What's the dífference between paying $10,000 cash for a car or paying a loan of $200 per month for 60 months?

Answer:

Quadratic formula:

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

Step-by-step explanation:

If you want to solve something that looks like [tex]ax^2+bx+c=0[/tex], the answer will have this form [tex]x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}[/tex].

This is called the quadratic formula.

Answer:

2.04 seconds

Step-by-step explanation:

Falling objects near the surface of the earth have an acceleration of -9.81 m/s².

Acceleration is the change in velocity over change in time:

a = (v − v₀) / t

-9.81 = (-30.0 − (-10.0)) / t

-9.81 = -20.0 / t

t = 2.04

It takes 2.04 seconds.

Answer:

C and D

Step-by-step explanation:

The quadratic formula is

x= (-b±√b²-4ac)/2a

The formula uses the numerical coefficients in the quadratic equation.

The general quadratic equation is ax²+bx+c where a, b and c are the numerical coefficients

So, lets try and see;

A.

[tex]5x+4=3x^4-2\\\\=3x^4-5x-2-4\\=3x^4-5x-6\\a=3,b=-5,c=-6[/tex]

But due to the fact that  in this equation you have x⁴, the equation is not a quadratic equation thus can not be solved using this formula

B

[tex]-x^2+4x+7=-x^2-9\\\\\\=-x^2+x^2+4x+7+9\\=4x+16[/tex]

C

[tex]9x+3x^2=14+x-1\\\\\\=3x^2+9x-x-14+1\\\\=3x^2+8x-13\\\\\\a=3,b=8,c=-13\\[/tex]

D.

[tex]2x^2+x^2+x=30\\\\\\=3x^2+x-30\\\\\\a=3,b=1,c=-30[/tex]

From the checking above, the equations will be C and D

Answer:

Option C and D

Step-by-step explanation:

To find : After being rearranged and simplified, which of the following equations could  be solved using the quadratic formula? Check all that apply.

Solution :

Quadratic equation is [tex]ax^2+bx+c=0[/tex] with solution [tex]x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]

A. [tex]5x+4=3x^4-2[/tex]

Simplifying the equation,

[tex]3x^4-2-5x-4=0[/tex]

[tex]3x^4-5x-6=0[/tex]

It is not a quadratic equation.

B. [tex]-x^2+4x+7=-x^2-9[/tex]

Simplifying the equation,

[tex]-x^2+4x+7+x^2+9=0[/tex]

[tex]4x+16=0[/tex]

It is not a quadratic equation.

C. [tex]9x + 3x^2 = 14 + x-1[/tex]

Simplifying the equation,

[tex]3x^2+9x-x-14+1=0[/tex]

[tex]3x^2+8x-13=0[/tex]

It is a quadratic equation where a=3, b=8 and c=-13.

[tex]x=\frac{-8\pm\sqrt{8^2-4(3)(-13)}}{2(3)}[/tex]

[tex]x=\frac{-8\pm\sqrt{220}}{6}[/tex]

[tex]x=\frac{-8+\sqrt{220}}{6},\frac{-8-\sqrt{220}}{6}[/tex]

[tex]x=1.13,-3.80[/tex]

D. [tex]2x^2+x^2+x=30[/tex]

Simplifying the equation,

[tex]3x^2+x-30=0[/tex]

It is a quadratic equation where a=3, b=1 and c=-30.

[tex]x=\frac{-1\pm\sqrt{1^2-4(3)(-30)}}{2(3)}[/tex]

[tex]x=\frac{-1\pm\sqrt{361}}{6}[/tex]

[tex]x=\frac{-1+19}{6},\frac{-1-19}{6}[/tex]

[tex]x=3,-3.3[/tex]

Therefore, option C and D are correct.

