What are the steps in solving word problems involving multiplication of simple
fractions with other operations?

Answer:

12.32 units

Step-by-step explanation:

Use the distance formula to find the lengths between two points:

\(d=\sqrt{(x_{2} -x_{1})^2+(y_{2}-y_{1})^2 }\)

the distance between (-2, -1) and (-5,-1) is

\(\sqrt{(-2-(-5))^2+(-1-(-1))^2}\)

=\(\sqrt{3^2+0^2}\)

=\(\sqrt{9}\)

=3

the distance between (-2, -1) and (-3, -4) is

\(\sqrt{(-2-(-3))^2+(-1-(-4))^2}\)

=\(\sqrt{1^2+3^2}\)

=\(\sqrt{10}\)

≈3.16

the distance between (-6, -4) and (-3, -4) is

\(\sqrt{(-6-(-3))^2+(-4-(-4))^2}\)

=\(\sqrt{(-3)^2+0^2}\)

=\(\sqrt{9}\)

=3

the distance between (-6, -4) and (-5, -1) is

\(\sqrt{(-6-(-5))^2+(-4-(-1)^2}\)

=\(\sqrt{(-1)^2+3^2}\)

=\(\sqrt{10}\)

≈3.16

Now, add all the sides:

3+3.16+3+3.16=12.32 units

Note: A property of a parallelogram is that it has two pairs of congruent sides, so for this problem, you could solve for 2 different sides, double them, then add and you will get the same thing.

The answer of the given question based on the Newton's law is , the scenario demonstrates Newton's third law of motion.

Newton's laws of motion are set of fundamental principles that describe  behavior of a objects in motion. They were formulated by Sir Isaac Newton in the 17th century and are considered to be the foundation of classical mechanics. It consists of three laws of motion they are , Newton's First Law of Motion , Newton's Second Law of Motion , Newton's Third Law of Motion. These laws explain how objects move and interact with one another, and they have numerous applications in physics, engineering, and other fields.

The scenario given describes Newton's third law of motion, which states that for every action, there is an equal and opposite reaction.

In this case, the action is the force exerted by the car on the wall, and the reaction is the force exerted by the wall on the car. According to Newton's third law, these forces are equal in magnitude but opposite in direction, which means that the car and the wall exert the same amount of force on each other in opposite directions.

Therefore, the scenario demonstrates Newton's third law of motion.

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The cross sectional area of the cargo trailer floor, which is a composite figure consisting of a square and an isosceles triangle, indicates that the volume of the inside of the trailer is about 3,952 ft³.

A composite figure is a figure comprising of two or more regular figures.

The possible cross section of the trailer, obtained from a similar question on the internet, includes a composite figure, which includes a rectangle and an isosceles triangle.

Please find attached the cross section of the Cargo Trailer Floor created with MS Word.

The dimensions of the rectangle are; Width = 6 ft, length = 10 ft

The dimensions of the triangle are; Base length 6 ft, leg length = 4 ft

Height of the triangular cross section = √(4² - (6/2)²) = √(7)

The cross sectional area of the trailer, A = 6 × 10 + (1/2) × 6 × √(7)

A = 60 + 3·√7

Volume of the trailer, V = Cross sectional area × Height

V = (60 + 3·√7) × 6.5 = 3900 + 19.5·√7

Volume of the trailer = (3,900 + 19.5·√(7)) ft³ ≈ 3952 ft³

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Answer:

8 - x

Step-by-step explanation:

Answer:

The surface area of the given triangular prism = 920 square cm.

In the given triangular prism

S1 = 25 cm

S2 = 25 cm

S3 = 14 cm = base

Height = 24 cm

length = 10 cm

Perimeter = S1 + S2+ S3

                 = 25+25+14

                 = 64 cm

Area of base =  length x base

                      = 10x14

                       = 140 square cm

Since we know that,

Surface area of triangular prism = (Perimeter × Length) + (2 × Base Area)

                                                     = (64x10) + 2x140

                                                     = 640 + 280

                                                     = 920 square cm

Hence,

The required surface area =  920 square cm.

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a. The formula B = 703w/h^2 can be solved for w by rearranging the equation as w = B * h^2 / 703.
b. For a person who is 64 inches tall and has a body mass index of 21.45, the weight can be calculated by substituting the values into the formula w = B * h^2 / 703, where B is 21.45 and h is 64 inches.

a. To solve the formula B = 703w/h^2 for w, we can rearrange the equation to isolate w on one side of the equation. Multiply both sides of the equation by h^2, then divide both sides by 703. The resulting equation is w = B * h^2 / 703.
b. To find the weight of a person who is 64 inches tall and has a body mass index of 21.45, we can substitute the values into the formula w = B * h^2 / 703. In this case, B is 21.45 and h is 64 inches. Plugging these values into the equation, we get w = 21.45 * 64^2 / 703. Evaluating this expression will give us the weight in pounds.

