Find The Values Of The Trigonometric Ratios For X Using The Graph Of The Unit Circle. (2024)

[tex]\begin{gathered} y=(x+2)^2-9 \\ y=(x+2)(x+2)-9 \\ y=x^2+2x+2x+4-9 \\ y=x^2+4x-5 \end{gathered}[/tex]

The quadratic equation is

[tex]y=x^2+4x-5[/tex]

The solutions to the question can all be gotten from the graph of the quadratic equation;

The axis of symmetry is the point on the x-axis that divides the vertex of the parabola into two equal parts.

Therefore, the axis of symmetry occurs at x = -2

The graph curves downwards, hence it has a minimum point.

The minimum point occurs at y = -9

The y-intercept is the point where the curve cuts the y-axis when x = 0

Thus, the y-intercept occurs at y = -5

The x-intercept is the point where the curve cuts the x-axis at y = 0. The x-intercept also means the root of the equation

The x-intercept occurs at x = -5 and x = 1.

The domain of the equation is the set of all real values of x that will give real values for y.

Hence, the domain is from minus infinity to plus infinity. All real values of x satisfy the equation.

Domain =

[tex](-\infty,+\infty)[/tex]

The range of a quadratic graph is the set of all real values of y that you can get by inputting real values of x

The graph ranges from y greater than or equal to -9

Therefore, the range is ;

[tex]y\ge-9[/tex]

Answer:

A. Average rate of change = -30

B. x ≥ 0

Explanation:

Part B.

The average rate of change from x = a to x = b can be calculated as

[tex]\frac{f(b)-f(a)}{b-a}[/tex]

In this case, we want to calculate the average rate of change from x = 3 to x = 5, so f(3) = 120 and f(5) = 60. Then

[tex]\frac{f(5)-f(3)}{5-3}=\frac{60-120}{2}=-\frac{60}{2}=-30[/tex]

So, the average rate of change is -30. So, from x = 3 to x = 5, when we increase the price by $1, the profit decreases by $30.

Part C.

The domain is the set of values that the variable x can take. In this case, x is the price of the pens, so it has to be greater than or equal to 0.

Therefore, the constraints for the domain are

x ≥ 0

Here, we want to know the time it took the ballon to reach the ground from the given height

Now, the height on the ground is at 0 ft

Thus, at this point, h= 0

Solution

The answer is

Answer

The answer is -2.

Explanation

In trying to factorize the expression given by grouping,

Translation

We are given the rule for a translation: 1 unit down and 6 units left. It can be written as:

(x,y) --> (x - 6 , y -1 )

We need to subtract 6 units to the x-coordinate and subtract 1 unit to the y-coordinate.

The point L=(-2,6) will map to L' when applied the translation described above.

The coordinates of L' are:

L' = (-2-6 , 6 - 1 ) = (-8 , 5)

The coordinates of L' are (-8,5)

We can express this as a system of equations.

Let x be the liters of the 50%-acid solution and y the liters of the 75%-acid solution.

We then need 2 equations to solve this.

One equation is the total amount of liters (70 liters) that is equal to the sum of the amount of each solution:

[tex]x+y=70[/tex]

The second equation is the final concentration, that is a weighted average of the concentration of each solution:

[tex]0.5\cdot x+0.75\cdot y=0.7\cdot70=49[/tex]

Then, we can solve this by substitution:

[tex]x+y=70\longrightarrow y=70-x[/tex][tex]\begin{gathered} 0.5x+0.75y=49 \\ 0.5x+0.75(70-x)=49 \\ 0.5+52.5-0.75x=49 \\ -0.25x=49-52.5 \\ -0.25x=-3.5 \\ x=\frac{3.5}{0.25} \\ x=14 \end{gathered}[/tex]

Then, we can calculate y as:

[tex]y=70-x=70-14=56[/tex]

Answer: we need 14 liters of the 50% acid solution and 56 liters of the 75% acid solution.

Given:

A bus has 15 rows of seats. Each row has 4 seats.

So, a total of 60 seats.

