for each of the indefinite integrals below, choose which of the following substitutions would be most helpful in evaluating the integral. enter the appropriate letter (a,b, or c) in each blank. do not evaluate the integrals. A. x= 6 tan θ B. x= 6 sin θ C. x= 6 sec θ 1. Integral (x^2 - 36)^5/2 dx 2. Integral (x^2 dx/ (root 36+x^2) 3. Integral (dx/ (root 36+x^2)^3 4. Integral (root x^2+36) dx 5. Integral x^2 (root 36+x^2)

Answer:

x-3 =0 or x-8 =0

x= +3 or x = +8

On solving the system of linear equations 3x + 3y = 45 and x + 2y = 24, the cost for each shirt is obtained as $6, and cost for each pair of pants is obtained as $9.

What is a linear equation?

A linear equation is one that has a degree of 1 as its maximum value. No variable in a linear equation, thus, has an exponent greater than 1. A linear equation's graph will always be a straight line.

Let the cost of shirts be x and the cost of pair of pants be y.

The number of shirts bought by person A = 3

The number of pair of pants bought by person B = 3

The total cost for person A = $45

The equation for person A is -

3x + 3y = 45....(1)

The number of shirts bought by person B = 1

The number of pair of pants bought by person B = 2

The total cost for person B = $24

The equation for person B is -

x + 2y = 24....(2)

Multiply equation (2) by 3 -

3(x + 2y) = 24 × 3

3x + 6y = 72....(3)

Subtraction equation (1) from equation (3) -

3x + 6y - (3x + 3y) = 72 - 45

3x + 6y - 3x - 3y = 27

3y = 27

y = 9

Substitute the value of y in equation (2) -

x + 2(9) = 24

x + 18 = 24

x = 24 - 18

x = 6

Therefore, the cost of shirts and pants is x = 6 and y = 9 respectively.

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The slopes of the two lines that form them must be equal (m1 = m2). This implies that the two lines are parallel.

1) Biconditional statement: If two lines are parallel, then the alternate interior angles formed by a transversal are equal.

Formula: m∠1 = m∠3 and m∠2 = m∠4

Calculation: If two lines are parallel with a transversal intersecting them, the slopes of the two lines must be equal (m1 = m2) and the alternate interior angles formed by the transversal must have the same measure (m∠1 = m∠3 and m∠2 = m∠4).

2) Biconditional statement: If two lines are parallel, then the corresponding angles formed by a transversal are equal.

Formula: m∠1 = m∠4 and m∠2 = m∠3

Calculation: If two lines are parallel with a transversal intersecting them, the slopes of the two lines must be equal (m1 = m2) and the corresponding angles formed by the transversal must have the same measure (m∠1 = m∠4 and m∠2 = m∠3).

3) Biconditional statement: If two angles are equal, then the lines that form them are parallel.

Formula: m1 = m2

Calculation: If two angles have the same measure, then the slopes of the two lines that form them must be equal (m1 = m2). This implies that the two lines are parallel.

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Answer:

100

Step-by-step explanation:

total angles of any triangle is 180

it tells you one of the base angles is 40, so 2 of them are 80

so the top one is 180-80 =100

An equation that represents the linear function shown in the table or mapping diagram is y = -x/4 - 3.

In order to write an equation that represents the linear function, we would use the point-slope form. Mathematically, the point-slope form of a straight line can be calculated by using this mathematical expression:

y - y₁ = m(x - x₁) or y - y₁ = (y₂ - y₁)/(x₂ - x₁)(x - x₁)

Where:

Next, we would determine the linear function representing the line which passes through the data points (-4, 6) and (8, 3) by using the point-slope form as follows:

y - y₁ = (y₂ - y₁)/(x₂ - x₁)(x - x₁)

y - 6 = (3 - 6)/(8 - (-4))(x - (-4))

y - 6 = (3 - 6)/(8 + 4)(x + 4)

y - 6 = -3/12(x + 4)

y - 6 = -x/4 - 9

y = -x/4 - 9 + 6

y = -x/4 - 3

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The copayment amount, given that it is 20 % of the amount is $ 840

The benefit amount is  $ 3, 360

A copayment for health insurance is a defined dollar amount that is determined by the insurance plan and used to divide the cost of covered treatments between the plan and the consumer.

