Lesson 8 Practice Problems
1. A number line
can represent positions that are north and south of a truck stop on a
highway. Decide whether you want positive positions to be north or south of the
truck stop. Then plot the following positions on a number line.
a. The truck
stop
b. 5 miles north of the truck
stop
C. 3.5 miles south of the truck stop

The formula for the pressure of a gas is given by the Ideal Gas Law, which states that P = nRT/V, where P is the pressure, n is the amount of gas in moles, R is the ideal gas constant, T is the temperature in kelvin, and V is the volume in cubic centimeters (cc).

Given the initial conditions, we can find the value of the ideal gas constant R:
1170 kpa = (6 moles)(R)(260 K) / (720 cc)
R = (1170 kpa)(720 cc) / (6 moles)(260 K)
R = 8.27 kpa·cc/mol·K
Now, we can use this value of R to find the pressure under the new conditions:
P = (3 moles)(8.27 kpa·cc/mol·K)(280 K) / (180 cc)
P = 3898.6 kpa·cc / 180 cc
P = 2166 kpa

Therefore, the pressure of the gas when the number of moles is 3, the temperature is 280 K, and the volume is 180 cc is 2166 kpa.

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The measure of angle IHJ is 131 degree.

The angle subtended by an arc at the center is twice the angle subtended at the circumference . More simply, the angle at the center is double the angle at the circumference.

From the figure,

3) Given that, arc ML=53 degree

∠LHM = 1/2 ×53

= 26.5

Arc MI = 89°

∠MHI=1/2 × 89°

= 44.5°

Here, ∠IHJ+∠MHI+∠LHM+∠LHK+∠KHJ=360°

∠IHJ+44.5°+26.5°+88°+70°=360°

∠IHJ=360°-229°

∠IHJ=131°

4) Arc WV= 1/2 ×55° = 27.5°

Here, Arc WT = Arc TU + Arc UV + Arc WV

= 50°+100°+27.5°

= 177.5°

5) Here, HG= 1/2 ×40 =20°

Arc FH= Arc FG+ Arc HG°

= 60°+20°

= 80°

Therefore, the measure of angle IHJ is 131 degree.

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If you have a 3600 m³ cube, then you can fit 3600 kiloliters in the cube.

A kiloliter (kL) is defined as a unit of volume which is used to measure liquids. It is equal to 1,000 liters, and it is used to measure large volumes of water, oil, and other liquids.

We know that 1 cubic meter is equal to 1 kiloliter,

We have to convert 3600 meter cube to Kiloliter,

So , we multiply by 3600,

On multiplying,

We get,

⇒ 3600 meter cube = 3600 × 1 Kiloliter,

⇒ 3600 meter cube = 3600 Kiloliter,

Therefore, 3600 Kiloliter can fill in the cube having the volume as 3600 meter cube.

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For the logarithmic function y = log (5x+6), the domain is (-6/5, [infinity]).

The domain of a logarithmic function is the set of all values for which the function is defined. In the case of y = log(5x+6), the function is only defined for values of x that make the expression inside the logarithm, 5x+6, greater than zero. This is because the logarithm of a negative number or zero is not defined.

To find the domain analytically, we need to solve the inequality 5x+6 > 0 for x:

5x+6 > 0

5x > -6

x > -6/5

This means that the domain of the function is all values of x greater than -6/5. In interval notation, this can be written as (-6/5, infinity).

So the domain of the logarithmic function y = log(5x+6) is (-6/5, infinity).

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The domains of each function and each composite function are as follows:

domain of f: (-∞, ∞)
domain of g: (-∞, ∞)
domain of f ∘ g: (-∞, ∞)
domain of g ∘ f: (-∞, ∞)

To find the composite functions f ∘ g and g ∘ f, we need to substitute the function g(x) into f(x) for f ∘ g, and the function f(x) into g(x) for g ∘ f.

(a) f ∘ g = f(g(x)) = f(x3 + 1) = 3(x3 + 1) − 8 = 3x3 + 3 − 8 = 3x3 − 5

(b) g ∘ f = g(f(x)) = g(3x − 8) = (3x − 8)3 + 1 = 27x3 − 72x2 + 64x − 511

The domain of a function is the set of all values of x for which the function is defined.

