Paula is using the formula
A = 550(1 + 0.05)t to represent the
amount of money A in her savings
account after t years.
Determine whether each statement
is true or false

The faster pipe can fill and drain the pool at a rate of 640.34 cubic feet per hour.

Reese is installing a rectangular pool in her backyard that is 30 feet long, 14 feet wide, and has an average depth of 8 feet. To fill and drain the pool, she is using two pipes. Let's call the slower pipe's rate of filling and draining the pool "r" (in units of volume per hour). Then, according to the problem, the faster pipe's rate is 2r+1 (since it is "1 more than 2 times faster" than the slower pipe).
If both pipes are open and working properly, we know it will take 3.5 hours to fill the pool. That means the total volume of the pool is:
V = length x width x depth
V = 30 ft x 14 ft x 8 ft
V = 3,360 cubic feet
We also know that when both pipes are open, they can fill the pool in 3.5 hours. That means the combined rate of filling the pool is:
V / t = (r + 2r+1)
3360 / 3.5 = 3r+1
960 = 3r+1
959 = 3r
r = 319.67 cubic feet per hour
So the slower pipe can fill and drain the pool at a rate of 319.67 cubic feet per hour. To find the rate of the faster pipe, we just need to substitute this value into our equation for the faster pipe's rate:
2r+1 = 2(319.67) + 1
2r+1 = 640.34
Therefore, the faster pipe can fill and drain the pool at a rate of 640.34 cubic feet per hour.

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The number of cones that can be filled with the ice cream from the container is 10.

Let's start with the container. We are given that it is a right circular cylinder with a diameter of 12 cm and a height of 15 cm. To find the volume of this cylinder, we use the formula:

Volume of cylinder = πr²h

where r is the radius of the cylinder (which is half of the diameter), and h is the height. Substituting the given values, we get:

Volume of cylinder = π(6 cm)²(15 cm) = 540π cubic cm

So the container has a volume of 540π cubic cm.

Now, let's move on to the cones. We are given that the cones have a height of 12 cm and a diameter of 6 cm. The cones have a hemispherical shape on the top, so we can consider them as a combination of a cone and a hemisphere. The formula for the volume of a cone is:

Volume of cone = (1/3)πr²h

where r is the radius of the base of the cone, and h is the height. Substituting the given values, we get:

Volume of cone = (1/3)π(3 cm)²(12 cm) = 36π cubic cm

The formula for the volume of a hemisphere is:

Volume of hemisphere = (2/3)πr³

where r is the radius of the hemisphere. Substituting the given values (the radius is half the diameter of the cone, which is 3 cm), we get:

Volume of hemisphere = (2/3)π(3 cm)³ = 18π cubic cm

So the total volume of each cone is:

Volume of cone + hemisphere = 36π + 18π = 54π cubic cm

To find out how many cones can be filled with the ice cream from the container, we divide the volume of the container by the volume of each cone:

Number of cones = Volume of container / Volume of each cone Number of cones

=> (540π cubic cm) / (54π cubic cm) Number of cones = 10

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Complete Question:

A container shaped like a right circular cylinder having diameter 12 cm and height 15 cm is full of ice cream. The ice cream is to be filled into cones of height 12 cm and diameter 6 cm, having a hemispherical shape on the top. Find the number of such cones which can be filled with ice cream.

Answer: 5(5w+6)

Step-by-step explanation:

Factor out a 5: 5(5w+6)

To check our work distribute: 25w+30

Answer: 5(5w+6)

Answer:

5 ( 5w + 5x )

Step-by-step explanation:

Just find a common factor in both terms: 5

5 ( 5w + 5x )

If you multiply again, you will see that the values of both expressions are the same.

A. The partial derivatives are not equal, F is not conservative, potential function f(x, y) = N

B. The partial derivatives are equal, F is conservative, potential function f(x, y) = -3xy - [tex]x^2[/tex] + C

C. The partial derivatives are not equal, F is not conservative, potential function f(x, y, z) = N

D. The partial derivatives are not equal, F is not conservative, potential function f(x, y, z) = N

E. The partial derivatives are not equal, F is not conservative, potential function f(x, y, z) = N

A. F(x, y) = (-6x – 7y) i +(-7x + 14y)j

To check if F is conservative, we compute the partial derivatives:

∂F/∂y = -6i - 7j

∂F/∂x = -7i + 14j

Since the partial derivatives are not equal, F is not conservative.