The copayment amount is 20 % of the cost which is $ 4, 200 so the copayment is:

= 20 % x 4, 200

= $ 840

This then means that the benefit amount would be;

= Cost - copayment

= 4, 200 - 840

= $ 3, 360

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Answer:

14

Step-by-step explanation:

We can divide 76,450 shavings by the average shavings per pencil, 5,425 to get an answer of 14.09... This can be rounded to 14.

A rectangle having perimeter 26 m has a length of 8m and breadth of 5m.

What is perimeter?

The complete length of a shape's boundary is referred to as the perimeter in geometry. A shape's perimeter is calculated by adding the lengths of all of its sides and edges. Its dimensions are expressed in linear units like centimetres, metres, inches, and feet.

The value for perimeter of a rectangle is - 26 m

Let x be the length of the rectangle and y be the breadth.

The formula for perimeter of rectangle is 2(l + b)

From the given information, it is known that -

Perimeter = 2(x + y) = 26 m...(1)

Difference between the length and breadth = x - y = 3 m....(2)

Use the first equation to solve for one of the variables in terms of the other -

2x + 2y = 26

x + y = 13....(3)

Add equation (2) and (3) -

x + y + x - y = 13 + 3

2x = 16

x = 16/2

x = 8 m

Now substitute the value of x in equation (2) -

x - y = 3

8 - y = 3

y = 8 - 3

y = 5 m

Therefore, the length is 8 m and the breadth is 5 m.

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If Bob is able to produce more donuts than Sue in one day, that means that Bob has an absolute advantage in the production pf donuts.

Absolute advantage is when a particular person is able to produce more amount of product than the other with the same amount of input of resources, for example, time. Comparative advantage on the other hand is based on the opportunity cost.

Comparative advantage is basically the ability to produce a good or a service for an opportunity cost which is lower than that of a competitor. Since, Bob is able to produce more donuts than Sue, Bob has an absolute advantage over Sue.

--The given question is incomplete, the complete question is

If Bob can produce more donuts in one day than Sue can produce in one day, then:

a. Bob has a comparative advantage in the production of donuts.

b. Sue has a comparative advantage in the production of donuts.

c. Bob has an absolute advantage in the production of donuts.

d. Sue has an absolute advantage in the production of donuts."--

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The graph of the inequality y > -10 is given by the image presented at the end of the answer.

The four inequality symbols, along with their meaning, are presented as follows:

The inequality for this problem is given as follows:

y > -10.

Which means that the solution is the numbers that are greater = to the right of -10, with an open interval.

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Answer:

see below

Step-by-step explanation:

Step 1

5(x – 3) = x + 13

Step 2

See if x is a solution

5 (6-3) ?= 6+13

5 (3) ?= 19

15 ?= 19

Step 3

This is not true so it is not a solution

Step 4

5(x – 3) = x + 13

Distribute

5x-15 = x+3

Subtract x from each side

5x-x -15 = x+13-x

4x-15 = 13

Add 15 to each side

4x-13+13 = 15+13

4x =28

Divide each side by 4

4x/4 = 28/4

x = 7

7 is a solution

Answer:

First you substitute 6 for x!

5(6 - 3) = 6 + 13

6 - 3 = 3

The equation is now:

5 * 3 = 6 + 13

5 * 3 = 15

6 + 13 = 19

The result is:

15 = 19 which is not a true statement.

x is not equal to 6.

Now lets try 7.