The domain of f is all real numbers, since there are no restrictions on the values of x. So the domain of f is (-∞, ∞).

The domain of g is also all real numbers, since there are no restrictions on the values of x. So the domain of g is (-∞, ∞).

The domain of f ∘ g is the same as the domain of g, since the values of x are first substituted into g(x) before being substituted into f(x). So the domain of f ∘ g is (-∞, ∞).

The domain of g ∘ f is the same as the domain of f, since the values of x are first substituted into f(x) before being substituted into g(x). So the domain of g ∘ f is (-∞, ∞).

Therefore, the domains of each function and each composite function are as follows:

domain of f: (-∞, ∞)
domain of g: (-∞, ∞)
domain of f ∘ g: (-∞, ∞)
domain of g ∘ f: (-∞, ∞)

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The solution set is: [tex]$\left{\frac{-1+\sqrt{5}}{3}, \frac{-1-\sqrt{5}}{3}\right}$[/tex]
Option (D) is correct.

What is a quadratic equation?

A quadratic equation is a type of equation in algebra that can be written in the form of ax^2 + bx + c = 0, where x is the unknown variable, and a, b, and c are constants with a not equal to zero

The equation is:

[tex]$-\frac{3}{2}x^2 = x + 1$[/tex]

We can rewrite this equation as:

[tex]$-\frac{3}{2}x^2 - x - 1 = 0$[/tex]

To solve for x, we can use the quadratic formula:

[tex]$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$[/tex]

Where a = -3/2, b = -1, and c = -1. Substituting these values into the quadratic formula, we get:

[tex]$x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(-\frac{3}{2})(-1)}}{2(-\frac{3}{2})}$[/tex]

Simplifying this expression, we get:

[tex]$x = \frac{1 \pm \sqrt{5}}{3}$[/tex]

Therefore, the solution set is:

[tex]$\left{\frac{-1+\sqrt{5}}{3}, \frac{-1-\sqrt{5}}{3}\right}$[/tex]

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Answer: 76

Step-by-step explanation:

58 and 18 are both numbers that don't have any variables, so they are like terms, so you just add them together

58+18=76

If you need to break it down a bit more:

50+10=60

8+8=16

60+16=76

The solution of the system of equations is ([2+9.48i]/2, 17+9.48i).

A system of linear equations consists of two or more equations made up of two or more variables such that all equations in the system are considered simultaneously. The solution to a system of linear equations in two variables is any ordered pair that satisfies each equation independently.

The given system of equations are y=(x-2)²+35 ----(i) and y=-2x+15 ----(ii).

Here, equation (i) = (ii)

(x-2)²+35= -2x+15

x²-4x+4+35= -2x+15

x²-4x+39= -2x+15

x²-4x+39+2x-15=0

x²-2x+24=0

x²-2x+24=0

By using quadratic formula, we get

x = [-b ± √(b² - 4ac)]/2a

x=[2±√((-2)² - 4×1×24)]/2×1

x=[2±√(-90)]/2

x=[2±9.48i]/2

Here, x=[2+9.48i]/2 and x=[2-9.48i]/2

So, y=2+9.48i+15=17+9.48i

Therefore, the solution of the system of equations is ([2+9.48i]/2, 17+9.48i).

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The scale factor from the original pool's dimensions to the new pool's dimensions is approximately 3.682.

The volume of a rectangular prism (such as a swimming pool) is given by the formula V = lwh, where l is the length, w is the width, and h is the height.

Let x be the scale factor from the original pool's dimensions to the new pool's dimensions. Then, the dimensions of the new pool can be expressed as lx, wx, and hx, where l, w, and h are the dimensions of the original pool.