Potential function f(x, y) = N

B. F(x, y) = -3yi – 2xj

To check if F is conservative, we compute the partial derivatives:

∂F/∂y = -3i

∂F/∂x = -2j

Since the partial derivatives are equal, F is conservative.

Potential function f(x, y) =[tex]-3xy - x^2 + C[/tex], where C is a constant.

C. F(x, y, z) = -3xi – 2yj+k

To check if F is conservative, we compute the partial derivatives:

∂F/∂x = -3i

∂F/∂y = -2j

∂F/∂z = k

Since the partial derivatives are not equal, F is not conservative.

Potential function f(x, y, z) = N

D. F(x, y) = (-3 sin y)i + (-14y – 3x cosy)j

To check if F is conservative, we compute the partial derivatives:

∂F/∂y = -3cosy j - 14i

∂F/∂x = -3cosy j

Since the partial derivatives are not equal, F is not conservative.

Potential function f(x, y) = N

E. [tex]F(x, y, z) = -3xi - 7yj + 7z^2k[/tex]

To check if F is conservative, we compute the partial derivatives:

∂F/∂x = -3i

∂F/∂y = -7j

∂F/∂z = 14zk

Since the partial derivatives are not equal, F is not conservative.

Potential function f(x, y, z) = N

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The problem is asking for the probability of construction given that Max had to stop for a passing train on his route to work. This can be solved using Bayes' theorem, which states that the probability of A given B is equal to the probability of B given A multiplied by the probability of A, divided by the probability of B.

In this case, let A be the event that there is construction on Max's route, and let B be the event that Max has to stop for a passing train. We are looking for the probability of A given B.

Using Bayes' theorem, we have:

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

We know that P(B) = 0.5, the probability that Max has to stop for a passing train. We also know that P(B|A) = 0.4, the probability that there is construction and Max has to stop for a passing train.

To find P(A), the probability of construction on Max's route, we need to use the complement of the event A, which is the probability that there is no construction:

P(not A) = 1 - P(A) = 1 - 0.4 = 0.6

Finally, we can plug in the values and solve for P(A|B):

P(A|B) = 0.4 * 0.4 / 0.5 = 0.32

Therefore, the probability that there was construction if Max had to stop for a passing train on his route to work is 0.32 or 32%.

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a.The radius of the hot tub is 4 feet. b. The amount of water that the hot tub can hold is 176.96 cubic feet c. The amount of water that Rita should put in the hot tub in order to follow the recommendation is 141.57 cubic feet.

a. Find the radius of the hot tub.

Given: Circumference = 25.12 feet

Formula: Circumference = 2 * pi * radius

1: Plug in the given circumference and the value of pi.

25.12 = 2 * 3.14 * radius

2: Solve for the radius.

radius = 25.12 / (2 * 3.14)

radius ≈ 4 feet

So, the hot tub's radius is 4 feet.

b. How much water can the hot tub hold?

Given: Radius = 4 feet, Depth = 3.5 feet

Formula: Volume = pi * radius^2 * depth

1: Plug in the radius, depth, and the value of pi.

Volume = 3.14 * (4^2) * 3.5

2: Calculate the volume.

Volume ≈ 176.96 cubic feet

So, the hot tub can approximately hold 176.96 cubic feet.

c. How much water should Rita put in the hot tub to follow the recommendation?

Given: Recommended capacity = 80% of full capacity

Formula: Recommended water = 0.8 * full capacity

1: Plug in the full capacity (volume) calculated in part b.

Recommended water = 0.8 * 176.96

2: Calculate the recommended water amount.

Recommended water ≈ 141.57 cubic feet

So, Rita should put approximately 141.57 cubic feet of water in the hot tub to follow the recommendation.

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Answer:

9

Step-by-step explanation:

First, let's move the variables to one side and the numbers to the other side:

[tex]\frac{2}{3}b+5=20-b\\[/tex]

subtract 5 from both sides:

[tex]\frac{2}{3}b=15-b\\[/tex]

add b to both sides:

[tex]\\1\frac{2}{3}b=15\\[/tex]

divide both sides by [tex]1\frac{2}{3}[/tex]:

[tex]b=9[/tex]

Hope this helps :)

The simplest formula for working out probability is the following:

Probability (P) = Number of favorable outcomes (F) / Total number of possible outcomes (T)

In this formula, the probability of an event occurring is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. Probability is a measure of the likelihood or chance of an event occurring. It is represented as a value between 0 and 1, where 0 indicates an impossible event and 1 indicates a certain event.