At the first stop, 25 people get on the bus.

So, the number of occupied seats is 25 and the number of unoccupied seats is 35.

At the second stop, 3 people get off of the bus, and 12 people get on the bus.

So, the number of occupied seats is 34 and the number of unoccupied seats is 26.

Hence, a number of empty seats are thereafter the second stop is 26.

Solution

For this case we have the following info given:

[tex]X\approx N(62,4)[/tex]

and we want to find this probability:

P(54And we can use the z score and we have:

[tex]z=\frac{54-62}{4}=-2[/tex][tex]z=\frac{66-62}{4}=1[/tex]

And we have this:

[tex]P(54And the final answer would be 81.9%

[tex]I\text{ = }\frac{tP}{60}[/tex]

Explanation:

Interest varies jointly with time and principle

let I = interest

time = t

Principle = P

Mathematically:

[tex]\begin{gathered} I\text{ }\alpha\text{ tP} \\ I\text{ = ktP} \\ where\text{ k = constant of proportionality} \end{gathered}[/tex]

when interest = 1000

time = 10 years

Principal = 6000

We will substitute these values in the formula we got above to get k:

[tex]\begin{gathered} 1000\text{ = k}\times10\times6000 \\ 1000\text{ = 60000k} \\ divide\text{ both sides by 60000:} \\ \frac{1000}{60000}\text{ = }\frac{60000k}{60000} \\ k\text{ = 1/60} \end{gathered}[/tex]

To get the equation of the variable, we will substitute the value of k in the formula:

[tex]\begin{gathered} I\text{ = }\frac{1}{60}\times t\times P \\ \\ Equation\text{ of the variable:} \\ I\text{ = }\frac{tP}{60} \end{gathered}[/tex]

Answer:

triangle

Explanation:

The base of a triangular pyramid is a triangle, so if this figure is sliced by a plane parallel to the base, the cross-section will have the shape of the base. Then, cross-section is also a triangle.

So, the answer is triangle

[tex]\begin{gathered} (g-f)(x)=g(x)-f(x) \\ =(x^2-7x+2)-(4x^2+11) \\ =(x^2-4x^2)-7x+(2-11) \\ =-3x^2-7x-9 \end{gathered}[/tex]

Step 1

Write your question.

[tex]\begin{gathered} 2\frac{3}{8}\text{ }\frac{.}{.}\text{ 7}\frac{5}{12} \\ \end{gathered}[/tex]

Step 2

Convert all mixed fractions to improper fractions.

[tex]=\text{ }\frac{19}{8}\text{ }\frac{.}{.}\text{ }\frac{89}{12}[/tex]

Step 3

Change division to multiplication and invert the fraction.

[tex]\begin{gathered} =\text{ }\frac{19}{8}\text{ x }\frac{12}{89} \\ =\text{ }\frac{19\text{ x 3}}{2\text{ x 89}} \\ =\text{ }\frac{57}{178} \end{gathered}[/tex]

From the given informatio, we can draw the following picture:

where x denote the shorter side of the rectangle and x+6 the longest one. Since the area of the rectangle is equal to the base times the height, we have

[tex]A=(x+6)x[/tex]

Since the area is 520 square centimeters, we have

[tex]520=(x+6)x[/tex]

Then, by distributing the variable x into the parentheses, we have the following equation:

[tex]x^2+6x=520[/tex]

or equivalently,

[tex]x^2+6x-520=0[/tex]

So, we can to apply the quadratic formula:

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

coming from the quadratic polynomial:

[tex]ax^2+bx+c=0[/tex]

By comparing this polynomial with ours, we can note that

[tex]\begin{gathered} a=1 \\ b=6 \\ c=-520 \end{gathered}[/tex]

so, we get

[tex]x=\frac{-6\pm\sqrt{6^2-4(1)(-520)}}{2}[/tex]

which gives

[tex]\begin{gathered} x=\frac{-6\pm\sqrt{2116}}{2} \\ x=\frac{-6\pm46}{2} \end{gathered}[/tex]

Then, we have 2 possible solutions:

[tex]\begin{gathered} x=\frac{-6+46}{2}=\frac{40}{2}=20 \\ and \\ x=\frac{-6-46}{2}=\frac{-52}{2}=-26 \end{gathered}[/tex]

However, since the dimension are always positive numbers, the searched side measure 20 cm. In other words, the shorter side measures 20cm and the longest one 20+6=26 cm. Therefore, the answer is: 26 centimeters.