We know that the original pool can hold 60,000 L of water, so we can set up the equation:

60,000 = lwh

Substituting in lx for l, wx for w, and hx for h, we get:

60,000 =[tex](lx)(wx)(hx)[/tex]

We also know that the new pool can hold 3,039,180 L of water, so we can set up another equation:

3,039,180 = [tex](lx)(wx)(hx)[/tex]

The result of dividing the second equation by the first equation is:

3,039,180/60,000 = [tex](lx)(wx)(hx)/(lx)(wx)(hx)[/tex]

Simplifying, we get:

50.653 =[tex]x^3[/tex]

The result of taking the cube root of both sides is:

x ≈ 3.682

Therefore, the scale factor from the original pool's dimensions to the new pool's dimensions is approximately 3.682.

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Answer:6 - 1 = 5

Step-by-step explanation: 12 / 2 = 6. 2 / 2 = 1.

The equation for the given tangent function is f(x) = 3 tan(x - π/2) - 1.

What is Trigonometric function ?

Trigonometric functions are mathematical functions that relate to angles of a right-angled triangle. The three most common trigonometric functions are sine, cosine, and tangent, which are denoted by sin, cos, and tan, respectively.

The general form of a tangent function is given by:

f(x) = A tan(B(x - C)) + D

where A is the amplitude, B is the frequency (inverse of the period), C is the horizontal shift, and D is the vertical shift.

Given the information, we have:

A = 3

period = π

frequency = 1/period = 1/π

horizontal shift = π/2 to the right

vertical shift = down 1

So, we can plug in the values into the general form and get:

f(x) = 3 tan(1(x - π/2)) - 1

Simplifying:

f(x) = 3 tan(x - π/2) - 1

Therefore, the equation for the given tangent function is f(x) = 3 tan(x - π/2) - 1.

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Answer:

70

Step-by-step explanation:

When we want to find 10% of a number, we move it's current decimal to the left one time.

700.

     ←

70.

10% of 700 is 70.

Other interactions that are confounded with the blocks are BCD and BCD.

(a) Constructing a one replicate of a 24 factorial design in factors A, B, C, and D in blocks of size 4 by confounding the interactions ABC and ACD with the blocks:

1. Number the blocks from 1 to 4.

2. Assign the factor levels to each block, so that the interactions ABC and ACD are confounded. One possible way to do this is as follows:

3. List the contents of two blocks, one of which contains the treatment combination (1) and the other contains treatment combination a.

(b) Other interactions that are confounded with the blocks are BCD and BCD.

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The correct measure is given as follows:

x = 6.

Vertical angles are angles that are opposite by the same vertex on crossing segments, hence they share a common vertex, thus being congruent, meaning that they end up having the same angle measure.

Over vertex C, the two angles are congruent, hence the value of x is obtained as follows:

12x - 9 = 6x + 27

6x = 36

x = 6.

(as the vertical angles are congruent, we can just equal the measures of the two angles and then solve the expression for the value of x).

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Answer:

1. plot the point (6,5) and label it Mr. Snuffles

2. 26 treats were eaten in total (21 not counting Mr.Snuffles)

Step-by-step explanation for plotting the point:

On the bottom line (miles walked) go to the number six, then go up five until it is on the five line for the side line (number of treats) plot the point here and then label the point Mr.Snuffles.

plot the point (6,5)

21 treats were eaten in total

Step-by-step explanation:

On the X axis, go to the number six, then go up five until it is on the five line for the side line (number of treats) plot the point here and then label the point Mr. Snuffles.

The end behavior of y as x goes to negative infinity in the equation y = -3x^5 + 4x^2 – 7 can be determined by looking at the leading term of the polynomial, which is -3x^5. As x goes to negative infinity, this term will dominate the other terms and determine the end behavior of the function.

Since the degree of the leading term is 5, which is an odd number, and the coefficient is negative, the end behavior of the function will be that y goes to positive infinity as x goes to negative infinity. This can be written in notation as:

lim(x→-∞) y = +∞

In conclusion, the end behavior of y as x goes to negative infinity in the equation y = -3x^5 + 4x^2 – 7 is that y goes to positive infinity.

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Answer:

5.30

Step-by-step explanation:

One hundredth is 0.01. 5.31-0.01=5.30.

sorry for answering late!