To calculate the probability, you need to determine the number of favorable outcomes, which are the desired outcomes or the outcomes you are interested in. Then, you divide that by the total number of possible outcomes, which is the number of equally likely outcomes in the given situation.

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the completed multiplication table is:  40  45  50  ,72  54  60 and 56  90  70 .by using common factor  logic we can solve this question.

what is  common factor ?

A common factor is a number that divides evenly into two or more other numbers. For example, the common factors of 12 and 18 are 1, 2, 3, and 6, because all of these numbers divide evenly into both 12 and 18.

In the given question,

From the given table, we can see that:

The first row reads 40, 45, 50 (which are multiples of 5).

The second row has an unknown number (let's call it x), 54, and 60.

The third row reads 56, an unknown number (let's call it y), 70 (which are also multiples of 7).

To find the missing numbers, we can use the fact that multiplication is commutative, meaning that the order of the factors does not matter. Therefore, we can fill in the missing numbers by looking for factors that are common to both the row and column headers.

Starting with the second row, we can see that the common factor between x and 54 is 9, since 9 x 6 = 54 and 9 x x = ?. So, the missing number in the second row is 9 x 10 = 90.

Moving on to the first column, we can see that the common factor between 40 and 56 is 8, since 8 x 5 = 40 and 8 x 7 = 56. So, the missing number in the second row, first column is 8 x 9 = 72.

Finally, we can find the missing number in the first row, second column by finding the common factor between 45 and 60, which is 15, since 15 x 3 = 45 and 15 x 4 = 60. So, the missing number in the first row, second column is 15 x 5 = 75.

Therefore, the completed multiplication table is:

40  45  50

72  54  60

56  90  70

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Answer:

  [tex]\displaystyle\textsf{a) }\binom{8}{4}\\\\\textsf{b) }\binom{-8}{4}[/tex]

Step-by-step explanation:

Given a translation vector ...

  [tex]\displaystyle \binom{g}{h}[/tex]

moves g to the right and h up, you want the vectors for 8 right, 4 up, and for 8 left, 4 up.

When we have g = units to the right, and we want 8 units to the right, we know that g = 8. Similarly, h = units up, and we want 4 units up, so h = 4.

Putting these values in the vector form, we have ...

a)   8 right, 4 up matches vector ...

  [tex]\displaystyle \boxed{\binom{8}{4}}[/tex]

b)  Left is the opposite of right, so 8 units left will be represented by ...

  g = -8

As before, 4 units up means h = 4.

  [tex]\displaystyle \boxed{\binom{-8}{4}}[/tex]

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The calculated value of x in the circle is 12.3

From the question, we have the following parameters that can be used in our computation:

The circle

∠QRS = 7x - 21

QS = 130

Using the theorem of intersecting chords, we have the following equation

∠QRS = 1/2 * QS

Substitute the known values in the above equation, so, we have the following representation

7x - 21 = 1/2 * 130

Evaluate

7x - 21 = 65

Evaluate the like terms

7x = 86

Divide by 7

x = 12.3

Hence, the value of x in the circle is 12.3

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Answer: ⬇️⬇️

Step-by-step explanation:

In the expression 5 x y/7, the value of y that would make a product greater than 5 is 8. 

HOW TO SOLVE ALGEBRAIC EXPRESSIONS?

According to this question, the following algebraic equation was given:

5 x y/7

This equation reveals that the result can only be equal to 5 when y is exactly 7.

This is because if y = 7, y/7 = 1.

 Therefore, the value of y that would make a product greater than 5 is 8.

a) We can conclude that there is significant evidence that the probability of heads is not 0.5.

b) A 95% confidence interval for the true probability of heads is (0.4872, 0.5262).

a) To test whether the probability of heads is significantly different from 0.5, we can use a two-tailed z-test with a significance level of 0.05. The null hypothesis (H₀) is that the probability of heads is 0.5, while the alternative hypothesis (Hₐ) is that it is not 0.5.

The test statistic is given by:

z = (x - np) / √(np(1-p))

where x is the number of heads observed (5067), n is the total number of coin tosses (10,000), and p is the hypothesized probability of heads under the null hypothesis (0.5).