5(7 - 3) = 7 + 13

7 - 3 = 4

7 + 13 = 20

5 * 4 = 20

Thus: x = 7

Step-by-step explanation:

Hope it helps! =D

50 fishing lures are carved in a week.

A) Big Lake Bob

Production Possibilities Schedule:

Fishing Lures    Duck Decoys

100                          0

90                          5

80                         10

70                         15

60                       20

50                       25

40                       30

30                       35

20                       40

10                       45

0                       50

50 fishing lures are carved in a week.

25 duck decoys are crafted in a week.

This implies that Big Lake Bob forgoes 10 fishing lures for every 5 duck decoys he carves.

Large Lake Schedule of Bob's productivity potentials

B) Bob's potential for production The schedule is a table of numbers that shows the various combinations of two items that can be created using the restricted resources (labor hours, capital, materials, etc.) and technology in Bob's economy.

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The smaller number is 25 and larger number is 37

Linear equations help in representing the relationship between variables such as x, y, and z, and are expressed in exponents of one degree. In these linear equations, we use algebra, starting from the basics such as the addition and subtraction of algebraic expressions.

Given here; The sum of two numbers is 62 and 3 times the smaller number exceeds twice the larger number by one.

let the smaller number be x and the larger number be y then we have

x+y=62

3x-2y=1

Thus solving this system of linear equations by method of elimination we get

5y=185

y=37

∴ x=25

Hence, The smaller number is 25 and larger number is 37

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The numeric values of the function R(t) = 328 - 296e^(-0.004t) at t = 0 and t = 30 are given as follows:

To find the numeric value of a function or of an expression, we replace each instance of the variable in the function or in the expression by the value at which we want to find the numeric value.

The function for this problem is defined as follows:

R(t) = 328 - 296e^(-0.004t)

At t = 0, the numeric value is found replacing the lone instance of t by zero, hence:

R(0) = 328 - 296e^(-0.004(0))

R(0) = 32.

At t = 30, the numeric value is found replacing the lone instance of t by 30, hence:

R(30) = 328 - 296e^(-0.004(30))

R(30) = 65.47.

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According to the empirical rule -

a) About 95% of organs will be in range of weights of organs 230 grams and 430 grams.

b) 68% percentage of organs weighs between 280 grams and 380 grams.

c) 32% percentage of organs weighs less than 280 grams or more than 380 grams.

d) 97.35% percentage of organs weighs between 180 grams and 430 grams.

What is empirical rule?

The empirical rule 68-95-99.7 tells us that:

68% of the data is expected to be within 1 standard deviation from the mean.

95% of the data is expected to be within 2 standard deviation from the mean.

99.7% of the data is expected to be within 3 standard deviation from the mean.

There is a bell shaped distribution ( assume as approximately normal) with mean of 330 g and standard deviation of 50 g.

a) This happens for an interval with ±2 standard deviations from the mean.

That is:

X1 = μ + z1 × σ                  X2 = μ + z2 × σ

X1 = 330 - 2 × 50             X2 = 330 + 2 × 50

X1 = 330 - 100                  X2 = 330 + 100

X1 = 230                           X2 = 430

Therefore, the range of weights of organs is 230 grams and 430 grams.

b) Calculate the z-scores for each value to know how many standard deviations are from the mean.

z1 = (x1 - μ)/σ                   z2 = (x2 - μ)/σ

z1 = (280 - 330)/50        z2 = (380 - 330)/50

z1 = -50/50                     z2 = 50/50

z1 = -1                              z2 = 1  

The values are 1 standard deviations from the mean each.

Therefore, the expected percentage is 68%.

c) This is the complementary of the point b.

Then, it is expected that (100 - 68)% = 32% of the organs weigh less than 280 grams or more than 380 grams.

d) Calculate the z-scores for each value to know how many standard deviations are from the mean.

z1 = (x1 - μ)/σ                   z2 = (x2 - μ)/σ

z1 = (180 - 330)/50        z2 = (430 - 330)/50

z1 = -150/50                     z2 = 100/50

z1 = -3                              z2 = 2  

The values are 1 standard deviations from the mean each.