Answer:

(x - 3)²/3² - (y + 4)²/2² = 1 ⇒ (0 , -4) , (6 , -4)

(x - 4)²/7² - (y + 6)²/5² = 1 ⇒ (-3 , -6) , (11 , -6)

(y + 5)²/5² - (x - 4)²/8² = 1 ⇒ (4 , 0) , (4 , -10)

(y + 7)²/7² - (x + 2)²/4² = 1 ⇒ (-2 , 0) , (-2 , -14)

(x + 1)²/9² - (y - 1)²/11² = 1 ⇒ (8 , 1) , (-10 , 1)

Step-by-step explanation:

* Lets revise the standard form of the equations of the hyperbola

- The standard form of the equation of a hyperbola with  center (h , k)

 and transverse axis parallel to the x-axis is  (x - h)²/a² - (y - k)²/b² = 1

- The coordinates of the vertices are (h ± a , k)

- The standard form of the equation of a hyperbola with center (h , k)

 and transverse axis parallel to the y-axis is  (y - k)²/a² - (x - h)²/b² = 1

- The coordinates of the vertices are (h , k ± a)

* Lets solve the problem

# (x - 3)²/3² - (y + 4)²/2² = 1

∵ (x - h)²/a² - (y - k)²/b² = 1

∴ a = 3 , b = 2 , h = 3 , k = -4

∵ The coordinates of the vertices are (h ± a , k)

∴ The coordinates of the vertices are (3 - 3 , -4) , (3 + 3 , -4)

∴ The coordinates of the vertices are (0 , -4) , (6 , -4)

* (x - 3)²/3² - (y + 4)²/2² = 1 ⇒ (0 , -4) , (6 , -4)

# (y - 1)²/2² - (x - 7)²/6² = 1

∵ (y - k)²/a² - (x - h)²/b² = 1

∴ a = 2 , b = 6 , h = 7 , k = 1

∵ The coordinates of the vertices are (h , k ± a)

∴ The coordinates of the vertices are (7 , 1 - 2) , (7 , 1 + 2)

∴ The coordinates of the vertices are (7 , -1) , (7 , 3)

* No answer for this equation

# (x - 4)²/7² - (y + 6)²/5² = 1

∵ (x - h)²/a² - (y - k)²/b² = 1

∴ a = 7 , b = 5 , h = 4 , k = -6

∵ The coordinates of the vertices are (h ± a , k)

∴ The coordinates of the vertices are (4 - 7 , -6) , (4 + 7 , -6)

∴ The coordinates of the vertices are (-3 , -6) , (11 , -6)

* (x - 4)²/7² - (y + 6)²/5² = 1 ⇒ (-3 , -6) , (11 , -6)

# (y + 5)²/5² - (x - 4)²/8² = 1

∵ (y - k)²/a² - (x - h)²/b² = 1

∴ a = 5 , b = 8 , h = 4 , k = -5

∵ The coordinates of the vertices are (h , k ± a)

∴ The coordinates of the vertices are (4 , -5 + 5) , (4 , -5 - 5)

∴ The coordinates of the vertices are (4 , 0) , (4 , -10)

* (y + 5)²/5² - (x - 4)²/8² = 1 ⇒ (4 , 0) , (4 , -10)

# (y + 7)²/7² - (x + 2)²/4² = 1

∵ (y - k)²/a² - (x - h)²/b² = 1

∴ a = 7 , b = 4 , h = -2 , k = -7

∵ The coordinates of the vertices are (h , k ± a)

∴ The coordinates of the vertices are (-2 , -7 + 7) , (-2 , -7 - 7)

∴ The coordinates of the vertices are (-2 , 0) , (-2 , -14)

* (y + 7)²/7² - (x + 2)²/4² = 1 ⇒ (-2 , 0) , (-2 , -14)

# (x + 1)²/9² - (y - 1)²/11² = 1

∵ (x - h)²/a² - (y - k)²/b² = 1

∴ a = 9 , b = 11 , h = -1 , k = 1

∵ The coordinates of the vertices are (h ± a , k)

∴ The coordinates of the vertices are (-1 + 9 , 1) , (-1 - 9 , 1)

∴ The coordinates of the vertices are (8 , 1) , (-10 , 1)

* (x + 1)²/9² - (y - 1)²/11² = 1 ⇒ (8 , 1) , (-10 , 1)

Answer:

(x - 3)²/3² - (y + 4)²/2² = 1 ⇒ (0 , -4) , (6 , -4)

(x - 4)²/7² - (y + 6)²/5² = 1 ⇒ (-3 , -6) , (11 , -6)

(y + 5)²/5² - (x - 4)²/8² = 1 ⇒ (4 , 0) , (4 , -10)

(y + 7)²/7² - (x + 2)²/4² = 1 ⇒ (-2 , 0) , (-2 , -14)

(x + 1)²/9² - (y - 1)²/11² = 1 ⇒ (8 , 1) , (-10 , 1)

Answer:

D. 6

Step-by-step explanation:

The result of 20 rolls in given in the statement we have to find how many times did the roll resulted in a result greater than the average number. So first we have to find the average of the 20 rolls.

