Match the items (example: a1)

a. Multiplicative Inverse
b. Additive Inverse
c. Commutative Property of Multiplication
d. Distributive Property
e. Additive Identity Property
f. Associative Property of Addition

To find the area of the shaded region, we need to determine the boundaries of the region and integrate the appropriate function over that interval.

From the given information, the boundaries of the shaded region are as follows:

The lower boundary is given by the curve y = x^3.

The upper boundary is given by the line y = 3.

The left boundary is the vertical line x = 1.

The right boundary is the vertical line x = 4.

To find the area, we need to integrate the difference between the upper and lower boundaries with respect to x over the interval [1, 4]:

Area = ∫[1, 4] (3 - x^3) dx

To evaluate this integral, we can expand the polynomial and then integrate:

Area = ∫[1, 4] (3 - x^3) dx

= ∫[1, 4] (3 - x^3) dx

= ∫[1, 4] 3 dx - ∫[1, 4] x^3 dx

= [3x] evaluated from 1 to 4 - [(1/4)x^4] evaluated from 1 to 4

= 3(4) - 3(1) - (1/4)(4^4) + (1/4)(1^4)

= 12 - 3 - 64/4 + 1/4

= 12 - 3 - 16 + 1/4

= -7 - 64/4 + 1/4

= -7 - 16 + 1/4

= -23 + 1/4

= -91/4

Therefore, the area of the shaded region is -91/4 square units.

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Answer:

y = \(\frac{4}{3}\) x + \(\frac{29}{3}\)

Step-by-step explanation:

assuming you require the equation of the perpendicular line.

the equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y- intercept )

y = - \(\frac{3}{4}\) x - 3 ← is in slope- intercept form

with slope m = - \(\frac{3}{4}\)

given a line with slope m then the slope of a line perpendicular to it is

\(m_{perpendicular}\) = - \(\frac{1}{m}\) = - \(\frac{1}{-\frac{3}{4} }\) = \(\frac{4}{3}\) , then

y = \(\frac{4}{3}\) x + c ← is the partial equation

to find c substitute (- 2, 7 ) into the partial equation

7 = - \(\frac{8}{3}\) + c ⇒ c = 7 + \(\frac{8}{3}\) = \(\frac{21}{3}\) + \(\frac{8}{3}\) = \(\frac{29}{3}\)

y = \(\frac{4}{3}\) x + \(\frac{29}{3}\) ← equation of perpendicular line

The equation of the line is y = (4x +29)/3

The line y = -3/4x - 3 has a slope of -3/4.  A line perpendicular to -3/4 has a slope of 4/3. So the slope of our new line will be 4/3.

We are told that the line goes through the point (-2, 7).  That is x = -2 and y = 7.  Now that we know a point and a slope, we can use the formula:

y2 - y1 = m(x2 - x1)

where,

x1 = -2

y1 = 7

m = 4/3 [slope]

Substituting in the formula,

y2 - 7 = 4/3(x2 - (-2))

y2 - 7 = 4/3*x2 + 4/3*2

3(y2 - 7) = 4(x2) + 8    

3y2 - 21 = 4(x2) +8

3y2 = 4(x2) + 8 +21

3y2 = 4(x2) +29

y2 = (4(x2) +29)/3          

Therefore, the equation of the line is y = (4x +29)/3

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The completer question is

'Find the equation of the line that is perpendicular to y = -3/4x - 3 and passes through the points (-2,7).'

Five nickels plus two dimes totaling

combining for a sum of $0.45.

three nickels and four dimes respectively

combining for a sum of $0.55.

two nickels and five dimes denoting x = 2, y = 5

combining for a sum of $0.60.

3 ways to achieve a total of 7 coins with nickels (N) and dimes (D), and their corresponding values are

Five nickels plus two dimes totaling x = 5, y = 2 respectively. The value of five nickels = $0.25

two dimes = $0.20

combining for a sum of $0.45.

three nickels and four dimes respectively depositing x = 3, y = 4

3 nickels = $0.15

4 dimes = $0.40

combining for a sum of $0.55.

two nickels and five dimes denoting x = 2, y = 5

2 nickels = $0.10

5 dimes = $0.50

combining for a sum of $0.60.