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Answer:

30 in 2

Step-by-step explanation:

What is the area of the figure below?

Answer:

  -1 ≤ x ≤ 2

Step-by-step explanation:

You want the solution to |x +1| +|x -2| = 3.

We find it convenient to solve these absolute value equations using a graphing calculator. When we subtract 3 from both sides, we have ...

  |x +1| +|x -2| -3 = 0

The solutions will show on the graph as places where the expression has a value of 0, that is, the x-intercepts.

The left-side expression has a value of 0 for all values of x between -1 and +2, inclusive. That is, the solution is ...

  -1 ≤ x ≤ 2

The absolute value function is piecewise defined:

  |x| = x . . . . for x ≥ 0

  |x| = -x . . . . for x < 0

That is, the behavior of the function changes at x=0.

In the given equation the absolute value function arguments are zero at ...

  x +1 = 0   ⇒   x = -1

  x -2 = 0   ⇒   x = 2

These x-values divide the domain of the equation into three parts.

In this domain, both arguments are negative, so the equation is actually ...

  -(x +1) -(x -2) = 3

  -2x +1 = 3

  -2x = 2

  x = -1 . . . . . . not in the domain

In this domain, the argument (x+1) is positive, but the argument (x-2) is negative. That means the equation is ...

  (x +1) -(x -2) = 3

  1 +2 = 3

True for all x in this domain.

In this domain, both arguments are positive, so the equation is ...

  (x +1) +(x -2) = 3

  2x -1 = 3

  2x = 4

  x = 2 . . . . in the domain (this point was excluded from x < 2).

The solution is -1 ≤ x ≤ 2.

The correct answer is option (b): increases, i.e., if the variability (p times q) increases and the sample size remains the same, then the standard error increases.

The standard error measures the variability of sample means around the true population mean. It is calculated as the standard deviation of the sample divided by the square root of the sample size. The formula for standard error is:

SE = s / √n, where s is the standard deviation and n is the sample size.

When the variability (p times q) increases, it means that there is more dispersion or spread of values in the population. This increased variability leads to a larger standard deviation (s) in the sample, assuming the sample size remains the same.

Since the standard error is calculated by dividing the standard deviation by the square root of the sample size, an increase in the standard deviation will result in a larger standard error. Therefore, if the variability increases and the sample size remains constant, the standard error will increase.

In summary, the standard error increases when the variability increases and the sample size remains the same. This is because a larger standard deviation implies more uncertainty in the sample mean estimate, resulting in a larger standard error.

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The value of the multiplier is **2**.

The multiplier is the ratio of the change in real GDP to the change in autonomous spending. It is calculated as follows:

```

Multiplier = 1/(1-MPC)

```

Where MPC is the marginal propensity to consume.

In this case, the slope of the AE curve is 0.80. This means that the MPC is 0.80. Therefore, the multiplier is 2.

A multiplier of 2 means that a change in autonomous spending will lead to a change in real GDP that is twice as large. For example, if autonomous spending increases by $100, then real GDP will increase by $200.

The slope of the AE curve is equal to 1/(1-MPC). Therefore, the slope of the AE curve can be used to calculate the value of the multiplier.

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Answer:

Please see the explanation.

Step-by-step explanation:

Let

\(f\left(x\right)=cosx\)

By the first principle

\(f\:'\left(x\right)=\lim _{h\to 0}\left(\frac{f\left(x+h\right)-f\left(x\right)}{h}\right)\)

          \(=\lim _{h\to 0}\left(\frac{cos\:\left(x+h\right)-cos\:x}{h}\right)\)

           \(=\lim _{h\to 0}\left[\frac{cos\:x\:cos\:h-sin\:x\:sin\:h\:-\:cos\:x}{h}\right]\)

           \(=\lim _{h\to 0}\left[\frac{-cos\:x\left(1-cos\:h\right)-sin\:x\:sin\:h\:}{h}\right]\)

           \(=\lim _{h\to 0}\left[\frac{-cos\:x\left(1-cos\:h\right)\:}{h}-\frac{sin\:x\:sin\:h}{h}\right]\)

           \(=-cosx\:\left(\lim \:_{h\to \:0\:}\frac{1-cos\:h}{h}\right)-sin\:x\:\lim \:\:_{h\to \:\:0}\:\left(\frac{sin\:h}{h}\right)\)

           \(=-cosx\:\left(0\right)-sinx\left(1\right)\)

            \(=-sin\:x\)

Answer:

3f=5

Step-by-step explanation:

f(x) = x+4

3f=4x

x<5

5<_x<7

7<_x<_10

Answer:

See below.