The formula for the average is:

[tex]\frac{\text{Sum of observations}}{\text{Total number of observations}}[/tex]

So, the formula for the given case will be:

[tex]Average = \frac{\text{Sum of results of 20 rolls}}{20}\\\\ = \frac{60}{20}\\\\ =3[/tex]

Thus, the average result from the 20 rolls is 3. Now we have to look for values greater than 3 in the rolls. These are:

6, 4, 5, 4, 5, 6

So, 6 values in total are greater than 3.

Hence, Gabe rolled 6 times above average.

Check the picture below.

the first one is simply a serie, using the previous term - 3 times the one after it.

the second one is just an arithmetic sequence, where you add the previous term plus the ordinal position.

the third one is also an arithmetic sequence, simply adding the previous term with 1.4.

recall that an arithmetic sequence is adding up, a geometric sequence is multiplying about.

Answer:

[tex]4^x-5=6[/tex] gives the solution [tex]x=\frac{\log(11)}{\log(4)}[/tex].

[tex]4^{x-5}=6[/tex] gives the solution [tex]x=\frac{\log(6)}{\log(4)}+5[/tex].

Step-by-step explanation:

I will solve both interpretations.

If we assume the equation is [tex]4^{x}-5=6[/tex], then the following is the process:

[tex]4^x-5=6[/tex]

Add 5 on both sides:

[tex]4^x=6+5[/tex]

Simplify:

[tex]4^x=11[/tex]

Now write an equivalent logarithm form:

[tex]\log_4(11)=x[/tex]

[tex]x=\log_4(11)[/tex]

Now using the change of base:

[tex]x=\frac{\log(11)}{\log(4)}[/tex].

If we assume the equation is [tex]4^{x-5}=6[/tex], then we use the following process:

[tex]4^{x-5}=6[/tex]

Write an equivalent logarithm form:

[tex]\log_4(6)=x-5[/tex]

[tex]x-5=\log_4(6)[/tex]

Add 5 on both sides:

[tex]x=\log_4(6)+5[/tex]

Use change of base formula:

[tex]x=\frac{\log(6)}{\log(4)}+5[/tex]

Answer:

6.292

Step-by-step explanation:

I got it right on the test.

Answer:

  d.  x-intercept: (-15/7,0); y-intercept: (0, -15)

Step-by-step explanation:

I find it convenient to start with the equation in standard form:

  7x + y = -15 . . . . . . use y = g(x); add 7x to both sides

Now, you can ...

  → find the x-intercept by setting y to zero and dividing by the x-coefficient.

  x = -15/7

  → find the y-intercept by setting x to zero and dividing by the y-coefficient.

  y = -15

The x- and y-intercepts for the function g(x) are (-15/7, 0) and (0, -15).

Answer:

Step-by-step explanation:

x-intercept is for y = 0

y-intercept is for x = 0

=======================================

We have G(x) = -7x - 15 → y = -7x - 15

x-intercept:

-7x - 15 = 0      add 15 to both sides

-7x = 15      divide both sides by (-7)

x = - 15/7 → (- 15/7, 0)

y-intercept:

y = -7(0) - 15

y = 0 - 15

y = -15 → (-15, 0)

Answer:

1. -10

2. [tex]x=1\pm\sqrt{2}i[/tex]

3. -1+2i

4. -3-7i

5. 13

6. rectangular coordinates are (-4.3,-2.5)

7. rectangular coordinates are (-2.5,4.3)

8. x^2 + y^2 = 8y

9. Polar coordinates of point (-3,0) are  (3,180°)

10. Polar coordinates of point (1,1) are  (√2,45°)

Step-by-step explanation:

1) Simplify (2+3i)^2 + (2-3i)^2

Using formula (a+b)^2 = a^2+2ab+b^2

=((2)^2+2(2)(3i)+(3i)^2)+((2)^2-2(2)(3i)+(3i)^2)

=(4+12i+9i^2)+(4-12i+9i^2)