The value of the variable 'x' in the isosceles right-angle triangle will be 2√2 units.

The Pythagoras theorem states that the sum of two squares equals the squared of the longest side.

The Pythagoras theorem formula is given as,

H² = P² + B²

The length of the hypotenuse is 4. And the triangle is an isosceles right triangle.

In the isosceles right triangle, the perpendicular and base of the triangle will be the same. Then the value of the variable 'x' is given as,

4² = x² + x²

2x² = 16

x² = 8

x = 2√2

The value of the variable 'x' in the isosceles right-angle triangle will be 2√2 units.

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To compare fractions using equivalent fractions, we need to find a common denominator, convert each fraction to an equivalent fraction with that denominator, and then compare the numerators directly.

To compare fractions 3/4 and 7/12 using equivalent fractions, we need to find a common denominator for both fractions. A common denominator is a number that is divisible by all the denominators of the fractions being compared. In this case, the least common denominator (LCD) for 4 and 12 is 12.

To convert 3/4 to an equivalent fraction with a denominator of 12, we need to multiply both the numerator and denominator by 3. This gives us the equivalent fraction of 9/12.

To convert 7/12 to an equivalent fraction with a denominator of 12, we need to multiply both the numerator and denominator by 1. This gives us the equivalent fraction of 7/12.

Now that both fractions have the same denominator, we can compare them directly. We can see that 9/12 is greater than 7/12 since 9 is greater than 7. Therefore, 3/4 is greater than 7/12.

This method allows us to compare fractions with different denominators and make accurate comparisons based on the relative size of the numerators.

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Answer:

Step-by-step explanation:

First we can determine that this is a special 45-45-90 triangle, since the two legs are congruent, meaning the two unknown angles are congruent as well:

180 = 90 + 2(theta)

theta = 45

Knowing this, we can remember that the hypotenuse of this special triangle is equal to the leg*sqrt(2).

This means that 1 = x(sqrt(2)).

[tex]x=\frac{1}{\sqrt{2} }[/tex]

Hope this helps!

The value of (f· g)(x) from the given functions is  -x^4 - x^2 - x.

Functions are the fundamental part of the calculus in mathematics. The functions are the special types of relations. A function in math is visualized as a rule, which gives a unique output for every input x.

Here,

The given functions are f(x)=x³+x+1 and g(x)=-x.

(f·g)(x)= f(x) × g(x)

= (x³+x+1) × (-x)

=  -x^4 - x^2 - x

so, e get,

(g·f)(x)= g(x) × f(x)

= (-x) × (x³+x+1)

= -x^4 - x^2 - x

Therefore, the value of (f· g)(x) from the given functions is  -x^4 - x^2 - x.

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Answer:

scatter plot with points plotted at 1 comma 2, 1 comma 3, 3 comma 2, 3 comma 3, 4 comma 2, 4 comma 5, 5 comma 6, 6 comma 6, 7 comma 8, 8 comma 7, 9 comma 10, and 10 comma 9, with a line drawn through the points 4 comma 5 and 5 comma 6.

Step-by-step explanation:

The scatter plot shows a collection of points on a coordinate plane. To determine which of the given graphs shows a line that fits the data, we need to analyze the points and their distribution.

From the given points, we can observe that there is no clear linear relationship between the x and y variables. However, we can see that there is a cluster of points between (4,2) and (7,8), and the rest of the points are scattered around this cluster. Therefore, any line that fits the data should pass through this cluster of points.

Now, let's analyze each option:

The first option shows a line passing through (6,6) and (7,7). While these two points are part of the cluster, the line does not seem to fit the rest of the data very well.

The second option shows a line passing through (4,5) and (5,6). These points are also part of the cluster, and the line seems to fit the data reasonably well.

The third option shows a line passing through (3,3) and (4,4). These points are near the edge of the cluster, but the line seems to fit the data reasonably well.

The fourth option shows a line passing through (1,2) and (7,8). While these two points are part of the cluster, the line does not seem to fit the rest of the data very well.