Step-by-step explanation:

\(\frac{1}{12} +\frac{1}{6} = \frac{2 + 1}{12} = \frac{1}{4}\)

\(\frac{1}{4} - \frac{1}{8}=\frac{2-1}{8} = \frac{1}{8}\)

\(\frac{2}{3} +\frac{1}{3} = 1\)

\(\frac{1}{2} - \frac{1}{6} = \frac{3-1}{6} = \frac{1}{3}\)

Answer:

1/12 + 16 = 1/4

1/4 - 1/8 = 1/8

2/3 + 1/3 = 1

1/2 - 1/6 = 1/3

Step-by-step explanation:

1/12 + 1/6

LCM of denominators 12 and 6 = 12

1/6

Multiply 2 to both numerator and denominator,

1/6 * 2/2

= ( 1*2 )/( 6*2 )

= 2/12

1/12 + 1/6

= 1/12 + 2/12

= ( 1 + 2 )/12

= 3/12

12 = 3*4

3/12

= 3/( 3*4 )

= 1/4

1/4 - 1/8

LCM of denominators 4 and 8 = 8

1/4

Multiply 2 to both numerator and denominator,

1/4 * 2/2

= ( 1*2 )/( 4*2 )

= 2/8

1/4 - 1/8

= 2/8 - 1/8

= ( 2 - 1 )/8

= 1/8

2/3 + 1/3

= ( 2 + 1 )/3

= 3/3

= 1

1/2 - 1/6

LCM of denominators 2 and 6 = 6

1/2

Multiply 3 to both numerator and denominator,

1/2 * 3/3

= ( 1*3 )/( 2*3 )

= 3/6

1/2 - 1/6

= 3/6 - 1/6

= ( 3 - 1 )/6

= 2/6

6 = 2*3

2/6

= 2/( 2*3 )

= 1/3

1. The function f(x) = x³(x + 2)⁸ is increasing on the intervals x = -2/9 to x = +∞ and positive on the interval x = 0 to x = +∞.

2. The function achieves its minimum at x = -4, where the minimum value is -16,384.

3. For the function f(x) = x√(x² + 16) on the interval -4 ≤ x ≤ 4, it is concave up for all x and there is no inflection point.

4. The minimum for the function f(x) = x√(x² + 16) occurs at x = -4, where the minimum value is -16, and the maximum occurs at x = 4, where the maximum value is 16√2.

To determine where the function f(x) = x³(x + 2)⁸ is increasing, we need to find the intervals where its derivative is positive.

First, let's find the derivative of f(x):

f'(x) = d/dx [x³(x + 2)⁸]

      = 3x²(x + 2)⁸ + x³(8)(x + 2)⁷(1)

      = 3x²(x + 2)⁷[(x + 2) + 8x]

      = 3x²(x + 2)⁷(9x + 2)

To find where f(x) is increasing, we need to find the intervals where f'(x) > 0.

1. Set f'(x) > 0:

3x²(x + 2)⁷(9x + 2) > 0

To determine the intervals, we can examine the sign changes in each factor. First, note that the factor 3x² is always positive, so it doesn't affect the sign of the expression.

For (x + 2)⁷, it will be positive for all x values since it is raised to an odd power.

For (9x + 2), we can find where it changes sign by solving the inequality:

9x + 2 > 0

9x > -2

x > -2/9

So, we have the following intervals:

1. x < -2/9

2. x > -2/9

Now, we need to determine the sign of f'(x) within each interval to find where f(x) is increasing.

1. For x < -2/9:

In this interval, (x + 2)⁷ is positive, and (9x + 2) is negative. Therefore, f'(x) is negative.

2. For x > -2/9:

In this interval, both (x + 2)⁷ and (9x + 2) are positive. Therefore, f'(x) is positive.

Therefore, the function f(x) = x³(x + 2)⁸ is increasing on the interval x = -2/9 to x = +∞.

To find where the function is positive, we need to analyze the sign of f(x) itself. Since the function involves multiplying terms, the sign of f(x) will depend on the signs of each term.

Let's consider the three factors: x, x³, and (x + 2)⁸.

1. For x < 0:

Therefore, f(x) is negative for x < 0.