We have a line described with the equation

[tex]y=0[/tex]

This describes a horizontal line in the plane, at the height of the x-axis, let's remember that

the horizontal line is the one whose trajectory represents the direction of the horizon as we perceive it. We can also say that the horizontal lines are lines with zero slope.

On the other hand, we must remember that parallel lines are those that, while traveling in the same direction, remain separated at exactly the same distance from each other for exactly the same time.

That is to say that for a line to be parallel we must have another horizontal line.

Now let's determine which equations represent parallel lines one by one.

[tex]y=-2[/tex]

The above equation represents a slope of zero, i.e. it is a horizontal line so it is a parallel line.

[tex]y=x[/tex]

The above equation represents a slope of one, i.e. it is not a horizontal line so it is not a parallel line.

[tex]y=-8[/tex]

The above equation represents a slope of zero, i.e. it is a horizontal line so it is a parallel line.

[tex]x=\frac{1}{4}y[/tex]

The above equation represents a slope of one, i.e. it is not a horizontal line so it is not a parallel line.

[tex]y=-8[/tex]

In conclusion, the answer is that y = -2 and y = -8 are parallel lines.

Solution

For this case we need to take in count that:

[tex]\pi=180[/tex]

Then we can do this:

[tex]\frac{\pi}{5}=36\text{degrees}[/tex]

And we can convert -155° on this way:

[tex]\frac{x}{-155}=\frac{\pi}{180}[/tex]

Solving for x we got:

[tex]x=-155\cdot\frac{\pi}{180}=-\frac{31}{36}\pi[/tex]

Okay, here we have this:

Considering the provided information, we are going to calculate the requested height of the cylinder, so we obtain the following:

Then to find the requested height we will clear the height of the formula for the volume of a cylinder, therefore we have:

V=h * π * r^2

h=V/(π * r^2)

Replacing with the given values:

h=V/(π * (d/2)^2)

h= 125.6 in^3/(π * (8in/2)^2)

h= 125.6 in^3/(π * (4in)^2)

h= 125.6 in^3/(π * 16in^2)

h= 125.6 in^3/(3.14 * 16in^2)

h= 125.6 in^3/(50.24 in^2)

h= 2.5 inches

Finally we obtain that height of the soup can is 2.5 inches.

The number 60 has the following distinct positive integer factors:

1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

So we have 12 distinct factors.

Given data:

The points obtained after getting correct answer is c=150.

The points losses after getting incorrect answer is i=250.

The expression for total questions is,

[tex]\begin{gathered} c+i=20 \\ i=20-c \end{gathered}[/tex]

The expression for total marks is,

[tex]150c-250i=200[/tex]

Substitute (20-c) for i in the above expression.

[tex]\begin{gathered} 150c-250(20-c)=200 \\ 150c-5000+250c=200 \\ 400c=5200 \\ c=13 \end{gathered}[/tex]

The numbers of incorrect questions are,

[tex]\begin{gathered} i=20-13 \\ =7 \end{gathered}[/tex]

Thus, the numbers of correct questions are 13, and numbers of incorrect questions are 7.

Given,

The initial balance in the account is - 45 dollars.

The amount he deposite in the account is $7.

The balance of the account after Jonathan's deposit is,

[tex]\text{Balance of the account=initial balance+deposite amount}[/tex]

Substituting the values then,

[tex]\text{Balance of the account=-}45+7=-38[/tex]

Hence, the balance of the account after Jonathan's deposit is $-38.

Answer:

To find coordinate of A, it is (6,-6) in polar coordinate.

we know that,

The polar coordinate system as a method in which a point is described by its distance from a fixed point at the center of the coordinate space known as a pole and by the measurement of the angle formed by a fixed-line and a line from the pole through the given point.

