find ℒ{f(t)} by first using a trigonometric identity. (write your answer as a function of s.) f(t) = sin(5t) cos(5t)

Clockwise rotation 90° about the center of (4,3)

From the given figure,

A, B, C: (1,6) (3,8) (5,6) transform to A'B'C' (7,6) (9,4) (7,2)

Apply clockwise 90° rotation formula:  

x' = (y - y₀) + x₀    

Where  x represent s original position and

            x₀ represents  center

Therefore,

⇒ y' = - (x - x₀) + y₀

For A (1,6) ⇒ A' (7 , 6)

7 = (6 - y₀) + x₀               x₀ - y₀ = 1   ... (1)

6 = - (1 - x₀) + y₀     ⇒           x₀ + y₀ = 7   ... (2)

From (1) + (2):  

2 x₀ = 8   ⇒          x₀ = 4

From (2)-(1):

2 y₀ = 6     ⇒         y₀ = 3

Rotation center is  (4 , 3).      

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The indicated derivatives (√f · 4) = 8i.

To summarize:

f(4) = -2

f'(4) = -3

(fg)'(4) = -16

(√f · 4) = 8i.

Let's find the indicated derivatives using the given information:

f(4):

Since the function value f(4) is given as -2, we know that f(4) = -2.

f'(4):

The derivative f'(4) is given as -3, so we know that f'(4) = -3.

(fg)'(4):

To find the derivative of the product of two functions, we can use the product rule. The product rule states that if we have two functions f(x) and g(x), then the derivative of their product is given by (fg)'(x) = f'(x)g(x) + f(x)g'(x).

Using this rule, we have:

(fg)'(4) = f'(4)g(4) + f(4)g'(4)

= (-3)(2) + (-2)(5)

= -6 - 10

= -16.

Therefore, (fg)'(4) = -16.

(√f) · (4):

We need to differentiate (√f) with respect to x and then evaluate it at x = 4.

Let's denote h(x) = √f(x). Then, h'(x) = (1/2)(f(x))^(-1/2) * f'(x).

Evaluating h'(4), we have:

h'(4) = (1/2)(f(4))^(-1/2) * f'(4)

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

= (1/2)(-1/√2) * (-3)

= -3/(2√2).

Now, we can evaluate (√f) · (4) at x = 4:

(√f · 4) = h(4) · 4 = √f(4) · 4 = √(-2) · 4 = 2i · 4 = 8i.

Therefore, (√f · 4) = 8i.

To summarize:

f(4) = -2

f'(4) = -3

(fg)'(4) = -16

(√f · 4) = 8i.

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This is the equation of a circle is (x - 14)² + (y - 5)² = 1 with center at (14, 5) and radius 1.

Starting with the x terms:

x² - 28x

= x² - 28x + 196 - 196

= (x - 14)² - 196

And now for the y terms:

y² - 10y

= y² - 10y + 25 - 25

= (y - 5)² - 25

Substituting these into the original equation gives:

(x - 14)² - 196 + (y - 5)² - 25 + 220 = 0

Simplifying gives:

(x - 14)² + (y - 5)² = 1

This is the equation of a circle with center at (14, 5) and radius 1.

To graph this, plot the point (14, 5) and draw a circle with radius 1 around it.

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Answer: -31/15

Step-by-step explanation:

Answer:

-31/15

Step-by-step explanation:

To find the flow of the velocity field f=4y^2 i + (8xy)j along each of the paths from (0,0) to (4,8), we need to integrate the vector field along the paths. Let's consider two paths: (i) a straight line path from (0,0) to (4,8) and (ii) a curved path along the parabola y=x^2 from (0,0) to (4,16).

(i) For the straight line path, we have the parametric equations x=t, y=2t. Substituting these into the velocity field, we get f(t)=4(2t)^2 i + (8t)(2t)j = 16t^2 i + 16t^2 j. Integrating f(t) with respect to t from 0 to 4, we get the flow along the straight line path as:

∫f(t) dt = ∫16t^2 i + 16t^2 j dt = [4t^3 i + 4t^3 j] from 0 to 4

= 64i + 64j

(ii) For the curved path along the parabola y=x^2, we have the parametric equations x=t, y=t^2. Substituting these into the velocity field, we get f(t)=4(t^2)^2 i + (8t)(t^2)j = 4t^4 i + 8t^3 j. Integrating f(t) with respect to t from 0 to 4, we get the flow along the curved path as:

∫f(t) dt = ∫4t^4 i + 8t^3 j dt = [t^5 i + 2t^4 j] from 0 to 4

= 1024i + 512j

Therefore, the flow of the velocity field along the straight line path from (0,0) to (4,8) is 64i + 64j, and the flow along the curved path along the parabola y=x^2 from (0,0) to (4,16) is 1024i + 512j.