2. For x = 0:

- f(x) = 0³(0 + 2)⁸ = 0

3. For x > 0:

Therefore, f(x) is positive for x > 0.

Hence, the function is positive on the interval x = 0 to x = +∞.

To find where the function achieves its minimum, we need to check the critical points and the endpoints of the given interval.

For f(x) = x³(x + 2)⁸, we have one critical point when f'(x) = 0.

Setting f'(x) = 0:

3x²(x + 2)⁷

(9x + 2) = 0

This equation has two solutions: x = 0 and x = -2/9.

Now, let's check the endpoints:

For x = -4:

f(-4) = (-4)³((-4) + 2)^8 = (-4)³(-2)⁸ = 64(-256) = -16,384

For x = 4:

f(4) = (4)³((4) + 2)⁸ = (4)³(6)^8 = 64(46,656) = 2,979,584

Comparing the values:

f(-4) = -16,384

f(0) = 0

f(-2/9) ≈ -0.019

f(4) = 2,979,584

The minimum value occurs at x = -4, where f(x) = -16,384.

Now, let's consider the function f(x) = x√(x² + 16) defined on the interval -4 ≤ x ≤ 4.

To determine the intervals of concavity, we need to find the second derivative, f''(x), and analyze its sign.

First, let's find the second derivative:

f(x) = x√(x² + 16)

f'(x) = √(x² + 16) + x * (1/2)(x² + 16)^(-1/2) * 2x

      = √(x² + 16) + x² / √(x² + 16)

      = (x² + 2(x² + 16)) / √(x² + 16)

      = (3x² + 32) / √(x² + 16)

f''(x) = [(3x² + 32) * (√(x² + 16))] - [(x² + 2(x² + 16)) * (1/2)(x² + 16)^(-1/2) * 2x)] / (x² + 16)

      = [(3x² + 32) * (√(x² + 16))] - [x * (x² + 2(x² + 16))] / (x² + 16)

      = [(3x² + 32) * (√(x² + 16))] - [(3x⁴ + 32x) / (x² + 16)]

To determine the intervals where f(x) is concave up and concave down, we need to find where f''(x) > 0 and where f''(x) < 0.

Let's analyze f''(x):

f''(x) = [(3x² + 32) * (√(x² + 16))] - [(3x⁴ + 32x) / (x² + 16)]

Since both the numerator and the denominator are positive for all x, the fraction is positive for all x

Thus, f''(x) > 0 for all x, meaning f(x) is concave up for all x in the interval -4 ≤ x ≤ 4.

There is no interval of concavity change, as f''(x) is always positive.

The inflection point for this function occurs where f''(x) changes sign, but since f''(x) is always positive, there is no inflection point within the interval -4 ≤ x ≤ 4.

The minimum and maximum for this function occur at the endpoints of the interval.

The minimum occurs at x = -4, where f(x) = (-4)√((-4)² + 16) = -4√(16) = -4 * 4 = -16.

The maximum occurs at x = 4, where f(x) = 4√(4² + 16) = 4√(16 + 16) = 4√(32) = 4 * 4√2 = 16√2.

Therefore, the minimum for the function f(x) = x√(x² + 16) occurs at x = -4, and the maximum occurs at x = 4.

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Answer:

(- 1, 3 )

Step-by-step explanation:

2x + 3y = 7 → (1)

- 2x + 4y = 14 → (2)

add (1) and (2) term by term to eliminate x

(2x - 2x) + (3y + 4y) = (7 + 14)

0 + 7y = 21

7y = 21 ( divide both sides by 7 )

y = 3

substitute y = 3 into either of the 2 equations and solve for x

substituting into (1)

2x + 3(3) = 7

2x + 9 = 7 ( subtract 9 from both sides )

2x = - 2 ( divide both sides by 2 )

x = - 1

As a check

substitute the values of x and y into the left side of both equations and if equal to the right side in both equations then the values of x and y are true.

(1) 2(- 1) + 3(3) = - 2 + 9 = 7 ← true

(2) - 2(- 1) + 4(3) = 2 + 12 = 14 ← true

then (- 1, 3 ) is the solution to the system

Answer:(- 1, 3 )

Step-by-step explanation:

Answer:

-6a-48

Step-by-step explanation:

Analysis, we can conclude that W = {f ∈ V : f(a) = f(b)} is Indeed a subspace of V

To determine whether the set W = {f ∈ V : f(a) = f(b)} is a subspace of V, we need to check three properties:

The zero vector is in W.

W is closed under vector addition.

W is closed under scalar multiplication.