Therefore, the expected percentage is 68%.

The values are 3 standard deviations from the mean each. Between 180 and 330 there is 99.7%/2 = 49.85% of the data.

The values are 2 standard deviations from the mean each. Between 330 and 430 there is 95%/2 = 47.5% of the data.

Then, between 180 grams and 430 grams there is (49.85% + 47.5)% = 97.35% of the data.

Therefore, the expected percentage is 97.35%.

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Answer:

The biologist goes to his 150 favorite hiking places and looks at an acre along each trail: Random Sampling.

The biologist orders the 15 million acres by latitude and then surveys every 100000th acre on the list: Stratified Sampling.

The biologist classifies the Sierras into 10 different ecological types and then makes sure that the proportion of each ecological type from the sample is the same as the proportion of that of the population: Stratified Sampling.

The biologist goes to 15 diverse locations that have 10 acres each and surveys all 10 acres for each of these 15 locations: Cluster Sampling.

The biologist enters the 15 million acres into a data base and had a computer randomly select 150 of these acres: Random Sampling.

The answers include the following below:

This is referred to as the discounted price at which goods or services are being sold.

Since we were told that the discount is 15%, then the sale price will be (100-85)% of $30 which is 85% of $30.

This therefore means that the equation to find the sale price will be 85/100 × 30 = 0.85(30).

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We have the product of 4 negative numbers, by the rule of signs, we can see that the product is positive.

Here we need to reember the rule of signs for the product, it is:

(-)*(-) = +

(+)*(+) = +

(+)*(-) = -

(-)*(+) = -

So the product between two negative numbers is positive, now let's look to our product.

(-1.2)*(-3.5)*(-2.7)*(-0.4)

We can rewrite this as:

(-1.2*-3.5)*(-2.7*-0.4)

The negative sings cancell, then we get:

(-1.2*-3.5)*(-2.7*-0.4)

(4.2)*(1.08)

Now we have the product of two positive numbers, this will be positive.

(4.2)*(1.08) = 4.536

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An inequality with two variables has a graph, which is a collection of points that shows all possible solution is x≤ 5 3/10.

The boundary line between the coordinate plane's two halves, one of which represents the inequality's solutions, is created by a linear inequality.

x + 1 1/2 ≤ 6 8/10 :

Solution : x ≤ 5 3/10

Decimal : x ≤ 5.3

Interval  notation : (-∝ , 53/10)

Steps:

x + 1 1/2 ≤ 6 8/10 :

Convert mixed Number to improper Fraction : 1 1/2  = 3/2

1 1/2

Convert mixed numbers to improper fraction ; a (b/c ) = a. c +b /c

1 1/2 = 1.2 + 1 /2

=3/2.

x + 1 1/2 ≤ 6 8/10 :

Convert mixed Number to improper Fraction : 6 8/10 = 68/10

6 8/10

Convert mixed numbers to improper fraction ; a (b/c ) = a. c +b /c

6 8/10 = 6.10 + 8 /10

=68/10

x+ 3/2 ≤ 68/10

Cancel the Common factor :2

x + 3/2 ≤ 34/5

Move 3/2 to the right side

x ≤ 53/10

Convert  improper Fraction  to  mixed Numbers: 53/10 = 5 3/10

53/10 = 5 Reminder 3

Convert to mixed Number : Quotient (Remainder / Divisor)

53/10 = 5 3/10

x≤ 5 3/10

x≤ 5 3/10.

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If f(x) = 3x and g(x)=x-2, then the value of f(g(4)) is 6.

What is a function?

A function is a mathematical object that assigns exactly one output to each input of a specified type. In other words, a function is a rule that takes in certain inputs (or "arguments") and produces a unique output. The inputs to a function are called the domain, while the set of possible outputs is called the range.