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The perimeter of the square is a rational number of meters. However, the product of a rational number and an irrational number is always irrational.

If the area of a square is 7 square meters, then each side of the square is √7 meters long. The perimeter of the square is simply the sum of the four sides, which is 4√7 meters.

To determine whether 4√7 is a rational or irrational number, we need to check if √7 is rational or irrational. Since √7 is not a perfect square, it cannot be expressed as a fraction of two integers. Therefore, √7 is an irrational number.

However, the product of a rational number and an irrational number is always irrational. Since 4 is a rational number and √7 is irrational, the product 4√7 is also an irrational number.

Therefore, the perimeter of the square is an irrational number of meters.

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The modern square of opposition includes four types of propositions: A (universal affirmative), E (universal negative), I (particular affirmative), and O (particular negative). The given proposition is an E proposition, which states that "It is false that some lunar craters are volcanic formations."

To determine the validity of the immediate inference that "Therefore, no lunar craters are volcanic formations," we need to consider the opposite proposition, which is an A proposition that states "All lunar craters are not volcanic formations."

According to the modern square of opposition, the immediate inference from E to E (universal negative to universal negative) is invalid. Therefore, the given immediate inference is also invalid from the Boolean standpoint.

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Within the range 0 ≤ x ≤ 3, the average rate of change of f(x) is 6.

Calculate the difference between the function values at the interval's endpoints and divide by the interval's length to determine the average rate of change of a function inside the interval.

In this case, the interval is [0, 3]. So, we need to find the values of f(0) and f(3) and calculate the difference, and then divide by 3 - 0 = 3.

[tex]f(0) = 2(0)^2 - 4 = -4\\\\f(3) = 2(3)^2 - 4 = 14[/tex]

The difference is: f(3) - f(0) = 14 - (-4) = 18

So, the average rate of change within the interval [0, 3] is:

average rate of change = (f(3) - f(0))/(3 - 0) = 18/3 = 6

Therefore, the average rate of change of f(x) within the interval 0 ≤ x ≤ 3 is 6.

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The given sequence, 3+6+9+12+15+18, can be written in summation notation as ∑(3n), where n starts from 1 and goes up to 6. The formula for the nth term of the sequence is 3n, which means that the first term is 3, the second term is 6, and so on. By using summation notation, we can simplify the expression and represent the entire sequence in a concise and efficient manner.

Summation notation, also known as sigma notation, is a mathematical shorthand that represents the sum of a sequence of numbers. It involves writing the summands inside the summation symbol (∑) and specifying the range of values that the index variable takes. In this case, the summand is 3n and the index variable goes from 1 to 6. Thus, the summation notation for the given sequence is ∑(3n), 1 ≤ n ≤ 6.

In conclusion, the expression 3+6+9+12+15+18 can be written in summation notation as ∑(3n), where n goes from 1 to 6. This allows us to represent the sequence in a compact form and easily find the sum of larger sequences by adjusting the range of the index variable. Summation notation is a useful tool in mathematics that simplifies the process of writing and solving mathematical problems.

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Answer:

a) 345

b) 0.0000002

Step-by-step explanation:

a)  We can multiply the 2.3 and 1.5 and 10^4 and 10^-2 together and simplify at the end.

Step 1:  Working out 2.3 * 1.5:

2.3 * 1.5 = 3.45

Step 2:  Working out 10^4 * 10^-2:

The product rule of exponents states that when you're multiplying the same bases with different exponents, you add the exponents.

So, 10^4 * 10^-2 = 10^(4 + (-2)) = 10^(4 - 2) = 10^2

Step 3:  Simplifying:  

Thus, we have 3.45*10^2 = 3.45 * 100 = 345

Step 4:  Checking our answer:  

We can check by doing the operations inside each parentheses instead of taking them out and seeing whether we still get 345 (i.e., the answer we got when we multiplied common terms instead):

(2.3 * 10^4) * (1.5 * 10^-2)

(2.3 * 10000) * (1.5 * 0.01)

23000 * 0.015

345

Thus, our answer is correct and 345 is the standard form of (2.3 * 10^4) * (1.5 * 10^-2)

b)  Similar to our process for part a), we can divide 3.6 by 1.8 and then divide 10^-5 by 10^2 and simplify at the end.