We know that i^2=-1

=(4+12i+9(-1))+(4-12i+9(-1))

=(4+12i-9)+(4-12i-9)

=(-5+12i)+(-5-12i)

=5+12i-5-12i

=-10

2. Solve x^2-2x+3 = 0

Using quadratic formula to find value of x

a=1, b=-2 and c=3

[tex]x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\\x=\frac{-(-2)\pm\sqrt{(-2)^2-4(1)(3)}}{2(1)}\\x=\frac{2\pm\sqrt{4-12}}{2}\\x=\frac{2\pm\sqrt{-8}}{2}\\x=\frac{2\pm2\sqrt{-2}}{2}\\x=\frac{2\pm2\sqrt{2}i}{2}\\x=2(\frac{1\pm\sqrt{2}i}{2})\\x=1\pm\sqrt{2}i[/tex]

3. If u =1+3i and v =-2-i what is u+v

u+v = (1+3i)+(-2-i)

u+v = 1+3i-2-i

u+v = 1-2+3i-i

u+v = -1+2i

4. if u = 3-4i and v = 3i+6 what is u-v

u-v = (3-4i)-(3i+6)

u-v = 3-4i-3i-6

u-v = 3-6-4i-3i

u-v = -3-7i

5. if u=(3+2i) and v=(3-2i) what is uv?

uv = (3+2i)(3-2i)

uv = 3(3-2i)+2i(3-2i)

uv = 9-6i+6i-4i^2

uv = 9-4i^2

i^2=-1

uv = 9-4(-1)

uv = 9+4

uv = 13

6. Convert (5, 7π/6)  to rectangular form

To convert polar coordinate into rectangular coordinate we use formula:

x = r cos Ф

y = r sin Ф

r = 5, Ф= 7π/6

x = r cos Ф

x = 5 cos (7π/6)

x = -4.3

y = r sin Ф

y = 5 sin (7π/6)

y = -2.5

So rectangular coordinates are (-4.3,-2.5)

7. Convert (5, 2π/3)  to rectangular form

To convert polar coordinate into rectangular coordinate we use formula:

x = r cos Ф

y = r sin Ф

r = 5, Ф= 2π/3

x = r cos Ф

x = 5 cos (2π/3)

x = -2.5

y = r sin Ф

y = 5 sin (2π/3)

y = 4.33

So rectangular coordinates are (-2.5,4.33)

8. Convert r=8cosФ to rectangular form

r.r = (8 cos Ф)r

r^2 = 8 (cosФ)(r)

Let (cosФ)(r) = y and we know that r^2 = x^2+y^2

x^2 + y^2 = 8y

9. Convert(-3,0) to polar form

We need to find (r,Ф)

r = √x^2+y^2

r = √(-3)^2+(0)^2

r =√9

r = 3

and tan Ф = y/x

tan Ф = 0/-3

tan Ф = 0

Ф = tan^-1(0)

Ф = 0°

As Coordinates are in 2nd quadrant, so add 180° in the given angle

0+180 = 180°

So,Polar coordinates of point (-3,0) are  (3,180°)

10) Convert (1,1) to polar form

We need to find (r,Ф)

r = √x^2+y^2

r = √(1)^2+(1)^2

r =√2

and tan Ф = y/x

tan Ф = 1/1

tan Ф = 1

Ф = tan^-1(1)

Ф = 45°

As Coordinates are in 1st quadrant, so Ф will be as found

So,Polar coordinates of point (1,1) are  (√2,45°)

Answer:

4x - 8

Step-by-step explanation:

Cost of producing x soccer balls = h(x) = 5x + 6 in thousands of dollars

Revenue generated from x soccer balls = k(x) = 9x - 2 in thousands of dollars

We need to calculate the profit for x soccer balls.

Profit = p (x) = Revenue - Cost

p(x) = (9x - 2) - (5x + 6)

p(x) = 9x -2 - 5x- 6

p(x) = 4x - 8

Thus the profit from x soccer balls in thousands of dollars would be 4x - 8.

Answer:4x-8

Step-by-step explanation:

Answer:

The required standard form of  ellipse is [tex]\frac{x^2}{25}+\frac{y^2}{34}=1[/tex].

Step-by-step explanation:

The standard form of an ellipse is

[tex]\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1[/tex]

Where, (h,k) is center of the ellipse.

