in
w = 3 in
h = 2/3 in

Plugging these values into the formula for the volume, we get:
V = lwh
V = 2 in × 3 in × 2/3 in
V = 4 in³

This confirms that the volume of the prism is indeed 12 in³, and that one of the dimensions (height) is a fraction.

Now, let's try another set of dimensions:
l = 1 in
w = 12 in
h = 1/12 in

Again, plugging these values into the formula for the volume, we get:
V = lwh
V = 1 in × 12 in × 1/12 in
V = 1 in³

This set of dimensions also satisfies the condition that the volume of the prism is 12 in³, with one of the dimensions (height) being a fraction.

Therefore, the possible dimensions of the prism are:
- Length = 2 in, width = 3 in, height = 2/3 in
- Length = 1 in, width = 12 in, height = 1/12 in.

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