In the polar coordinate system, the coordinates of a point are represented as,

[tex](r,\theta)[/tex]

where r is the distance of the point from the pole, and θ is the measure of the angle.

we know that,

[tex]r^2=x^2+y^2[/tex][tex]\theta=tan^{-1}(\frac{y}{x})[/tex]

we get,

(x,y)=(6,-6)

x=6, y=-6

Using this we get,

[tex]r^2=6^2+(-6)^2[/tex][tex]r^2=36+36[/tex][tex]r^2=72[/tex][tex]r=8.48[/tex]

Then,

[tex]\theta=tan^{-1}(\frac{-6}{6})[/tex]

[tex]\theta=tan^{-1}(-1)[/tex][tex]\theta=-\frac{\pi}{4}[/tex]

we get the polar coordinate of A is,

[tex](r,\theta)=(8.48,-\frac{\pi}{4})[/tex]

Given:

Isabella bought 2 apples and 4 mangoes for 28 cents. Mia bought 8 apples and 4 mangoes for 34 cents.

Required:

We need to find the cost of single apple.

Explanation:

First of all let assume a as a apple and m as a mango so now the equations are as

[tex]\begin{gathered} 2a+4m=28 \\ 8a+4m=34 \end{gathered}[/tex]

now use substitution method

[tex]4m=28-2a[/tex]

put this in another equation and we get

[tex]\begin{gathered} 8a+28-2a=34 \\ 6a=6 \\ a=1 \end{gathered}[/tex]

Final answer:

So the cost for single apple is 1 cent.

First, completing the square, we have:

[tex]undefined[/tex]

Given

Answer

CT = 12

SX= 16-12 = 4

MT = 5

DY = 7

What do they mean to write 5 less than r as an expression?

Explanation:

From the given data,

[tex]r-5[/tex]

Thus, the expression is (r-5)

Hey :)

[tex]\star\sim\star\sim\star\sim\star\sim\star\sim[/tex]

Put a minus sign between 5 and r. And put 5 after the minus.

[tex]r-5[/tex]. This is what the expression is. If you do everything correctly, then ur answer should be [tex]r-5[/tex].

So, the calculations showed that the answer is x=5. I hope i could provide a good explanation and a correct answer to you. Thankyou for taking the time to read my answer.

here for service,

silennia

[tex]\star\sim\star\sim\star\sim\star\sim\star\sim[/tex]

Given data:

A class has 12 boys and 21 girls.

The GPA of the whole class is 2.96.

The GPA of the boys is 2.40.

to find the GPA of girls,

let x be the GPA of girls.

Then,

(12+21)(2.96) = 12(2.40) + 21x

33(2.96) = 28.8 + 21x

97.68 = 28.8 + 21x

68.88 = 21x

x = 68.88/21

x = 3.28.

If B² -4AC > 0, the conic is a hyperbola. If B² -4AC < 0, the conic is a circle, or an ellipse. If B² - 4AC = 0, the conic is a parabola.

So in this case the conic section in question is a hyperbola. Therefore it is false.

Explanation

We must solve this divition and write it in reduce form:

[tex]\frac{7}{18}\div\frac{9}{2}=[/tex]

Remember that dividing by a fraction is equivalent to multiplying its inverse:

[tex]\frac{7}{18}\div\frac{9}{2}=\frac{7}{18}\cdot\frac{2}{9}[/tex]

Now we can multiply both numerators and denominators separately:

[tex]\frac{7}{18}\cdot\frac{2}{9}=\frac{7\cdot2}{18\cdot9}=\frac{14}{162}[/tex]

We can simplify that fraction. We can divide both the numerator and the denominator by 2:

[tex]\frac{14\div2}{162\div2}=\frac{7}{81}[/tex]

7/81 can't be simplified since 7 is a prime number so this is a fraction in reduced form.

Answer

Then the answer is:

[tex]\frac{7}{81}[/tex]