It is given that the center of the circle is (0,0), so the standard form of the ellipse is

[tex]\frac{x^2}{a^2}+\frac{y^2}{b^2}=1[/tex]             .... (1)

If a>b, then coordinates of vertices are (±a,0), coordinates of co-vertices are (0,±b) and focus (±c,0).

[tex]c^2=a^2-b^2[/tex]            .... (2)

If a<b, then coordinates of vertices are (0,±b), coordinates of co-vertices are (±a,0) and focus (0,±c).

[tex]c^2=b^2-a^2[/tex]            .... (3)

It is given that co-vertex of the ellipse at (5, 0); focus at (0, 3). So, a<b we get

[tex]a=5,c=3[/tex]

Substitute a=5 and c=3 these values in equation (3).

[tex]3^2=b^2-(5)^2[/tex]

[tex]9=b^2-25[/tex]

[tex]34=b^2[/tex]

[tex]\sqrt{34}=b[/tex]

Substitute a=5 and [tex]b=\sqrt{34}[/tex] in equation (1), to find the required equation.

[tex]\frac{x^2}{5^2}+\frac{y^2}{(\sqrt{34})^2}=1[/tex]

[tex]\frac{x^2}{25}+\frac{y^2}{34}=1[/tex]

Therefore the required standard form of  ellipse is [tex]\frac{x^2}{25}+\frac{y^2}{34}=1[/tex].

Answer:

Arc length is  [tex]\frac{14\pi}{3}[/tex] ,  or,  14.7

Step-by-step explanation:

AB is an arc intercepted by 140 degree angle. The formula for length of an arc is given by

[tex]AL=\frac{\theta}{360}*2\pi r[/tex]

Where

AL is the arc length

[tex]\theta[/tex]  is the angle (in our case, 140)

r is the radius of the circle (which is 6)

Substituting, we get:

[tex]AL=\frac{\theta}{360}*2\pi r\\AL=\frac{140}{360}*2\pi (6)\\AL=\frac{7}{18}*12\pi\\AL=\frac{14\pi}{3}[/tex]

In decimal (rounded to tenths) -  14.7

Answer:

Δ ABC was dilated by a scale factor of 1/2, reflected across the y-axis

and moved through the translation (3 , 2)

Step-by-step explanation:

* Lets explain how to solve the problem

- The similar triangles have equal ratios between their

  corresponding side

- So lets find from the graph the corresponding sides and calculate the

  ratio, which is the scale factor of the dilation

- In Δ ABC :

∵ The length of the horizontal line is x2 - x1

- Let A is (x1 , y1) and B is (x2 , y2)

∵ A = (-4 , -2) and B = (0 , -2)

∴ AB = 0 - -4 = 4

- The corresponding side to AB is ED

∵ The length of the horizontal line is x2 - x1

- Let E is (x1 , y1) , D is (x2 , y2)

∵ E = (5 , 1) and D = (3 , 1)

∵ DE = 5 - 3 = 2

∵ Δ ABC similar to Δ EDF

∵ ED/AB = 2/4 = 1/2

∴ The scale factor of dilation is 1/2

* Δ ABC was dilated by a scale factor of 1/2

- From the graph Δ ABC in the third quadrant in which x-coordinates

 of any point are negative and Δ EDF in the first quadrant in which

 x-coordinates of any point are positive

∵ The reflection of point (x , y) across the y-axis give image (-x , y)

* Δ ABC is reflected after dilation across the y-axis

- Lets find the images of the vertices of Δ ABC after dilation and

 reflection  and compare it with the vertices of Δ EDF to find the

 translation

∵ A = (-4 , -2) , B = (0 , -2) , C (-2 , -4)

∵ Their images after dilation are A' = (-2 , -1) , B' = (0 , -1) , C' = (-1 , -2)

∴ Their image after reflection are A" = (2 , -1) , B" = (0 , -1) , C" = (1 , -2)

∵ The vertices of Δ EDF are E = (5 , 1) , D = (3 , 1) , F = (4 ,0)

- Lets find the difference between the x-coordinates and the

 y- coordinates of the corresponding vertices

∵ 5 - 2 = 3 and 1 - -1 = 1 + 1 = 2

∴ The x-coordinates add by 3 and the y-coordinates add by 2

∴ Their moved 3 units to the right and 2 units up

* The Δ ABC after dilation and reflection moved through the

  translation (3 , 2)

Answer:

X=5, X=9

Step-by-step explanation:

The first one has two sides that are equal length, so the angles opposite of those sides are equal. This means that there are 2 73 degree angles. A triangle only had 180 degrees, so the last angle is equal to 34 degrees. When you set 6x+4=34, x is equal to 5.