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Answer:

Step-by-step explanation:

3

The best approximation of the area of the circle is 113.04 square inches.

A circle is a shape with all points equidistant from a central point, known as the center. The distance from the center to any point on the edge of the circle is called the radius.

It can be calculated using the formula π \(r^2\), where r is the radius.

Since the diameter of the circle is 12 inches, the radius would be half of that which is 6 inches.

So, the area would be approximately 3.14 * \(6^2\) = 113.04 square inches.

Therefore, The best approximation of the area of the circle is 113.04 square inches.

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β2 indicates the difference in the intercept for D=1 versus D=0 in the regression model. The correct interpretation is B.

In the given regression model, β2 represents the coefficient associated with the interaction term between the continuous variable X and the binary variable D.

The correct answer is B. β2 indicates the difference in the intercept for D=1 versus D=0. In other words, it captures the change in the intercept of the regression line when the binary variable D changes from 0 to 1 while holding other variables constant. This means that the relationship between the dependent variable Y and the continuous variable X differs depending on the value of the binary variable D.

Option A is incorrect because the sign of β2 can be positive or negative, depending on the nature of the relationship between Y and the interaction term (Xi × Di).

Option C is also incorrect because β2 does not represent the difference in means in Y between the two categories of the binary variable D. Instead, it represents the difference in the intercepts or the baseline values of the regression line.

Therefore, the correct interpretation is that β2 indicates the difference in the intercept for D=1 versus D=0 in the regression model.  

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To find the Laplace transform of the following functions:

(a) To find the Laplace transform of f(t) = 2te^t, we can use the formula:

L{t^n} = n! / s^(n+1)

Applying this formula to f(t) = 2te^t:

L{2te^t} = 2 * L{t} * L{e^t}

The Laplace transform of t is given by L{t} = 1/s^2, and the Laplace transform of e^t is 1/(s-1).

Substituting these values:

L{2te^t} = 2 * (1/s^2) * (1/(s-1))

         = 2 / (s^2 * (s-1))

Therefore, the Laplace transform of f(t) = 2te^t is 2 / (s^2 * (s-1)).

(b) To find the Laplace transform of g(t) = e^t cos(2t), we can use the property:

L{e^at cos(b*t)} = s - a / (s - a)^2 + b^2

Applying this property to g(t) = e^t cos(2t):

L{e^t cos(2t)} = s - 1 / (s - 1)^2 + 2^2

              = s - 1 / (s - 1)^2 + 4

Therefore, the Laplace transform of g(t) = e^t cos(2t) is (s - 1) / ((s - 1)^2 + 4).

(c) To find the Laplace transform of h(t) = 67 sin(2πt) cos(2πt), we can use the property:

L{sin(a*t) cos(a*t)} = 1/2 * s / (s^2 - a^2)

Applying this property to h(t) = 67 sin(2πt) cos(2πt):

L{67 sin(2πt) cos(2πt)} = 1/2 * s / (s^2 - (2π)^2)

                        = 1/2 * s / (s^2 - 4π^2)

Therefore, the Laplace transform of h(t) = 67 sin(2πt) cos(2πt) is 1/2 * s / (s^2 - 4π^2).

(d) To find the Laplace transform of y(t) = e^(5t-2)u(t-1), we can use the time-shifting property:

L{e^(a(t-b))u(t-b)} = e^(as) / (s - a)

Applying this property to y(t) = e^(5t-2)u(t-1):

L{e^(5t-2)u(t-1)} = e^(-2s) / (s - 5)

Therefore, the Laplace transform of y(t) = e^(5t-2)u(t-1) is e^(-2s) / (s - 5).

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Answer:

115 lbs = 52.16 kilograms

Step-by-step explanation:

115 pounds is a little over 52kg

Answer:

Pemdas:

Step-by-step explanation:

-3*6x then -3*10

-18x+30= -0.6

The formula which represents h in terms of p, s, d, and g is:

then the correct option is B.

Answer:

i guess it might be x=7

Step-by-step explanation:

Answer:

If angle B is 60 and angle C is 70, We can make an equation by:

180 - (60+70)

The answer for this equation will be 50.

According to the upper equation, angle A is 50, angle B is 60, and angle C is 70.

The amount of Chemical  B needed is 1.6 g.