Step 1:  Working out 3.8 / 1.8:  

3.6 / 1.8 = 2

Step 2:  Working out 10^-5 / 10^2:  

The quotient rule of exponents states that when we divide the same bases with different exponents, we subtract the exponents.

Thus, 10^-5 / 10^2 = 10^(-5 - 2) = 10^-7

Step 3:  Simplifying:

2 * 10^-7 = 2 * 0.0000001 = 0.0000002

Step 4:  Checking our answer:

We can check by doing the operations inside each parentheses instead of taking them out and seeing whether we still get 0.0000002 (i.e., the answer we got when we multiplied common terms instead):

(3.6 * 10^-5) / (1.8 * 10^2)

(3.6 * 0.00001) / (1.8 * 100)

0.000036 / 180

0.0000002

Thus, our answer is correct and 0.0000002 is the standard form of (3.6 * 10^-5) / (1.8 * 10^2)

The first year in which the taxpayer could receive a qualified distribution from the Roth IRA would be 2022.

To determine this, we need to look at the five-year rule for Roth IRA distributions. This rule states that a taxpayer must wait five years from the year of their first contribution to a Roth IRA before they can take a qualified distribution (i.e., a tax-free distribution of earnings and contributions).

Since the taxpayer made their first contribution for the 2018 tax year, the five-year clock starts on January 1, 2018. Therefore, the earliest year in which they could receive a qualified distribution is 2022.

It is important to note that there are other rules and exceptions that could affect when a taxpayer can take distributions from a Roth IRA,

such as age and disability, and that tax implications should also be considered when making decisions about Roth IRA contributions and distributions.

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Graphing the equation y = -x/3 + 4.

- Find the y-intercept

- Find the slope.

We have,

The equation can be graph y = -x/3 + 4, follow these steps:

- Identify the y-intercept:

Set x = 0 and solve for y:

y = -0/3 + 4 = 4

So the y-intercept is (0, 4).

- Identify the slope:

The slope is the rate at which the line rises or falls as it moves horizontally. The slope = -1/3.

Now,

Starting at the y-intercept of (0, 4), use the slope to find another point on the line. The slope of -1/3 means that for every 3 units to the right, the line goes down 1 unit.

So from the y-intercept, move 3 units to the right and 1 unit down to get the point (3, 3).

Plot this point.

Then,

Use a straight edge to draw a line through the two points.

This line is the graph of the equation y = -x/3 + 4.

Thus,

The equation y = -x/3 + 4 is graphed.

- Find the y-intercept

- Find the slope.

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After analysing the given data we conclude that the height of the streetlight is 29.4 feet, under the condition that a six-foot man places a 15 foot shadow. At the same time a streetlight casts an 80-foot shadow.

Now Let us consider the height of the streetlight "h".
The given angle of elevation is 52.5 degrees. This projects that the angle between the horizontal line and the line of sight to the top of the streetlight is 52.5 degrees.
We can apply the tangent function to evaluate h. tan(52.5) = h/20.
Evaluating  for h, we get h = 20 × tan(52.5) = 29.4 feet (rounded to one decimal place).
Therefore, the height of the streetlight is approximately 29.4 feet.
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The complete question is
A six-foot man casts a 15 foot shadow. At the same time a streetlight casts an 80-foot shadow.
The same six-foot tall man wants to indirectly measure the streetlight in screen 3. But it is a cloudy day and there are no shadows. So holding his phone by his eye, he uses the "level" feature on the Measure app to sight the top of the streetlight. Standing 20 feet away he finds an angle of elevation of 52.5 degrees.
Write and solve an equation to determine the height of the streetlight.

Answer:

-6.29

Step-by-step explanation:

Add two negative integers together is just as simple as two positive integers together.

Let's for a little bit ignore that negative. What is 4.058 + 2.232? 6.290 or just 6.29. Now slap a negative in front of it!

Your answer is -6.29.

Different rules arise when adding a negative integer with a positive integer. We can cross the bridge when we get there :)

Answer:

mean : 48.142

median: 52

mode: there is no mode

Step-by-step explanation:

I dont know about the explaining part

Solving an exponential equation, we can see that it will take 14.4 years.

We know that Darin invests $4000 in an account that earns 4.8% annual interest compounded continuously.