Plugging in the values, we get:

z = (5067 - 5000) / √(10,000 * 0.5 * 0.5) = 2.20

The P-value for this test is the probability of getting a z-score greater than 2.20 or less than -2.20, which is approximately 0.0287. Since the P-value is less than the significance level of 0.05, we reject the null hypothesis and conclude that there is significant evidence that the probability of heads is not 0.5.

b) To find a 95% confidence interval for the true probability of heads, we can use the formula:

p ± z*√(p(1-p)/n)

where p is the sample proportion (5067/10000), n is the sample size (10,000), and z is the critical value from the standard normal distribution corresponding to a 95% confidence level (1.96).

Plugging in the values, we get:

p ± 1.96*√(p(1-p)/n) = 0.5067 ± 0.0195

So a 95% confidence interval for the true probability of heads is (0.4872, 0.5262). This means that we can be 95% confident that the true probability of heads falls within this interval based on the observed sample proportion.

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Answer:

54% × 50000= 27000, the answer is 27000

The measure of the angle is 40 degrees.

Let x be the measure of the angle and y be its complement.

The sum of an angle and its complement is 90 degrees, so we have:

[tex]x+y=90[/tex]

Also, we know that "the measure of an angle of it is 160 less than 4 times its complement", which can be written as:

[tex]x=4y-160[/tex]

Now we can substitute the first equation into the second equation:

[tex]4y-160+y=90[/tex]

Simplifying and solving for y, we get:

5y = 250

y = 50

Substituting y = 50 into the first equation gives:

x + 50 = 90

x = 40

Therefore, the measure of the angle is 40 degrees.

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It is not possible for Zachary to buy a burger, a souvenir, and a pass with the given prices and have $4.50 left over.

Let's represent the cost of the souvenir as x.

Then, according to the problem:

The cost of the burger is 1/6 of the cost of the souvenir, which is (1/6)x.

The cost of the pass is 4/3 times the cost of the souvenir, which is (4/3)x.

The total cost of the burger, souvenir, and pass is equal to the amount Zachary brought to the state fair, which is $38.25.

Zachary had $4.50 left over after buying these items, so the cost of the burger, souvenir, and pass must be $33.75.

Putting all this information together, we can write an equation:

(1/6)x + x + (4/3)x = 33.75

Simplifying the left side of the equation:

(7/6)x = 33.75

Multiplying both sides by 6/7:

x = 30

Therefore, the cost of the souvenir is $30, the cost of the burger is (1/6) * 30 = $5, and the cost of the pass is (4/3) * 30 = $40.

To check that these values are correct, we can add them up:

30 + 5 + 40 = 75

And we can subtract the total cost from the amount Zachary brought:

38.25 - 75 = -36.75

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There is a probability that 105 of the 350 students would select May 9 for the dance.

To solve the above we need to find the proportion of students in the sample who voted for each date and this can be done by:

May 2:  18/70

  = 0.257

May 9: 39/70

  = 0.557

May 16: 13/70

   = 0.186

if the whole population of 350 students voted, then

May 2: 0.257 x 350

   =  90

May 9: 0.557 x 350

 = 195

May 16: 0.186 x 350

  = 65

From the above calculation, we can see that if 105 students out of 350) were to choose a date, the most likely date that they will select is  May 9, since it is the one that has the highest proportion of votes in the sample.

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The revenue of the chairs sold as per given R(x) = 0.004x³ -0.04x² +0.6x for different conditions are,

The current daily revenue is $2646.

Increase in revenue for 92 chairs sold every day is $185.16

Marginal revenue is $90.6 per lawn chair for 92 chairs sold every day.

Estimated revenue for R(91), R(92) and R(93) is equal to $2555.4 , $2464.8, and $2374.2 respectively.

Daily revenue from sale of x chairs is,

R(x) = 0.004x³ -0.04x² +0.6x

The current daily revenue can be found by evaluating the function at x = 90,

R(90) = 0.004(90)³ - 0.04(90)² + 0.6(90)

        = 2916 - 324 + 54

        = $2646

Increase in revenue,

⇒ difference between the revenue from selling 92 lawn chairs and the revenue from selling 90 lawn chairs,

R(92) - R(90)

= [0.004(92)³ - 0.04(92)² + 0.6(92)] - [0.004(90)³ - 0.04(90)² + 0.6(90)]

= 3114.752 -338.56 + 55.2 - 2646

= 2831.16 - 2646

= $185.16

Revenue would increase by$185.16 if 92 lawn chairs were sold each day.