The second triangle is an equilateral triangle, so every angle is equal to 60 degrees. We can set 7x-3=60. Add 3 to isolate x. 7x=63. Divide by 7 to solve for x. x=9.

Answer:

Give Zdomi Brainliest now <3

Step-by-step explanation:

Answer:

This conduit fill table is used to determine how many wires can be safely put in conduit tubing.The rows going across is the size of the conduit and the type. The columns going down shows the gauge of wire that is being used. The results are the numbers of wires of that gauge, that can be run through that size, of that kind of conduit such as EMT, IMC, and galvanized pipe. This chart is based on the 2017 NEC code.

Step-by-step explanation:

Answer:

Her total run is 0.81 miles.

Step-by-step explanation:

Consider the provided information.

The provided information can be visualized by the figure 1.

The path she covers represent a right angle triangle, where the length of two legs are given as 0.19 and 0.28.

Use the Pythagorean theorem to find the length of missing side.

[tex]a^2+b^2=c^2[/tex]

Where, a and b are the legs and c is the hypotenuse of the right angle triangle.

The provided lengths are 0.19 and 0.28.

Now, calculate the missing side.

[tex](0.19)^2+(0.28)^2=(c)^2[/tex]

[tex]0.0361+0.784=(c)^2[/tex]

[tex]0.1145=c^2[/tex]

[tex]\sqrt{0.1145}=c[/tex]

[tex]c\approx{0.34}[/tex]

Thus, the total distance is:

0.34 + 0.19 + 0.28 = 0.81

Therefore, her total run is 0.81 miles.

Answer:

about 0.81 miles

Step-by-step explanation:

Alyssa's route can be considered a right triangle with legs of length 19 and 28 (hundredths). The Pythagorean theorem tells us the hypotenuse (x) will satisfy ...

x^2 = 19^2 +28^2

x^2 = 1145

x = √1145 ≈ 34 . . . . hundredths of a mile

Then Alyssa's total route is ...

0.19 + 0.28 + 0.34 = 0.81 . . . . miles

Answer:

the coordinates of C' = (2,1)

Step-by-step explanation:

The coordinates of point C can be found by looking at the graph.

Coordinates of C are C= (6,3)

If ABCD is dilated by a factor of 1/3 then the coordinates of C' can be found by multiplying the coordinates of C by 1/3

C = (6,3)

C' =(1/3*6,1/3*3)

C' = (2,1)

So, the coordinates of C' = (2,1)

Answer:

=49.15 cm²

Step-by-step explanation:

To find the area of the triangle we use the sine formula

A= 1/2ab Sin∅

where A is the area a and b are the lengths of two sides that intersect at a point and ∅ is the angle between them.

a=10 cm

b= 12 cm

∅=55°

A= 1/2×10×12×Sin 55

=49.15 cm²

ANSWER

[tex]Area =49.1 {cm}^{2} [/tex]

EXPLANATION

The area of triangle given included angle and length of two sides can be calculated using the formula:

[tex]Area = \frac{1}{2} ab \sin(C) [/tex]

Where C=55° is the included angle and a=12 cm , b=10cm are the known sides.

We plug in these values into the formula to get,

[tex]Area = \frac{1}{2} \times 12 \times 10 \sin(55 \degree) [/tex]

[tex]Area =49 .14912266[/tex]

Rounding to the nearest tenth, the area is

[tex]Area =49.1 {cm}^{2} [/tex]

Answer:

[tex]x = 0 , \pi , 2\pi[/tex]

Step-by-step explanation:

The given equation is:l

[tex] \tan^{2} (x) \sec^{2} x + 2 \sec^{2} x - \tan^{2} x = 2[/tex]

Add -2 to both sides of the equation to get:

[tex] \tan^{2} (x) \sec^{2} x + 2 \sec^{2} x - \tan^{2} x - 2 = 0[/tex]

We factor the LHS by grouping.