This refers to a diagram that shows a series of one or more points, lines, line segments, curves, or areas that represents the variation of a variable when compared with that of one or more other variables.

The quantity of Chemical B is plotted against the quantity of  Chemical A.

Chemical B  is plotted on the vertical axis using the scale of 1 cm to represent 2 units. Chemical A is plotted on the horizontal axis using the scale of 1 cm to represent 1 unit.

Therefore, on the graph, the line passing through the origin of the graph shows that when  1.4 g of Chemical A is used  1.6 g of Chemical B is needed.

Chemical B  is plotted on the vertical axis using the scale of 1  cm to represent 2 units.

To get the amount of Chemical B is  on vertical axis =  2 grams

                                                                                             5 boxes

                                                                                       =0.4 g

Tracing the mass of Chemical A used 1.4 g it meets the line passing through the center at 4th line (4th box).

To get the amount of Chemical B is  on vertical axis = amount of Chemical B per box * the number of line (boxes).

To get the amount of Chemical B is  on vertical axis = 0.4 g * 4 lines (boxes)

                                                                                        = 1.6 g

Hence The amount of Chemical  B needed is 1.6 g.

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Answer:

90 bag

Step-by-step explanation:

no of bags will he have left=213-123

=90 bags of candy

Answer:

1.9 x 10 to the 7th power

Step-by-step explanation:

1.9 x 10 to the 7th power

Percentage of the population remains after another 15 years if the decrease is exponential . Let P be the initial population of the sparsely populated area. After 20 years, the number of inhabitants decreases by half. This means that the population after 20 years will be P/2.

If the decrease is exponential, then the population after t years will be P(1/2)^(t/20).We want to find the percentage of the population that remains after another 15 years, which means we want to find the value of P(1/2)^(15/20) as a percentage of P.

Simplifying this expression, we get:P(1/2)^(15/20) = P(1/2)^(3/4) = P(1/sqrt(2))^3 = P(1/1.414)^3 ≈ 0.352 P

Therefore, the percentage of the population that remains after another 15 years is approximately 35.2%.

To summarize, if the population of a sparsely populated area decreases by half in 20 years and the decrease is exponential, then the percentage of the population that remains after another 15 years is approximately 35.2%.

This can be found by using the formula P(1/2)^(t/20)

to calculate the population after t years, where P is the initial population and t is the time elapsed in years, and then plugging in t = 15 and simplifying.

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Answer:

Let C be the position of the meteorological station, B the position of the balloon and CB be the cable.

Let the height of the balloon from the ground be AB=hmetres.

In a right angled triangle ACB

BC

AB

=sin60

∘

⇒

200

h

=

2

3

since

sin60

∘

=

2

3

⇒h=200×

2

3

⇒h=100

3

∴h=100×1.732=173.2 since

3

=1.732

Hence, the height of the balloon above the ground is 173.2m

By inscribing the given information in a right triangle, we will see that the ballon is 167.7 meters above the ground.

Think in the situation as in a right triangle, you have a hypotenuse that is equal to the length of the cable, you know one angle of 57°, and the opposite cathetus to this angle represents the height at which the ballon is, this is what we want to find.

Using the relation:

sin(θ) = (opposite cathetus)/hypotenuse.

Where:

Replacing that we get:

sin(57°) = height/200m

sin(57°)*200m = height = 167.7m

So the balloon is 167.7 meters above the ground.

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Answer:

2/3

Step-by-step explanation:

Scaling can be reversed by using reciprocal factors. Because the scale factor from S to M is 2/3​​, the scale factor from M to S 2/3.

The scale factor from M to S will be 2/3.

You must specify the extent of the shape's enlargement when describing one.

The scale factor is the ratio of two dimensions such that one figure is large and another is small.

The scale factor is done due to the unpractical measurement of any figure.

For example, 50 meters is very big so we cannot measure it but if we can convert it into 50 mm such that the scale ratio is maintained then it will be easy to draw.

Given,

Scale factor M to S

M/S = 3/2

By cross multiplication

2M = 3S

S/M = 2/3

S/M will be scale factor S to M.

Hence "The scale factor from M to S will be 2/3".