Let's analyze each property:

Zero vector: The zero vector in V is the constant function f(x) = 0 for all x in [a, b]. This function satisfies f(a) = f(b) = 0, so the zero vector is in W.

Vector addition: Suppose f1 and f2 are two functions in W. We need to show that their sum, f1 + f2, is also in W. Let's evaluate (f1 + f2)(a) and (f1 + f2)(b):

(f1 + f2)(a) = f1(a) + f2(a) = f1(b) + f2(b) = (f1 + f2)(b)

Since (f1 + f2)(a) = (f1 + f2)(b), the sum f1 + f2 satisfies the condition for W. Therefore, W is closed under vector addition.

Scalar multiplication: Let f be a function in W and c be a scalar. We need to show that the scalar multiple cf is also in W. Let's evaluate (cf)(a) and (cf)(b):

(cf)(a) = c * f(a) = c * f(b) = (cf)(b)

Since (cf)(a) = (cf)(b), the scalar multiple cf satisfies the condition for W. Therefore, W is closed under scalar multiplication.

Based on the above analysis, we can conclude that W = {f ∈ V : f(a) = f(b)} is indeed a subspace of V

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The correct option is A. m ≤ (8)(20)

The inequality which gives the possible values of 'm' is m ≤ (8)(20).

A declaration of an order relationship between two numbers or algebraic expressions, such as greater than, greater than or equal to, less than, or less than or equal to.

An inequality is a relation that compares two numbers or other mathematical expressions in an unequal way. The majority of the time, size comparisons between two numbers on the number line are made.

According to the question;

Terrence's car contains 8 gallons of fuel.

Terrence can drive the car 'm' miles using the fuel currently in the car.

The car can drive 20 miles per gallon of fuel,(which is maximum fuel capacity of the car to drive).

Then,

The total miles 'm' covered by the car is 8×20 which is maximum capacity of the car to travel.

Thus, total miles covered by the car are less than the maximum value which is given by the inequality-

m ≤ (8)(20)

Therefore, the inequality which gives the possible values of 'm' is m ≤ (8)(20).

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The complete question is-

Terrence's car contains 8 gallons of fuel. He plans to drive the car 'm' miles using the fuel currently in the car. If the car can drive 20 miles per gallon of fuel, which inequality gives the possible values of 'm'?

answer choices

A. m ≤ (8)(20)

B. m ≥ (8)(20)

C. 8 ≤ 20m

D. 8 ≥ 20 m

Answer:

Mean or expected value (μ ) = 1400

Step-by-step explanation:

Step(i):-

Given that the probability of success p = 0.7

Given number of trials n = 2000

Let 'X' be the random variable in binomial distribution

Now , we have to find the mean or expected value of the binomial distribution

Step(ii):-

 Mean = n p

 μ    = 2000× 0.7

  μ    = 1400

No Solutions

5−4+7x+1 =

x + 2

One Solution

5−4+7x+1 =

x - 2

Answer:

Step-by-step explanation:

First off, we need to remember that the arc measure is equal to the angle measure inside the circle. That means m(arc) MN is 70 degrees. Knowing this, we can find the measures of the other arcs:

m(arc)NP = 180 - 70 - 30 = 80

m(arc)QR = 70 based on vertical angles principle

m(arc)MR = 80+30 = 110 based on vertical angles and angle sum principles

Now we can list the answers using this:

1. 70

2. 70

3. 180

4. 210

5. 110

6. 110

Hope this helps!

Answer:

The answers are in the picture I got it correct as you can see.

The actual nominal rate of return is approximately 6.68%. None of the provided answer options exactly match this value, but the closest option is 6.77%.

To calculate the actual nominal rate of return, we use the Fisher equation:

Nominal Rate = (1 + Real Rate) * (1 + Inflation Rate) - 1

Given:

Real Rate = 3.87% (0.0387 as a decimal)

Inflation Rate = 2.75% (0.0275 as a decimal)

Now, let's plug in the values:

Nominal Rate = (1 + 0.0387) * (1 + 0.0275) - 1

Nominal Rate = 1.0387 * 1.0275 - 1

Nominal Rate = 1.0668 - 1

Nominal Rate = 0.0668 or 6.68% (rounded to two decimal places)

So, the actual nominal rate of return is approximately 6.68%. None of the provided answer options exactly match this value, but the closest option is 6.77%.

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10y−7z

Plug in the numbers for y and z.

(10)(9)−(7)(3)

= 69

Answer:

69

Step-by-step explanation:

10y-7z

Plug in y and z

10(9)-7(3)

Solve using order of operations

90-21=69