The formula for a continuous copound is:

[tex]f(t) = A*e^{r*t}[/tex]

Where A is initial amount and r is the rate of interest, in this case we have:

A = $4000

r = 0.048

Then the formula is.

[tex]f(t) = 4000*e^{0.048*t}[/tex]

It will be doubled when f(t) = 8000, then we need to solve:

[tex]8000 = 4000*e^{0.048*t}\\\\8000/4000 = e^{0.048*t}\\2 = e^{0.048*t}\\\\ln(2) = 0.048*t\\\\t = ln(2)/0.048 = 14.4[/tex]

So it will take 14.4 years.

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The volume of the frustum of a pyramid with square base of side 9, square top of side 4, and height 5 is approximately 64.39 cubic units.

To find the volume of the frustum, we need to use the formula V = (1/3)h(A1 + A2 + √(A1A2)), where V is the volume, h is the height, A1 is the area of the base, A2 is the area of the top, and √(A1A2) is the geometric mean of the areas. In this case, the height is 5, the base and top are both squares with sides of 9 and 4, respectively, and we can calculate the areas using A = s^2. Thus, A1 = 81 and A2 = 16. Plugging these values into the formula, we get V = (1/3)(5)(81 + 16 + √(81*16)) ≈ 64.39 cubic units. Therefore, the volume of the frustum of the pyramid is approximately 64.39 cubic units.

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Answer:

  y < 3/4x -2

Step-by-step explanation:

You want the inequality expression that corresponds to the given graph.

The boundary line rises 3 squares for each 4 to the right. Its slope is ...

  m = rise/run = 3/4

The boundary line crosses the y-axis at y = -2. Its y-intercept is ...

  b = -2

The slope-intercept form of the equation of the boundary line is ...

  y = mx +b

  y = 3/4x -2

The shading is below the dashed line, so the line is not part of the solution set. Only y-values less than those on the line are in the solution set.

The inequality that describes the graph is ...

  y < 3/4x -2

The half-life of the goo is approximately 18.75 minutes. The formula for g(t), the amount of goo remaining at time t, is g(t) = 132 * (1/2)^(t/18.75).

To find the half-life of the goo, we can use the formula for exponential decay: A(t) = A0 * (1/2)^(t/h), where A(t) is the amount of radioactive substance at time t, A0 is the initial amount, h is the half-life, and t is time. We are given A0 = 132 grams, A(75) = 2.0625 grams, and we need to solve for h. Plugging in these values, we get:

2.0625 = 132 * (1/2)^(75/h)

Solving for h, we get h ≈ 18.75 minutes.

The formula for g(t) is g(t) = A0 * (1/2)^(t/h). Plugging in A0 = 132 and h = 18.75, we get g(t) = 132 * (1/2)^(t/18.75). This formula gives us the amount of goo remaining at time t, where t is measured in minutes.

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The exact value of the expression cos^-1(cos(11π/6)) is π/ 6 The inverse cosine function (cos^-1) returns the angle whose cosine is a given value.

In this case, we are given the cosine of 11π/6, which is -√3/2 (since cosine is negative in the third quadrant where 11π/6 lies). The value of π/6 is the angle whose cosine is -√3/2. Therefore, the exact value of the expression is π/6.

This means that for any angle θ, the cosine of (θ + 2nπ) is the same as the cosine of θ, where n is any integer. In other words, the cosine function repeats itself every 2π radians.

The inverse cosine function, however, returns a unique angle between 0 and π whose cosine is a given value. In this case, since we are given a negative cosine value, we know that the angle must lie in the second or third quadrant.

By evaluating the cosine function for angles in those quadrants, we can determine which angle has the given cosine value. In this case, we find that the angle is π/6, which is in the third quadrant where cosine is negative.

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1.823 radians  are the coordinates of A and A,  The coordinates of B' are; B' = (-3 * 4, -2 * 4) = (12, -8)  

a) To enlarge a triangle with a scale factor of 4, we need to first enlarge the original triangle by a scale factor of 1, and then reflect it across the y-axis.

The coordinates of A can be found using the law of sines:

sin(A) = (a/2) / sin(c)

where a is the semi-perimeter of the original triangle, c is the semi-perimeter of the enlarged triangle, and sin(A) is the length of side A.