The given functions are f(x) = 3x and g(x)=x-2.

To find f(g(4)), we first need to evaluate g(4). The function g is defined as g(x) = x - 2, so g(4) = 4 - 2 = 2.

Next, we need to evaluate f(2), where f is defined as f(x) = 3x. So, f(2) = 3 * 2 = 6.

Therefore, f(g(4)) = f(2) = 6.

Hence, if f(x) = 3x and g(x)=x-2, then the value of f(g(4)) is 6.

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Answer:

39.9

Step-by-step explanation:

Rectangle: area = length × width

Semicircle: area = 0.5πr²

total area = 9 × 6 - 0.5 × 3.14159 × 3²

total area = 39.9

The probability of a claim of at least $4000 is 0.35, option(d), the expected value of C is $2770 option (c), and the best interpretation is If the company insures a large number of these cars, they can expect the cost per car to average approximately Exp(C), option(d).

Given probability distribution

C       $0       $1000      $4000       $10000

P(C)   0.60       0.05        0.13             0.22

The probability that the insurance company will have to pay a claim of at least $4000 for a randomly selected car is

= (1-0.60-0.05) = 0.35

so the expected value of C

= 0x0.60 + 1000x0.05 + 4000x0.13 + 10000x0.22

= 50+520+2200 = $2770

Now, the best interpretation of expected value:

If the company insures a large number of these cars, they can expect the cost per car to average approximately Exp(C).

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The complete expression of the algebra is;

4a(-2a³ + 4a + 3) + 2a(3a³ - 2a + 7) = -2a⁴ + 12a² + 26a

We are given the expression as;

4a(-2a³ + 4__ + 3) + 2a(__a³ - 2a + _) = -2a* + 12a² + 26a

Expanding the left hand side gives;

-8a⁴ + __(16a) + 12a + __(2a⁴) - 4a² + __(2a)

Grouping like terms gives;

-8a⁴ + __(2a⁴) + 12a + __(2a) + __(16a) - 4a²

Factorizing out exponents gives;

a⁴(-8 + __(2)) + a(12 + __(2)) + a²((16/a) - 4)

Comparing this with the right hand side gives;

a⁴(-8 + __(2)) = -2a*  -----(1)

((16/a) - 4)a² = 12a²    -----(2)

(12 + __(2))a = 26a   ----(3)

From eq 1, we can see that;

* = 4

From eq 2;

((16/a) - 4) = 12

16/a = 12 + 4

a = 16/16

a = 1

From eq 3;

(12 + __(2)) = 26

___ * 2 = 26 - 12

___ = 14/2

___ = 7

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Answer:

90 km/h

Step-by-step explanation:

speed = distance/time

speed = 225 km / 2.5 h

speed = 90 km/h

Answer:

90km/hr

Step-by-step explanation:

To find the speed, take the distance and divide it by the time.

225 km/ 2.5 hours

90km/hr

Answer:

12,000

Step-by-step explanation:

How can we do this?

We can do this by taking the number 11,900,

11,900

looking at the hundredths place,

11,900

and figuring out if it is closer to 10, or 0. 9 is closer to 10, so we round up.

12,000

Therefore, 11,900 rounded to the nearest thousand is 12,000.

Answer:

31.81

Step-by-step explanation:

the dotted line ² = (22²-16²)

x = sq root of [dotted line²  +(44-16)² ]

= sq root of [22²-16²+28²]

31.81

Answer:

Step-by-step explanation

The order in which you shoot the scenes will not matter in terms of the final outcome of the movie.

The mathematical property that one can use to answer this question is scheduling.

From the information, in order to produce the movie, one needs to shoot a scene for a high-speed chase and need to shoot a scene for a bank robbery.

Here, the film production crew will be putting the movie together. However, it may impact the logistics of the production process and the scheduling of the crews. It is not a mathematical property that I used to answer this question. It is simply a matter of logistics and scheduling.

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