The marginal revenue is the derivative of the revenue function,

R'(x) = 0.012x² - 0.08x + 0.6

Marginal revenue when 90 lawn chairs are sold daily,

we can evaluate the derivative at x = 90,

R'(90) = 0.012(90)² - 0.08(90) + 0.6

          = $90.6

When 90 lawn chairs are sold daily, the marginal revenue is $90.6 per lawn chair.

Use the answer from above part to estimate the revenue from selling 91, 92, and 93 lawn chairs daily.

Assume that the marginal revenue is approximately constant in a small interval around 90,

Use the linear approximation,

R(91) ≈ R(90) + R'(90)(1)

       = $2646 + $90.6

       = $2555.4

R(92) ≈ R(90) + R'(90)(2)

        = $2646 + 2($90.6)

        = $2464.8

R(93) ≈ R(90) + R'(90)(3)

         = $2646 + 3($90.6)

         = $2374.2

If 91, 92, and 93 lawn chairs were sold daily,

The estimated daily revenue would be $2555.4 , $2464.8, and $2374.2 respectively.

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The above question is incomplete, I answer the question in general according to my knowledge:

Pierce Manufacturing determines that the daily revenue, in dollars, from the sale of x town chairs is

R(x) = 0.004x³ -0.04x² +0.6x .

Currently, Pierce sells 90 lawn chairs daily.

a) What is the current daily revenue?

b) How much would revenue increase if 92 lawn chairs were sold each day?

c) What is the marginal revenue when 90 lawn chairs are sold daily

d) Use the answer from part (c) to estimate R(91), R(92) and R(93)

Answer:

75 pounds of fudge & 80 pounds of toffee per month.

Step-by-step explanation:

Let qty of fudge be x pounds & qty of toffee be y pounds per month

They make an average of 155 pounds of fudge and toffee combined each month

x+y=155 (1)

They sell a pound of fudge for $8.40 and a pound of toffee for $7.28. Mandy's averages $1,212.40 each month in fudge and toffee sales.

So the cost of monthly fudge&toffee is 8.40*x+7.28*y=1,212.40

8.40x+7.28y=1212.40 ;(2)

multiply eq (1) by 8.40 & then subtract 2 from 1

8.40x+8.40y=155*8.40

8.40x+8.40y=1,302 (1')

8.40x+7.28y=1212.40 ;(2)

-----------------------

(8.40-7.28)y=1,302-1,212.40

1.12y=89.60

y=89.60/1.12=80 pounds of toffee

In order to solve for x, take the value for y and substitute it back into either one of the original equations.

x+80=155

x=155-80=75 pounds

So they make 75 pounds of fudge & 80 pounds of toffee per month.

The slope of the line is negative, the correct answer is (A) m is negative.

The formula for equation of line is expressed as;

y = mx + b

Where m is slope and b is y-intercept.

Given that; that the graph of a linear function y = mx + 2 is a straight line with slope m and y-intercept (0,2).

Also, the line passes through the point (4,0), we can use this point to find the value of the slope m.

0 = m(4) + 2

Solve for m

0 = 4m + 2

4m = -2

m = -2/4

m = -1/2

Hence, the slope m is negative.

Option A is the correct answer.

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The distance which this light travel per second is equal to 3 × 10⁸ meters per seconds.

In Mathematics and Science, speed is the distance covered by a physical object per unit of time.

In Mathematics and Science, the speed of any a physical object can be calculated by using this formula;

Speed = distance/time

By making distance the subject of formula, we have:

Distance, d(t) = speed × time

Distance = (9.45 × 10¹⁵ meters per year) × (1 year/ 3.15 × 10⁷ seconds)

Distance = 3 × 10⁸ meters per seconds.

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Complete Question:

Light travels 9.45 × 10¹⁵ meters in a year. There are about 3.15 × 10⁷ seconds in a year. How far does light travel per second?

The total number of Huns that migrated to the West is 111.

To find the total number of Huns that migrated to the West, we can use the expression:

Total number of Huns = Number of groups x Number of people per group

For the front and back sections, there were 3 groups of 5 people each, so the total number of Huns in each section is:

3 x 5 = 15

For the middle section, there were 9 groups of 9 people each, so the total number of Huns in the middle section is:

9 x 9 = 81

To find the total number of Huns that migrated to the West, we add up the number of Huns in each section:

15 + 81 + 15 = 111

Therefore, the total number of Huns that migrated to the West is 111.