Substituting the given values, we get:

sin(A) = (4/2) / sin(6)

= 2 / 3 sin(6)

sin(6) = (a/2) / sin(A)

= (4/2) / 2 / sin(2)

= (4/2) / 2 * sin(2) / sin(A)

= (4/2) * sin(A) / sin(2)

= 4 * sin(A) / 3

Therefore, the length of side A is:

A = sin^-1(4 * sin(A) / 3)

= sin^-1(4) - sin^-1(sin(A) / 3)

= 1.823 radians

To find the coordinates of A', we can reflect the point A across the y-axis. The reflection is given by the equation:

(x, y) -> (-x, y)

Substituting the coordinates of A, we get:

A' = (-1.823, 1.823)

b) To find the coordinates of B', we need to find the coordinates of B first. We can do this by reflecting the point B across the y-axis using the equation:

(x, y) -> (-x, -y)

Substituting the coordinates of B, we get:

B = (-3, -2)

The coordinates of B' are:

B' = (-3, -2)

Since the triangle is enlarged with a scale factor of 4, the coordinates of B' will be multiplied by 4. Therefore, the coordinates of B' are:

B' = (-3 * 4, -2 * 4) = (12, -8)  

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The curves intersect at the points (0, -2, 24) and (2, 4, 36).

To find the intersection point of the curves r1(t) and r2(s), we need to equate their respective components and solve for the parameters t and s.

r1(t) = (t, 2 - t, 24 + t²)
r2(s) = (6 - s, s - 4, s²)

To find the point of intersection between the curves r1(t) and r2(s), we need to set the equations equal to each other and solve for t and s.

Step 1: From equation 1, t = 6 - s.
Step 2: Substitute t in equation 2:
2 - (6 - s) = s - 4
s - 4 = s - 2
s = 2

Setting the x-coordinates of the curves equal to each other, we get:

t = 6 - s

Setting the y-coordinates of the curves equal to each other, we get:

2 - t = s - 4

Simplifying this equation, we get:

t + s = 6

Finally, setting the z-coordinates of the curves equal to each other, we get:

24 + t² = s²

Substituting t = 6 - s into this equation, we get:

24 + (6 - s)² = s²

Expanding and simplifying, we get:

s² - 12s + 48 = 0

This quadratic equation can be factored as:

(s - 6)(s - 8) = 0

Therefore, s = 6 or s = 8.

Step 3: Substitute the value of s back into equation 1 to find t:
t = 6 - 2
t = 4

Substituting these values of s into the equation t + s = 6, we get:

t = 0 when s = 6

t = 2 when s = 8

Step 4: Now, substitute the values of t and s into either r1 or r2 to find the intersection point:
r1(4) = (4, 2 - 4, 24 + 4²) = (4, -2, 24 + 16) = (4, -2, 40)

Therefore, the curves intersect at the points (0, -2, 24) and (2, 4, 36).

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u(b) = b^2/2,  we can evaluate the integral from u(a) to u(b), giving us the new definite integral in terms of u.

To find the new limits of integration, we need to express the integral in terms of the new variable u. Using the change of variables formula, we have:

du/dx = x/2

dx = 2du/x

Substituting into the integral, we get:

∫ f(x) dx = ∫ f(x(u)) dx/du * 2du/x

Since u = x^2/2, we have x = √(2u). Substituting this into the integral, we get:

∫ f(x(u)) dx/du * 2du/√(2u)

Simplifying, we have:

∫ f(x(u)) √2 du

Now, we need to determine the new limits of integration in terms of u. If the original limits were a and b, then the new limits are:

u(a) = a^2/2

u(b) = b^2/2

Therefore, we can evaluate the integral from u(a) to u(b), giving us the new definite integral in terms of u.

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Answer:

Obtuse, scalene

Step-by-step explanation:

No same values for sides. So, scalene. Since 7^2 > 6^2 + sqrt5 ^2 , this triangle is obtuse.

The partial fraction decomposition of the rational function 4x/(16x^2 + 40x + 25) is given by the sum of two terms: A/(4x + 5) + B/(4x + 5)^2, where A and B are constants that can be solved using algebraic manipulation

To find the partial fraction decomposition of the given rational function, we first factor the denominator into two linear factors: (4x + 5)(4x + 5). Then, we can write the function as a sum of two terms with undetermined coefficients:

4x/(16x^2 + 40x + 25) = A/(4x + 5) + B/(4x + 5)^2

To solve for A and B, we can multiply both sides by the common denominator (4x + 5)^2 and simplify:

4x = A(4x + 5) + B

Expanding and equating coefficients of like terms, we get:

4x = 4Ax + 5A + B

0 = 16A + 4B

Solving for A and B, we get:

A = -1/16

B = 1/16

Therefore, the partial fraction decomposition of the given rational function is: 4x/(16x^2 + 40x + 25) = -1/(16(4x + 5)) + 1/(16(4x + 5)^2)

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The appropriate measure of variability for this data set is the IQR, and its value is 350.

The range is one measure of variability, but it is heavily influenced by extreme values, which makes it less reliable. The IQR (interquartile range) is a better measure of variability because it is not affected by extreme values. Therefore, the appropriate measure of variability for this data set is the IQR.

To calculate the IQR, we first need to find the median, which is the middle value in the ordered list of data:

200, 450, 600, 650, 700, 800

The median is (600 + 650) / 2 = 625.

Next, we find the values that mark the 25th and 75th percentiles of the data set. The 25th percentile is the value that is greater than 25% of the values in the data set, and the 75th percentile is the value that is greater than 75% of the values in the data set. We can use the following formula to find these values:

25th percentile = (n + 1) × 0.25

75th percentile = (n + 1) × 0.75

where n is the number of values in the data set. In this case, n = 6, so:

25th percentile = (6 + 1) × 0.25 = 1.75

75th percentile = (6 + 1) × 0.75 = 5.25

We round these values up and down to get the indices of the corresponding values in the ordered list:

25th percentile index = 2

75th percentile index = 5

The values at these indices are 450 and 800, respectively. Therefore, the IQR is:

IQR = 800 - 450 = 350

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Based on the normal model N(100,15) describing IQ scores, we can use the standard normal distribution to answer these questions. To find the percentage of people with IQs over 80, we need to calculate the Z-score for 80: Z = (80-100)/15 = -1.33. Using a standard normal distribution table, we find that the area to the right of Z = -1.33 is 0.0918, which means about 9.18% of people have IQs over 80.

To find the percentage of people with IQs under 90, we calculate the Z-score for 90: Z = (90-100)/15 = -0.67. Using the same table, we find that the area to the left of Z = -0.67 is 0.2514, which means about 25.14% of people have IQs under 90.

To find the percentage of people with IQs between 112 and 123, we need to calculate the Z-scores for 112 and 123: Z1 = (112-100)/15 = 0.80 and Z2 = (123-100)/15 = 1.53. Using the table, we find the area to the left of Z1 is 0.7881 and the area to the left of Z2 is 0.9370. Therefore, the percentage of people with IQs between 112 and 123 is approximately 14.89%.

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The translation rule is (x, y) → (x + (-2) , y + 4)

We have,

From the figure,

We see the coordinates of Figure I.

(4, -5), (2, 1), and (-3, -3) _____(1)

We see that the coordinates of Figure Ii.

(2, -1), (0, 5), and (-5, 1) _____(2)

Now,

From (1) and (2),

Taking the corresponding coordinates.

(4, -5) and (2, -1)

(2, 1) and (0, 5)

(-3, -3) and (-5, 1)

We see that,

x coordinates is substrated by 2 and y coordinate is added by 4.

So,

The translation rule is (x, y) → (x + (-2) , y + 4)

Thus,

The translation rule is (x, y) → (x + (-2) , y + 4)

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Answer:

y - 3 = -(x + 1)

y - 3 = -x - 1

y = -x + 2

Answer:

Pick options, 1,2,3,4 but DO NOT select option 5!!!

Step-by-step explanation:

The basic equation is y=mx+b where slope is m and b is the y intercept.

So our y intercept for this equation is 2. The line has a negative slope bc as increases, y decreases. (The line is pointing down.) Those are two good clues to start.

Let's calc the slope. slope = rise/run = (y2-y1)/(x2-x1)

(-1,3) and (1,1) are shown on the graph

slope = (3-1)/(-1-1)

= 2/-2 = -1

Slope = -1

Our equation is now y=-x+2

So let's find everything that equals y=-x+2

y-3 = -x-1 is the same as

y = -x+2

So pick the first option, y-3 = -x-1

(y+1) = -(x-3)

y+1 = -x+3

y=-x+2

So pick the 2nd option, (y+1) = -(x-3)

(y-3) = -(x+1)

y-3 = -x-1

y=-x-1+3

y=-x+2

So pick the 3rd option, (y-3) = -(x+1)

We already know to pick the 4th option, y=-x+2

(y-3) = (x-1)

y= x-1+3

y=x-2

DON'T PICK THE 5th option, because this has the wrong slope and wrong intercept!