[tex]\sec^{2} x(\tan^{2} (x) + 2 ) - 1( \tan^{2} x + 2) = 0[/tex]

[tex](\sec^{2} x - 1)(\tan^{2} (x) + 2)= 0[/tex]

We now apply the zero product property to get:

[tex](\sec^{2} x - 1) = 0 \: \: or \: \: (\tan^{2} (x) + 2)= 0[/tex]

This implies that:

[tex]\sec^{2} x = 1 \: \: or \: \: \tan^{2} (x) = - 2[/tex]

[tex] \tan^{2} (x) = - 2 \implies \tan(x) = \pm \sqrt{ - 2} [/tex]

This factor is never equal to zero and has no real solution.

[tex]\sec^{2} x = 1[/tex]

This implies that:

[tex]\sec \: x= \pm\sqrt{1} [/tex]

[tex] \sec(x) = \pm - 1[/tex]

Recall that

[tex] \frac{1}{ \sec(x) } = \cos(x) [/tex]

We reciprocate both sides to get:

[tex] \cos(x) = \pm1[/tex]

[tex]\cos x = 1 \: or \: \cos x = - 1[/tex]

[tex]\cos x = 1 \implies \: x = 0 \: or \: 2\pi[/tex]

[tex]\cos x = - 1 \implies \: x = \pi[/tex]

Therefore on the interval

[tex]0 \leqslant x \leqslant 2\pi[/tex]

[tex]x = 0 , \pi , 2\pi[/tex]

Answer:

  Each bouquet included 2 foil balloons and 8 latex balloons.

Step-by-step explanation:

Let f represent the number of foil balloons in each bouquet. Then 10-f is the number of latex balloons. The problem statement tells us the cost of all of the bouquets is ...

  18(1.94f +0.17(10-f)) = 94.32

We can divide by 18 to get ...

  1.94f +1.70 -0.17f = 5.24

  1.77f = 3.54 . . . . . . . . . . . . subtract 1.70

  f = 3.54/1.77 = 2 . . . . . . . . divide by the coefficient of f

The number of latex balloons is 10-2 = 8.

Each bouquet included 2 foil and 8 latex balloons.

Answer:

B.

Step-by-step explanation:

It doesn't make sense for either x or y to be negative because we are talking about x representing the number of acres and y representing another number of acres and that they should add up to no more than 400.

So I'm not going to look at A or C.

B looks good 250+150=400

D almost looks good 1+400=401; the problem with this answer is that is more than 400

Option B (250, 150) is the only viable solution as it adheres to the constraints of the problem, summing to exactly 400 acres with both variables being non-negative.

The farmer has set a constraint for planting peas and carrots where the total acres used for planting both cannot exceed 400 acres.

Thus, we are looking for a solution where the sum of x (acres of peas) and y (acres of carrots) must be equal to or less than 400. The solution should also satisfy the requirement that both x and y must be non-negative since you cannot plant crops on a negative amount of land.

Among the options provided, option B (250, 150) is the viable solution. Here's why:

A.) (−125, 500): We cannot have negative acres for crops, so x cannot be negative. Also, y exceeds 400 acres on its own, violating the total area constraint.

B.) (250, 150): This solution sums up to 400 acres exactly, fitting within the constraint and with both x and y being non-negative.

C.) (400, −10): y cannot be negative, representing a nonsensical scenario for planting.

D.) (1, 400): The total acreage here is 401, which exceeds the maximum allowable acreage.

Answer:

117 miles

Step-by-step explanation:

First we need an equation for the situation.  The number of miles is our unknown, x.  If she is charged 18 cents per mile, that can be expressed as .18x.  The flat rate, what she is charged regardless of how many miles she drives, is 18.95.  In other words, even if she drives 0 miles, she is still charged 18.95 for the rental of the car.  C(x) is the amount she will pay after the flat rate plus the number of miles she drives.  Therefore, our equation is:

C(x) = .18x + 18.95

If she can only spend 40, then we replace C(x) with 40 and solve for x, the number of miles:

40 = .18x + 18.95

Begin by subtracting 18.95 from both sides to get:

21.05 = .18x

Now divide both sides by .18 to get that

x = 116.9 miles

Rounding, we have that she can drive

117 miles

with the amount of money she has to spend on a rental car.