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Answer: 452.39

Step-by-step explanation: R= 6, H = 12. Volume formula is V= pi*r^2*h/3. For you, the equation is V = pi*6^2*12/3. 6^2 = 36, and 12/3 = 4. V = pi*36*4. Solving, we get V = 452.39.

This is an unfavorable variance.

To calculate the dollar variance, we subtract the actual amount from the budgeted amount:

Dollar variance = Budgeted amount - Actual amount = $106.00 - $100.00 = $6.00 (favorable)

The dollar variance of $6.00 suggests that the actual expenses were less than the budgeted expenses, which is a favorable variance.

To calculate the percentage variance, we use the following formula:

Percentage variance = (Budgeted amount - Actual amount) / Budgeted amount x 100%

Substituting the values, we get:

Percentage variance = ($106.00 - $100.00) / $106.00 x 100% = 5.66% (rounded to the nearest whole percent)

The percentage variance of 5.66% suggests that the actual expenses exceeded the budgeted expenses by 5.66%, which is an unfavorable variance.

It's important to note that the dollar variance and percentage variance provide different perspectives on the variance, and they should be considered together to fully understand the implications of the variance. In this case, the dollar variance is favorable, indicating that the company spent less than expected.

However, the percentage variance is unfavorable, indicating that the company's expenses were higher than budgeted. The company may use this information to identify areas where they can reduce expenses in the future or adjust their budgeting process to be more accurate.

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Answer:

Step-by-step explanation:

To solve the equation, we'll need to consider two cases:

Case 1: 2x - 3 is positive or zero

If 2x - 3 is positive or zero, then |2x - 3| = 2x - 3, and the inequality becomes:

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

Rearranging and simplifying:

x^2 - x - 6 < 0

Factoring:

(x - 3)(x + 2) < 0

The solutions to this inequality are:

-2 < x < 3

However, we still need to check that 2x - 3 is indeed positive or zero for this range of x. We can see that this is true for x in the range (-2, 3), so this is a valid solution.

Case 2: 2x - 3 is negative

If 2x - 3 is negative, then |2x - 3| = -(2x - 3), and the inequality becomes:

-(2x - 3) < 3 + x - x^2

Rearranging and simplifying:

x^2 - 3x - 6 < 0

Factoring:

(x - 3)(x + 2) > 0

The solutions to this inequality are:

x < -2 or x > 3

However, we still need to check that 2x - 3 is indeed negative for this range of x. We can see that this is true for x < -2, so this is a valid solution.

Putting these two cases together, we get the solution:

x < -2 or -2 < x < 3

I hope this helps! Let me know if you have any questions.

If there are 750 contestants on Monday, Jessica should expect around 75 of them to dunk a teacher.

Based on the table provided, we can see that the percentage of winners at the dunk-a-teacher booth varies from day to day.

On Monday, 10% of contestants were able to dunk a teacher.

Therefore, if there are 750 contestants on Monday, Jessica should expect around 75 of them to dunk a teacher.

This is calculated by multiplying 750 by 0.10, which gives us 75. It's important to note that this is just an estimation, as the actual number of winners may be slightly higher or lower than 75.

However, this gives Jessica a rough idea of what to expect at the booth on Monday.

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The car mechanic is correct in saying that they will have used over 80% of the oil in the tin after 9 weeks.

To determine if the car mechanic is correct, we first need to calculate how much oil they will use in 9 weeks.

450 millilitres of oil are used each week, so after 9 weeks, they will have used:

450 x 9 = 4050 millilitres

Next, we need to convert this to litres, since the oil tin is measured in litres.

There are 1000 millilitres in 1 litre, so:

4050 ÷ 1000 = 4.05 litres

Therefore, after 9 weeks, the car mechanic will have used 4.05 litres of oil.

Now we need to determine if this is over 80% of the total oil in the tin.

The tin contains 5 litres of oil, so we need to find 80% of 5:

5 x 0.8 = 4

So if the car mechanic has used more than 4 litres of oil in 9 weeks, they have used over 80% of the oil in the tin.

We know from earlier that they will have used 4.05 litres, which is slightly over 80%. Therefore, the car mechanic is correct in saying that they will have used over 80% of the oil in the tin after 9 weeks.