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Although a bigger random sample can aid in increasing the precision and decreasing the variability of a national opinion poll, it cannot ensure a decrease in bias, which is dependent on the efficacy of the sampling technique and the sample's representativeness.

A nationwide opinion poll's results are often less variable as the random sample size is increased since the bigger sample size more accurately represents the people being sampled.

It does not necessarily lessen bias, which is the term for systemic mistakes in the sampling process or polling methodology that may endure even with a larger sample size.

The accuracy of the poll's results can be improved by reducing random mistakes like sampling and measurement errors with a bigger sample size.

The standard error of the mean, a measurement of the variability of the sample mean from sample to sample, is often less the bigger the sample size.

A national opinion poll's measure of interest, the population mean, can be more accurately estimated with a smaller standard error.

If the sample is not completely random or representative of the population being investigated, bias may still exist in a bigger sample.

For instance, the findings may be biassed if the pollsters oversample a certain demographic group or geographic area.

Non-response bias can also happen if a certain group of people is less likely than others to reply to the survey.

As a result, while a bigger random sample can aid in increasing the precision and decreasing the variability of a national opinion poll, it cannot ensure a decrease in bias, which is dependent on the efficacy of the sampling technique and the sample's representativeness.

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This is a translation of 2 units to the left and 5 units up, so the correct option is the first one.,

Remember that a vertical translation of N units is:

g(x) = f(x) + N

if N < 0 the translation is down.

if N > 0 the translation is up.

And a horizontal translation of N units is:

g(x) = f(x + N)

If N > 0 the translation is to the right.

if N < 0 the translation is to the left.

Here we have the transformation:

f(x) = x²

g(x) = (x + 2)² + 5

So this is a translation of 2 units to the left and 5 units up.

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The probability that the number of lost-time accidents occurring over a period of 7 days will be exactly 3 is 0.7295

Binomial Distribution

The binomial distribution is the sum of a series of multiple independent and identically distributed Bernoulli trials. In a Bernoulli trial, the experiment is said to be random and can only have two possible outcomes- success or failure.

In a binomial distribution the probability of getting success must remain the same for the trials we are investigating. For example, when tossing a coin, the probability of flipping a coin is ½ or 0.5 for every trial we conduct, since there are only two possible outcomes.

Mean rate of lost time accident = 0.3 per day

probability that the number of lost-time accidents occurring over a period of 7 days will be no more than 3.

n = 7

Using binomial distribution formula :

P(x ≤3)

Probability of success (p) = 0.3

1 - p = 1- 0.3 = 0.7

nCx * p^x * (1 - p)^(n-x)

Using the binomial probability distribution calculator to save computation time :

P(x ≤3) = P(x = 0) + p(x = 1) + p(x = 2) + p(x = 3)

P(x ≤3) = 0.0403 + 0.1556 + 0.2668 + 0.2668

P(x ≤3) = 0.7295

0.7295

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under the restrictions that c is negative to ensure strict concavity, the utility function u(w) = -(b - w)c displays increasing absolute risk aversion.

To ensure that u(w) is strictly increasing, we need the derivative of u(w) with respect to w to be positive for all values of w. Taking the derivative, we have du(w)/dw = -c. For u(w) to be strictly increasing, -c must be positive, which implies c must be negative.

To ensure that u(w) is strictly concave, we need the second derivative of u(w) with respect to w to be negative for all values of w. Taking the second derivative, we have d²u(w)/dw² = 0. Since the second derivative is constant and negative, u(w) is strictly concave.

Now, let's examine the concept of increasing absolute risk aversion. If a utility function u(w) exhibits increasing absolute risk aversion, it means that as wealth (w) increases, the individual becomes more risk-averse.

In the given utility function u(w) = -(b - w)c, when c is negative (as required for strict concavity), the absolute risk aversion increases as wealth (w) increases. This is because the negative sign implies that the utility function is concave, indicating that the individual becomes more risk-averse as wealth increases.

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Answer:

C. 8

Step-by-step explanation:

f(5) =1/4(2)^5

because x=5

f(5)=1/4(32)

because 2^5 is 2*2*2*2*2=32

f(5)=8

because 1/4*32 or 32/4 is 8

Hope this helped :) I haven't even learned f(x) in school yet haha. I'm in 8th grade, but I took extracurriculars for math and I'm on the math team. Ergo, I know it's right!

the area inside the curve R is approximately 42.4115 square units.