Based on the analysis above, the second and third options seem to be the best fits for the data. However, the second option seems to be a slightly better fit than the third option since the line passes through two points that are closer to the center of the cluster. Therefore, the answer is:

scatter plot with points plotted at 1 comma 2, 1 comma 3, 3 comma 2, 3 comma 3, 4 comma 2, 4 comma 5, 5 comma 6, 6 comma 6, 7 comma 8, 8 comma 7, 9 comma 10, and 10 comma 9, with a line drawn through the points 4 comma 5 and 5 comma 6.

Answer: B

scatter plot with points plotted at 1 comma 2, 1 comma 3, 3 comma 2, 3 comma 3, 4 comma 2, 4 comma 5, 5 comma 6, 6 comma 6, 7 comma 8, 8 comma 7, 9 comma 10, and 10 comma 9, with a line drawn through the points 4 comma 5 and 5 comma 6

Step-by-step explanation:

The value of h(-9)= 26 using the function h(x)=−2x+8

And, the value of x for which h(x) = -10 is x = 9.

To evaluate h(-9), we simply plug in -9 for x in the function h(x) = -2x + 8:

h(-9) = -2(-9) + 8

h(-9) = 18 + 8

h(-9) = 26

So the value of h(-9) is 26.

To find the value(s) of x for which h(x) = -10, we can set the function equal to -10 and solve for x:

-2x + 8 = -10

-2x = -18

x = 9

So the value of x for which h(x) = -10 is x = 9.

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From left to right, function f has zeros at x = 0, x = 3, x = -3, x = 1/2, and x = -3/2.

An expression, rule, or law in mathematics that explains how one independent variable and one dependent variable are connected (the dependent variable).

The factors of the function f(x) can be found by factoring the expression:

f(x) = 2x⁴ - x³ - 18x² + 9x = x(2x³ - x² - 18x + 9)

To find the zeros of f(x), we need to find the values of x that make the expression in the parentheses equal to zero:

2x³ - x² - 18x + 9 = 0

We can use synthetic division or other methods to factor this polynomial and find its zeros. Alternatively, we can use the Rational Zeros Theorem to test possible rational zeros:

Possible rational zeros: ±1, ±3, ±9, ±1/2, ±3/2, ±9/2

Testing x = 1: 2(1)³ - (1)² - 18(1) + 9 = -8, not a zero

Testing x = -1: 2(-1)³ - (-1)² - 18(-1) + 9 = 28, not a zero

Testing x = 3: 2(3)³ - (3)² - 18(3) + 9 = 0, a zero

Testing x = -3: 2(-3)³ - (-3)² - 18(-3) + 9 = 0, a zero

Using polynomial division or factoring by grouping, we can factor the polynomial further:

2x³ - x² - 18x + 9 = (x - 3)(2x² + 5x - 3)

The quadratic factor can be factored using the quadratic formula or other methods:

2x² + 5x - 3 = (2x - 1)(x + 3)

Therefore, the zeros of f(x) are:

x = 0 (from the factor x)

x = 3 (from the factor x - 3)

x = -3 (from the factor x + 3)

x = 1/2 (from the factor 2x - 1)

x = -3/2 (from the factor 2x - 1)

So the completed statement is:

From left to right, function f has zeros at x = 0, x = 3, x = -3, x = 1/2, and x = -3/2.

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The slope of the curve at a particular point on the hyperbolic spiral, we need to take the derivative of the equation with respect to θ and evaluate it at the given value of θ. Slope of the curve found to be = 2pi. The correct answer is option C

The equation of the hyperbolic spiral is given by: r = 1/θ To express this equation in terms of x and y, we use the polar-to-rectangular coordinate transformation: x = r cos(θ) y = r sin(θ)

Substituting the equation for r, we get: x = (cos(θ))/θ y = (sin(θ))/θ Taking the derivative of y with respect to x using the chain rule, we get:

[tex](dy/dx) = (dy/dθ)/(dx/dθ) (dy/dx) = [(cos(θ)/θ^2) + (sin(θ)/θ)] / [(-sin(θ)/θ^2) + (cos(θ)/θ)] At t = π, θ = π/2.[/tex]

Therefore, the slope of the hyperbolic spiral at t = π is: (dy/dx) = 2/π The correct option to this value is C. 2pi

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The expanded form of the expression as a sum constant multiple of logarithms is:

log₄(xy⁵z⁴) = log₄(x) + 5log₄(y) + 4log₄(z)

There are four basic properties of logarithms:

logₐ(xy) = logₐx + logₐy.

logₐ(x/y) = logₐx - logₐy.

logₐ(xⁿ) = n logₐx.

logₐx = logₓa / logₓb.