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The mean amount of D-glucose in cockroach hindguts is 44.24 micrograms, with a standard deviation of 26.16 micrograms. The 96% confidence interval for the mean is (20.19, 68.29) micrograms.

We will first calculate the mean, standard deviation, and then the 96% confidence interval for the given data on D-glucose amounts in cockroach hindguts.

Data: 55.95, 68.24, 52.73, 21.5, 23.78

1. The mean is calculated as follows.

Mean = (55.95 + 68.24 + 52.73 + 21.5 + 23.78) / 5 = 221.2 / 5 = 44.24

The mean amount of D-glucose in cockroach hindguts is approximately 44.24 micrograms.

2. To find the standard deviation, we calculate the variance first:

Variance = [(55.95 - 44.24)^2 + (68.24 - 44.24)^2 + (52.73 - 44.24)^2 + (21.5 - 44.24)^2 + (23.78 - 44.24)^2] / (5 - 1) = 2739.736 / 4 = 684.934

Standard Deviation = √684.934 = 26.16 (rounded to two decimal places)

The standard deviation of D-glucose in cockroach hindguts is approximately 26.16 micrograms.

3. To calculate the 96% confidence interval, we need to find the margin of error:

Margin of Error = (Critical value * Standard Deviation) / √sample size

For a 96% confidence interval, the critical value (z-score) is approximately 2.05.

Margin of Error = (2.05 * 26.16) / √5 ≈ 24.05

Now, we can find the confidence interval:

Lower limit: Mean - Margin of Error = 44.24 - 24.05 = 20.19

Upper limit: Mean + Margin of Error = 44.24 + 24.05 = 68.29

The 96% confidence interval for the mean amount of D-glucose in cockroach hindguts approximately is (20.19, 68.29) micrograms.

Note: The question is incomplete. The complete question probably is: To study the metabolism of insects, researchers fed cockroaches measured amounts of a sugar solution. After 2, 5, and 10 hours, they dissected some of the cockroaches and measured the amount of sugar in various tissues. Five roaches fed the sugar D-glucose and dissected after 10 hours had the following amounts (in micrograms) of D-glucose in their hindguts: Amounts of D-glucose 55. 95, 68. 24, 52. 73, 21. 5, 23. 78. The researchers gave a 96% confidence interval for the mean amount of D-glucose in cockroach hindguts under these conditions. The insects are a random sample from a uniform population grown in the laboratory. We therefore expect responses to be Normal. What is the mean, standard deviation, and confidence interval?

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The given expression is sin⁻¹ (sin((-5t-6)). Since the argument of sin⁻¹ and sin is the same, we can simplify the expression as follows:
sin⁻¹ (sin((-5t-6))) = -5t-6
OR, -5t-6 = (-2π/π)(-5t-6/2) = -2π(2.5t+3)/π = -5π/2(2.5t+3)
Therefore, the answer is -5π/2(2.5t+3).

Given the expression: sin^-1(sin(-5t-6))
To find the angle -z/2, we can use the following properties:
1. sin⁻¹ (sin(x)) = x, if -π/2 ≤ x ≤ π/2 (i.e., x is in the range of the principal branch of the inverse sine function).
2. The sine function has a periodicity of 2π. Therefore, sin(x) = sin(x + 2nπ), where n is an integer.

Given angle: -5t - 6
We need to add 2nπ to this angle to bring it into the range of -π/2 to π/2:
⇒ -5t - 6 + 2nπ, where n is an integer.

Now, we apply the sine and inverse sine functions:
sin⁻¹ (sin(-5t - 6 + 2nπ))

Since sin^-1(sin(x)) = x when x is in the range of the principal branch, our final expression becomes:
-z/2 = -5t - 6 + 2nπ

In this expression, -z/2 represents the angle in radians, written as a multiple of r. To find the multiple, you simply have to solve for -z/2 in terms of r.
Therefore, the answer is: -z/2 = -5t - 6 + 2nπ.

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The measure of the exterior angle of the triangle is 128°.

Trigonometry deals with the relationship between the ratios of the sides of a right-angled triangle with its angles.

Trigonometric functions are the functions that denote the relationship between angle and sides of a right-angled triangle.

Sin θ = Opposite Side/Hypotenuse

Cos θ = Adjacent Side/Hypotenuse

Tan θ = Opposite Side/Adjacent

Recall that the measure of the exterior angle of a triangle is the sum of the opposite interior angles. That is:

x = 38 + 90

x = 128°

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