The maximum number of miles Meredith can drive without exceeding her $40 budget is 117 miles.

Total budget of Meredith on car rental = $40.

Initial charges: $18.95

Remaining budget for mileage charges: $40 - $18.95 = $21.05

Maximum number of miles = $21.05 / $0.18 per mile = 116.94 miles ≈ 117 miles

Answer:

22.2 ft²

Step-by-step explanation:

The area (A) of the sector is

A = area of circle × fraction of circle

   = πr² × [tex]\frac{50}{360}[/tex]

   = π × 7.13² × [tex]\frac{5}{36}[/tex]

   = π × 50.8369 × [tex]\frac{5}{36}[/tex]

   = [tex]\frac{50.8369(5)\pi }{36}[/tex] ≈ 22.2 ft² ( nearest tenth )

Answer:

Area of smaller sector = 22.2 ft²

Step-by-step explanation:

Points to remember

Area of circle = πr²

Where 'r' is the radius of circle

To find the area of circle

Here r = 7.13 ft

Area =  πr²

 =  3.14 * 7.13²

 = 159.63 ft²

To find the area of smaller sector

Here central angle of sector is 50°

Area of sector = (50/360) * area of circle

 = (50/360) * 159.63

 = 22.17 ≈ 22.2 ft²

Answer:

30 and 150

Step-by-step explanation:

Whether these are same side interior or same side exterior, the sum of them is 180 when the angles are on the same side of a transversal that cuts a pair of parallel lines.  Let's call the angles A and B.  If angle A is 5 times smaller than angle B, then angle B is 5 times larger than angle A.  So the angles are x and 5x.  They are supplementary so

x + 5x = 180 and

6x = 180 so

x = 30 and 5(30) = 150

Answer:

The correct answer option is B. 1/5.

Step-by-step explanation:

We are given four fractions in the answer options and we are to determine whether which one of them has terminating decimal as its decimal expansion.

Terminating decimal means a decimal value which has a finite amount of numbers and has an end to it.

[tex]\frac{1}{3} = 0.333333333[/tex]

[tex]\frac{1}{5} = 0.2[/tex]

[tex]\frac{1}{7} = 0.142357142[/tex]

[tex]\frac{1}{9} = 0.111111111[/tex]

Therefore, the correct answer is 1/5.

Answer:

The correct answer option is b) 1.95.

Step-by-step explanation:

We are to evaluate [tex] ln 7 [/tex].

For this, we can either log in the value directly in a scientific calculator for the the given value [tex] ln 7 [/tex] and get the answer in decimals.

Another way can be to rewrite the expression as:

log to the base [tex] e [/tex] or [tex] 7 [/tex] = x

or [tex] e ^ x = 7 [/tex]

which gives x = 1.95

Answer:

The length of this altitude is 5 cm.

Step-by-step explanation:

The length of the altitude = ?

Given the diagonal forms the altitude of the parallelogram. The figure is shown in image.

Given

The perimeter of the parallelogram = 50 cm

The length of one side is 1 cm longer than the length of the other.

Thus,

Let one side (a) is x cm, The other side (b) be (x + 1) cm

Perimeter of parallelogram = 2(a + b) = 2(x +(x + 1)) = 4x + 2 = 50 cm

Thus,

x = AB = CD = 12 cm

x + 1 = BC = AD = 13 cm

Using Pythagorean theorem to find the length of the altitude as:

ΔABC is a right angle triangle.

AB² + AC² = BC²

AC² = 13² - 12² = 5 cm

The length of this altitude is 5 cm.

Answer:

13

Step-by-step explanation:

If the length of a major axis of an ellipse is 13, the sum of the lengths of the red and blue line segments is 13.

P = point in the figure

F1 = focus

F2 = focus

PF1 + PF2 = 13

Answer:

140

Step-by-step explanation:

Rewording the problem will make it easier to write the equation you need to solve this. Think of it in simpler terms: "7 is 5% of how many?". "7" is just a 7; the word "is" means =; "5%" is expressed as its decimal equivalency (.05); the word "of" means to multiply; and "how many" is our unknown (x). Putting that all together in one equation looks like this:

7 = .05x

Solve for x by dividing both sides by .05 to see that

x = 140