According to the problem, we will use some of the basic logarithmic properties,

Given expression,

log₄(xy⁵z⁴)

We can use the properties of logarithms to expand the expression:

log₄(xy⁵z⁴) = log₄(x) + log₄(y⁵) + log₄(z⁴)

Using the properties of logarithms, we can separate the terms inside the logarithm as separate logarithms, where the multiplication of variables is represented as a sum of logarithms.

So, we have:

log₄(xy⁵z⁴) = log₄(x) + 5log₄(y) + 4log₄(z)

Therefore, the expanded form of the expression as a sum constant multiple of logarithms is:

log₄(xy⁵z⁴) = log₄(x) + 5log₄(y) + 4log₄(z)

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In the following question, among the conditions given, the circumference of a cylinder is "20π cm, or approximately 62.83 cm" with a 6 cm radius and a 4 cm height.

The formula for the circumference of a cylinder is:

Circumference = 2πr + 2πh

where r is the radius of the circular base and h is the height of the cylinder.

In this case, the radius is 6 cm and the height is 4 cm, so we can plug those values into the formula:

Circumference = 2π(6) + 2π(4)

Circumference = 12π + 8π

Circumference = 20π

So the circumference of the cylinder is 20π cm, or approximately 62.83 cm (if we use a calculator to approximate the value of π to two decimal places).

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Answer:

y ≈ 27

Step-by-step explanation:

using the tangent ratio in the right triangle

tan48° = [tex]\frac{opposite}{adjacent}[/tex] = [tex]\frac{30}{y}[/tex] ( multiply both sides by y )

y × tan48° = 30 ( divide both sides by tan48° )

y = [tex]\frac{30}{tan48}[/tex] ≈ 27 ( to the nearest whole number )

The area of rectangle S is 84.15 square unit.

Area is the region enclosed by the shape of an object. The space covered by a figure or any two-dimensional geometric shape in the plane is the area of ​​the shape, or

Area is a measure of the surface area of ​​a shape. To find the area of ​​a rectangle or square, you need to multiply the length and width of the rectangle or square.

Solution according to the given information in the question:

Area of rectangle R = 187

Ratio of rectangle R = 17

Ratio of rectangle S = 7.65

Area of rectangle S = (7.65/17)×187

                                 = 84.15 square unit

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Step-by-step explanation:

Step 1: Simplify the identity we need to find

[tex] \sin(2x) = 2 \sin(x) \cos(x) [/tex]

So we need to find sin and cos

Here, we are given tan(x)

A simple way to find sin (x) and cos(x) when given tan(x) is to use the definition of the trig functions of acute angles.

[tex] \tan( \alpha ) = \frac{y}{x} [/tex]

[tex] \cos( \alpha ) = \frac{x}{r} [/tex]

[tex] \sin( \alpha ) = \frac{y}{r} [/tex]

where r is

[tex]r = \sqrt{ {x}^{2} + {y}^{2} } [/tex]

Here, y is 4 and x is 3.

So

[tex]r = \sqrt{ {3}^{2} + {4}^{2} } = 5[/tex]

Since both cosine and sine are in quadrant 3, they are both negative.

Know we can plug in the knowns, since we know x,y, and r.

[tex] \cos( \alpha ) = - \frac{3}{5} [/tex]

[tex] \sin( \alpha ) = - \frac{4}{5} [/tex]

Now plug in the knowns for sin 2a

[tex] \sin(2 \alpha ) = 2( - \frac{3}{5} )( - \frac{4}{5} ) = \frac{24}{25} [/tex]