To find the area inside the curve R, we can use a double integral. The formula for finding the area of a region using a double integral is:

A = ∬R dA

where A is the area of the region R, and dA is an infinitesimal element of area in the region R.

In polar coordinates, dA can be expressed as:

dA = r dr dθ

where r is the radius and θ is the angle.

Substituting this into the formula for the area, we get:

A = ∫₀^2π ∫₀^(5+sinθ) r dr dθ

We can evaluate this integral by integrating first with respect to r and then with respect to θ:

A = ∫₀^2π [1/2 r²] from 0 to (5+sinθ) dθ

A = ∫₀^2π 1/2 (5+sinθ)² dθ

Expanding the square and simplifying, we get:

A = ∫₀^2π 1/2 (25 + 10sinθ + sin²θ) dθ

A = 1/2 ∫₀^2π (25 + 10sinθ + sin²θ) dθ

Using the trigonometric identity sin²θ = (1-cos2θ)/2, we can simplify this to:

A = 1/2 ∫₀^2π (25 + 10sinθ + 1/2 - 1/2cos2θ) dθ

A = 1/2 ∫₀^2π (27/2 + 5sinθ - 1/2cos2θ) dθ

Integrating each term separately, we get:

A = 1/2 [27/2θ - 5cosθ + 1/4sin2θ] from 0 to 2π

A = 1/2 [(27/2)(2π) - 5cos2π + 1/4sin2(2π) - (27/2)(0) - 5cos0 + 1/4sin0]

A = 1/2 (27π)

A = 13.5π

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Answer: 1

\((\frac{x^{a+b} }{x^{c} }) ^{a-b}.(\frac{x^{b+c} }{x^{a} }) ^{b-c} .(\frac{x^{c+a} }{x^{b} })^{c-a}\\\\= \frac{x^{(a+b)(a-b)} }{x^{c(a-b)} }.\frac{x^{(b+c)(b-c)} }{x^{a(b-c)} }.\frac{x^{(c+a)(c-a)} }{x^{b(c-a)} }\\\\=\frac{x^{a^{2}-b^{2} } }{x^{ac-bc} }.\frac{x^{b^{2}-c^{2} } }{x^{ab-ac} }.\frac{x^{c^{2}-a^{2} } }{x^{bc-ab} } \\\\=\frac{x^{a^{2}-b^{2}+b^{2}-c^{2}+c^{2}-a^{2} } }{x^{ac-bc+ab-ac+bc-ab} }\\\\=\frac{x^{0} }{x^{0} }=1\)

Step-by-step explanation:

Answer:

the answer is x=-3 and y=3

Answer:

A) 60 feet.

Step-by-step explanation:

We know that it’s a rectangular swimming pool with a length of 48 feet and a width of 36 feet.

In order for the hose to be extended from the southwest corner to the northewest corner for the shortest length, the hose must be perfectly straight.

Therefore, the the hose will form a hypotenuse.

In other words, the shortest length will simply be the length of the diagonal of the rectangle.

We can find the length of the diagonal using the Pythagorean Theorem, namely:

\(a^2+b^2=c^2\)

Where a and b are the side lengths, and c is the length of the hypotenuse or in this case the diagonal.

So, substitute 48 for a and 36 for b:

\((48)^2+(36)^2=c^2\)

Evaluate:

\(2304+1296=c^2\)

Add:

\(c^2=3600\)

Take the square root of both sides:

\(c=60\text{ feet}\\\)

Therefore, the shortest length for the hose is 60 feet.

9514 1404 393

Answer:

  C.  (x^2-1)^2 + (2x)^2 = (x^2 +1)^2

Step-by-step explanation:

Inexplicably, three of the four answer choices have minus signs in them. Those are all incorrect. The appropriate choice is ...

  C. (x^2-1)^2 + (2x)^2 = (x^2 +1)^2

__

Expanded, this is ...

  x^4 -2x^2 +1 +4x^2 = x^4 +2x^2 +1 . . . . the desired identity

Answer:

C.  (x^2-1)^2 + (2x)^2 = (x^2 +1)^2

Step-by-step explanation: