The look-out point of a lighthouse is 50 feet above sea level. A woman observes a boat in the water from the look-out point. The angle of depression to the boat in the water is 20°.

What is the distance from the base of the light house to the boat in the water?

Read the problem and answer choices. You want to get from ABCD to EFGH, so you need to figure out how to do that with reflection, translation, and dilation—in that order.

The reflection part is fairly easy. ABC is a bottom-to-top order, and EFG is a top-to-bottom order, so the reflection is one that changes top to bottom. It must be reflection across a horizontal line. The only horizontal line offered in the answer choices is the x-axis. Selection B is indicated right away.

The dimensions of EFGH are 3 times those of ABCD, so the dilation scale factor is 3. This means that prior to dilation, the point H (for example), now at (-12, -3) would have been at (-4, -1), a factor of 3 closer to the origin. H corresponds to D in the original figure, which would be located at (0, -2) after reflection across the x-axis.

So, the translation from (0, -2) to (-4, -1) is 4 units left (0 to -4) and 1 unit up (-2 to -1).

The appropriate choice and fill-in would be ...

... B. Reflection across the x-axis, translation 4 units left and 1 unit up, dilation with center (0, 0) and scale factor 3.

_____

You can check to see that these transformations also map the other points appropriately. They do.

The straight line joining the points A(3,-5) and B(6,k) has a gradient of 4.

Gradient is the slope

So the slope of the line joining the points A(3,-5) and B(6,k) is 4

Slope of line joining two points = [tex]\frac{y_2-y_1}{x_2-x_1}[/tex]

A(3,-5) and B(6,k) are (x1,y1)  and (x2,y2)

slope = [tex]\frac{y_2-y_1}{x_2-x_1}[/tex]

slope = [tex]\frac{k-(-5)}{6-3}=\frac{k+5)}{3}[/tex]

We know slope =4

 [tex]\frac{k+5)}{3}=4[/tex]

Cross multiply  and solve for k

k + 5 = 12

k = 7

The value of k = 7

Answer: x = -2 and y = 4

Step-by-step explanation:

We have given a system of equations

2x + 4y =12   ...........(1)

and

x + y = 2         .............(2)

Now to plot them in graph we need to find the points of the above linear equations .

From equation 1  ,we get

[tex]4y=12-2x\\\Rightarrow\ y=\frac{12-2x}{4}\\ \Rightarrow\ y=3-\frac{x}{2}\\\text{Put x=0, then} \\y=3-\frac{0}{2}=3\\\Rightarrow\ \text{Put x=6}\\y=3-\frac{6}{2} =3-3=0[/tex]

So get the points (0,3) and (6,0).

From equation 2 ,we get

[tex]y=2-x\\\text{Put x=0}\\y=2-0=2\\\text{Put x=2}\\y=2-2=0[/tex]

So we get the points (0,2) and (2,0).

So after plotting these points we get two lines intersecting at (-2,4).

Therefore our solution is x=-2 and y= 4.

Answer:

(-2, 4)

Step-by-step explanation:

The 1:3 ratio means that the distance from A to the point is 1/4 of the distance from A to B.

The difference of y-coordinates is 10-8 = 2. 1/4 of that is 2·1/4 = 1/2, so the point of interest will have y-coordinate 8 + 1/2 = 8 1/2. This apparently corresponds to the first selection:

... (6 1/2, 8 1/2)

Given

Lee converted 500 U.S. dollars to 625 Singapore dollars.

x represents U.S. dollars and s represents Singapore dollars.

Find out equations represents the relationship between the two currencies.

To proof

As given in the question

x represents U.S. dollars and s represents Singapore dollars.

converted 500 U.S. dollars to 625 Singapore dollars

500 x = 625 s

[tex]x = \frac{625}{500} s[/tex]

x = 1.25 s

This shows that the U.S dollars is equal to 1.25 times of singapore dollars.

Hence proved

x = 3

2x + y - z = 3 → (1)

- x + y + 2z = 0 → (2)

3x + 2y + z = 9 → (3)

we require to eliminate the y and z terms from the equations

(1) + (3) : 5x + 3y = 12 → (4)

multiply (1) by 2

4x +2y - 2z = 6 → (5)

(2) + (5) : 3x + 3y = 6 → (6)

(4) - (6) : 2x = 6 ⇒ x = 3

look at the shaded right-angled triangle on the left

8 is the hypotenuse n a is opposite to the 40-degree angle

by definition, sin = opposite side / hypotenuse

so sin40=a/8

rearranging a=8sin40

Two right-angled triangles share a common side with length a.

For the inverted triangle on the left, a is the opposite side to the 40-deg angle.

Its longest side, hypotenuse, is 8cm.

Use the sine function, sin40 = opposite/hypotenuse = a/8

Multiply 8 on both sides, a = 8sin40

Answer:

The Answer is B

Step-by-step explanation:

Subtract f-g

4x + 1 - x^2 - 5

Distribute the - sign

4x + 1 - x^2 + 5

Combine like terms

-x^2 + 4x + 1 + 5

-x^2 + 4x + 6

First we need to know how much the discount was so 5% is actually 0.05 then we need to multiply it by 53.75 which would equal 2.6875 then subtract it by 53.75 which would equal -51.0625.

B) .95(53.75)

This is a trick question because D) could also be correct if they said you could subtract it but it's not saying use the distributive property.

The answer would be B) .95(53.75)

Answer:

C. consecutive interior angles

Step-by-step explanation:

The subject angles are between lines a and b, so are interior (not exterior). They are on different corners of the intersection, so are not corresponding. They are on the same side of line c, so are not alternate. The share a side, but not a vertex, so they are consecutive. The appropriate choice is ...

... C. consecutive interior angles

Answer: C. consecutive interior angles

300

divide 9 × [tex]10^{6}[/tex] by 3 × [tex]10^{4}[/tex]

= [tex]\frac{9}{3}[/tex] × [tex]10^{6}[/tex] / [tex]10^{4}[/tex]

= 3 × [tex]10^{6-4}[/tex] = 3 × [tex]10^{2}[/tex] = 300

Let x be the bigger number and y be the smaller number.

One number is 11 more than twice another number. Thus,

x = 2y + 11

The sum of the numbers is twice their difference, thus,

x + y = 2(x-y), which simplifies to:

x + y = 2x - 2y

3y = x  (Now plug this back to the first equation:

3y = 2y + 11, and solve:

y = 11

Plug in y = 11 to the first equation:

x = 2(11) + 11 = 22 + 11 = 33

Thus the numbers are: x = 33, y = 11.

Your question answers itself.

You have shown the correct answer for the first drop-down: -161/289

You have shown the 3 incorrect answers out of the 4 choices for the second drop-down. The correct choice is the remaining one: -240/161.

_____

Even if all you know is that tan = sin/cos, you would suspect that answer choice based on the numerator of the cosine: 161. That value is likely the one in the denominator of the tangent value.

_____

cos(2θ) = 2cos(θ)² . . . . trig identity

... = 2(-8/17)² -1 = 128/289 -1 = -161/289

sin(2θ) = √(1-cos(2θ)²) . . . . trig identity

... = √(289² -161²)/289 = √57600/289 = 240/289

tan(2θ) = sin(2θ)/cos(2θ) . . . . trig identity

... = (240/289)/(-161/289) = -240/161

Answer:

2.

1.

Step-by-step explanation:

Angles 135° and (2x+15)° together make up a line (the transversal crossing m and n). Such angles are called a "linear pair" and their sum is always 180°. That means we can write the equation ...

... 135° + (2x+15)° = 180°

... 150 +2x = 180 . . . . . . . remove the degree symbol, combine terms

... 2x = 30 . . . . . . . . . . . . subtract 150

... x = 15 . . . . . . . . . . . . . . divide by 2

Angle 1 and angle (2x+15)° are on opposite sides of the transversal line, and are both between the parallel lines m and n. This makes them alternate interior angles. Such angles are congruent—they have the same measure. We know the measure of angle (2x+15)° is (2·15+15)° = 45°, so we know the measure of ∠1 is also 45°.

a) The sum of angles in a triangle is always 180°. This means ...

... (15x +10)° + (15x -10)° + (3x +15)° = 180°

... 33x +15 = 180 . . . . . . . drop the ° symbol, combine terms

... 33x = 165 . . . . . . . . . . subtract 15

... x = 5 . . . . . . . . . . . . . . . divide by 33

b) ∠A = (15x+10)° = (15·5 +10)°

... ∠A = 85°

(1)

(a)

the sum of the angles in a triangle = 180°, hence

3x + 15 + 15x - 10 + 15x + 10 = 180

33x + 15 = 180 ( subtract 15 from both sides )

33x = 165 ( divide both sides by 33 )

x = 5

(b) ∠A = 15x + 10 = (15 × 5 ) + 10 = 75 + 10 = 85°

(2)

2x + 15 + 135 = 180 ( straight angle )

2x + 150 = 180 ( subtract 150 from both sides )

2x = 30 ( divide both sides by 2 )

x = 15 ⇒ 2x + 15 = 45

∠1 = 45° ( alternate angles are congruent )

Answer:

parallel

Step-by-step explanation:

Translation moves each point the same amount in the same direction. Essentially, the original segment becomes the side of a parallelogram, whose other side is the image, and whose ends are the vectors specifying the translation.

In the attachment, we have designated (a, b) as point A, and (c, d) as point B. We had to choose specific values for these in order to plot them, but the description of the effect of translation applies no matter what the point coordinates are chosen to be.

Solution: We have to find the Frequency and Relative frequency of the given data:

Frequency is the number of times a number occurs.

Relative Frequency is the number of times a number occurs divided by the total number of items.

Therefore, the frequency and relative frequency are calculated as below:

Number       Frequency          Relative Frequency

20                       1                     [tex]\frac{1}{31} \times 100 =3.2\%[/tex]

21                        4                    [tex]\frac{4}{31} \times 100 =12.9\%[/tex]

22                       2                    [tex]\frac{2}{31} \times 100 =6.5\%[/tex]

23                       4                    [tex]\frac{4}{31} \times 100 =12.9\%[/tex]

24                       3                    [tex]\frac{3}{31} \times 100 =9.7\%[/tex]

25                       2                    [tex]\frac{2}{31} \times 100 =6.5\%[/tex]

26                       3                    [tex]\frac{3}{31} \times 100 =9.7\%[/tex]

27                       5                    [tex]\frac{5}{31} \times 100 =16.1\%[/tex]

28                       3                    [tex]\frac{3}{31} \times 100 =9.7\%[/tex]

29                       4                    [tex]\frac{4}{31} \times 100 =12.9\%[/tex]

Total                  31

Ok, so what this question is really asking you is: [tex]2/5=0.4[/tex]

What is [tex]6*0.4[/tex] and that equals [tex]6*0.4=2.4[/tex]

Now because the the number is 2.4 plot that point

The area of the sector of a circle with diameter 34 feet and an angle of π/5 radians is approximately 9.0095 square feet.

To find the area of the sector of a circle, we need to use the formula A = πr2θ/360, where A is the area, r is the radius, and θ is the angle in radians. The given diameter is 34 feet, so the radius is 17 feet. The angle is π/5 radians. Plugging these values into the formula, we get A = π(17)2(π/5)/360. Simplifying, we find A = 9.0095 square feet. Rounded to four decimal places, the area of the sector is 9.0095 square feet.

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To find the area of the sector of a circle, you use the formula A = 0.5 * r² * θ where r is the radius (half of the diameter) and θ is the angle in radians. After plugging in the provided values, you calculate the area to four decimal points.

The subject is related to finding the area of the sector of a circle. The formula to calculate the area of the sector is A = 0.5 * r² * θ, where r is the radius and θ is the angle in radians. In the given problem, the diameter of the circle is given as 34 feet, so the radius will be 17 feet, and the angle is given as π/5 radians.

Step 1: First, note down the radius from the given diameter, which would be 17 feet, as radius = diameter/2.

Step 2: Plug the value of the radius and the angle into the formula for the area of a sector. The calculation will look like this: A = 0.5 * (17)² * (π/5).

Step 3: Calculate the area by multiplying the values. The final answer will be rounded to four decimal places.

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Hey!

We have:

... 3x + 5y = 2

... 2x - 6y = 20

In order to get 5x - y , we will need to add both the equations.

3x+5y=2

2x-6y=20

____________________________________________________

...5x - y = 22

Hence, the required answer is 22.

Hope it helps!

Linear equations in two variables are solved to obtain the values of variables. Thus, the value of the equation [tex]5x - y[/tex] is [tex]\bold{22}[/tex].

Given equations are mentioned below:

[tex]3x + 5y = 2\\2x - 6y = 20[/tex]

We need to determine the values of the expression [tex]5x - y[/tex].

For calculating the value of the above expression, first we need the value of the variables x and y.

[tex]\begin {aligned}3x + 5y &= 2\\3x&=2-5y\\x&=\dfrac{2-5y}{3} \end{aligned}[/tex]

Now, substitute the value of the x in another expression and solve it further.

[tex]\begin{aligned}2 \times \dfrac{2-5y}{3} -6y&=20\\ \dfrac{4-10y}{3}-6y&=20\\4-10y-18y&=60\\28y&=-56\\y&=-2 \end{aligned}[/tex]

Now, calculate the value of x.

[tex]\begin{aligned}x&=\dfrac{2-5 \times -2}{3}\\&=\dfrac{12}{3}\\&=4 \end{aligned}[/tex]

Now, calculating the value of  [tex]5x - y[/tex].

[tex]5\times4 - (-2)=22[/tex]

Hence, the value of the expression [tex]5x - y[/tex] is 22.

Learn more about the linear equations in two variables here:

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The product of the two roots is c/a.

_____

Consider the equation

... a(x -p)(x -q) = 0

which has solutions x=p and x=q.

When multiplied out, it becomes ...

... a(x² -(p+q)x +pq) = ax² -a(p+q)x +apq = 0

When apq = c, the product pq is c/a.

A. -2/3 because you would go down two and then to the right three. I hope that helps.

Solution :

Given ,line that passes through the  pair of points (1,7) and (10,1)

The slope of the line passing through the points [tex](x_{1},y_{1} ) \:and \:(x_{2},y_{2})[/tex] is given by,

                         [tex]m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}[/tex]

Here,  [tex](x_{1},y_{1} ) =(1,7)\:and \:(x_{2},y_{2})=(10,1).[/tex]

[tex]\Rightarrow m=\frac{1-7}{10-1} \\\Rightarrow m=\frac{-6}{9}=\frac{-2}{3}[/tex]

Hence , slope of the line passing through the pir of points (1,7) and (10,1) is [tex] \frac{-2}{3}[/tex] ( first option)

The population is multiplied by 100% -2.2% = 97.8% each year. After 5 years, the population will have been multiplied by this value 5 times, so will be ...

... 3500×0.978⁵ ≈ 3132

The appropriate choice is ...

... C. 3132

The slope of your line is the x-coefficient: 5.

The slope of a perpendicular line is the negative reciprocal of that: -1/5.

[tex]\frac{6}{8}[/tex]

let x be the unknown value in the ratio, hence

[tex]\frac{6}{x}[/tex] = [tex]\frac{9}{12}[/tex] ( cross- multiply )

9x = 72 ( divide both sides by 9 )

x = 8

thus [tex]\frac{6}{8}[/tex] = [tex]\frac{9}{12}[/tex]

2/6 is remaining from Jillian's whole.

[tex]\frac{2}{6}[/tex] = [tex]\frac{1}{3}[/tex]

Since the denominators of both fractions are common , that is 6

To subtract, subtract the numerators leaving the denominator

[tex]\frac{5}{6}[/tex] - [tex]\frac{3}{6}[/tex] = [tex]\frac{5-3}{6}[/tex] = [tex]\frac{2}{6}[/tex]

This fraction may be simplified by dividing the numerator/ denominator by 2

[tex]\frac{2}{6}[/tex] = [tex]\frac{1}{3}[/tex] ← in simplest form

(a) 6% of the value of the investment is added each year. This means the value of the investment is multiplied by 1.06 each year. After 15 multiplications, the value is ...

... $3000×1.06¹⁵ ≈ $7189.67

(b) The interpretation of this result is ...

... After 15 years, the investment is worth $7189.67.

The future value of a $3000 investment at an annual interest rate of 6%, compounded annually for 15 years, is approximately $8093.97. This means that the original sum will grow to this amount after the set period.

To find the future value of an investment, we use the formula for compound interest which is A = P(1 + r/n)^(nt). Where:

Given are P = $3000, r = 6%/100 = 0.06 (annually), n = 1 (since it's compounded annually), and t = 15 years. Plugging these into the formula, we get A = $3000(1 + 0.06/1)^(1*15) = $3000(1.06)^15 = $8093.97.

Interpreting this, it means that investing $3000 at 6% compounded annually would grow to approximately $8093.97 after 15 years.

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f(3) = -9 for the function [tex]\( y = f(x) = -3x \) when \( x \) is 3.[/tex]

To find f(x) when ( x ) is 3 for the given function [tex]\( y = f(x) = -3x \),[/tex] you substitute 3 for ( x ) in the function:  [tex]\[ f(3) = -3 \times 3 \][/tex]

Now, calculate the value: [tex]\[ f(3) = -9 \][/tex]

Therefore, when ( x ) is 3, f(x) is -9 for the function [tex]\( y = f(x) = -3x \).[/tex]

In more detail, this means that if you plug ( x = 3 ) into the function, it will result in ( y = -9 ). The function ( y = -3x ) represents a linear relationship where the coefficient of ( x ) is -3. This indicates that for each unit increase in ( x ), ( y ) decreases by 3 units. In the specific case of ( x = 3 ), substituting this value into the function gives [tex]\( y = -3 \times 3 = -9 \).[/tex]

This kind of analysis is fundamental in understanding the behavior of linear functions. It provides insight into how the function's output (y) changes in response to changes in the input ( x ). In this case, when ( x ) increases by 1, ( y ) decreases by 3, leading to the slope of -3 in the function ( y = -3x ).

352 divided by 53 is 6.64150943

Hi TaeArmy23,

352 / 53

= 6.6415

= 6.6

Hope This Helps!

Answer:

(a) 0

(b) 2

(c) 0

(d) 1

Step-by-step explanation:

All of these can be solved easily by a graphing calculator. The attached graphs show the real solutions of those equations that have them. Here, you can also answer the question from the sign of the discriminant.

For y = ax² +bx +c

the value of the discriminant is

... b² -4ac

When the discriminant is negative, both solutions are complex. When 0, there is one solution. When positive, there are two real solutions.

(a) The discriminant is ...

... (-9)² -4(12)(4), a negative number. There are no real solutions.

(b) This needs to be rearranged to ...

... y = -x² -10x +2

Then the discriminant is ...

... (-10)² -4(-1)(2), a positive number. There are 2 real solutions.

(c) This needs to be rearranged to ...

... y = 5x² -x +9 . . . . add 7-3y

Then the discriminant is ...

... (-1)² -4(5)(9), a negative number. There are no real solutions.

(d) This equation is in vertex form, and the vertex is (4, 0). Since the vertex is the only x-intercept, there is one real solution.