What is the value of x?

Answer:

B. csc²(x)

Step-by-step explanation:

You can use the relations ...

sec(x) = 1/cos(x)

csc(x) = 1/sin(x)

cot(x) = cos(x)/sin(x)

to replace the functions in your expression. Then you have ...

sec²(x)·cot²(x) = (1/cos(x)·cos(x)/sin(x))² = (1/sin(x))² = csc²(x)

___

Alternate solution

You can also use the relation

cot(x) = csc(x)/sec(x)

Then ...

(sec(x)·cot(x))² = (sec(x)·csc(x)/sec(x))² = csc²(x)

Answer:

Yes! The correct answer is option B

Step-by-step explanation:

B.  csc^2x

Both [tex]f[/tex] and [tex]g[/tex] satisfy the conditions for Rolle's theorem, which then says there exists [tex]c\in[-1,3][/tex] (for [tex]f[/tex]) such that [tex]f'(c)=0[/tex], and [tex]c\in[-2,1[/tex] (for [tex]g[/tex]) such that [tex]g'(c)=0[/tex].

1.

[tex]f(x)=x^2-2x-8\implies f'(x)=2x-2[/tex]

[tex]f'(c)=2c-2=0\implies c=1[/tex]

2.

[tex]g(t)=2t-t^2-t^3=0\implies g'(t)=2-2t-3t^2[/tex]

[tex]g'(c)=2-2c-3c^2=0\implies c=\dfrac{-1\pm\sqrt7}3[/tex]

Answer:

nonagon: 140°

100-gon: 176.4°

Step-by-step explanation:

A regular polygon is one that has all sides the same length and all internal angles the same measure. A "nonagon" has nine (9) sides. An image of one is attached. There are some interesting relationships among the angles shown.

The figure can be divided into 9 congruent triangles. Each has a vertex at the center of the figure, and the other two vertices are each end of one side. The central angle (40°) is the supplement of the interior angle at each vertex of the nonagon (140°). Obviously, the central angles sum to 360° (one full circle), so each one has a measure that is 360° divided by the number of sides:

360°/9 = 40°

So, we can figure the interior angle (at the vertex) by subtracting from 180° the value of 360° divided by the number of sides.

For the 100-gon, with 100 sides, the interior angle will be ...

180° -360°/100 = 180° -3.6° = 176.4°

Final answer:

The interior angle of a regular nonagon is 140°, and for a regular 100-gon, it's 176.4°.

Explanation:

To find the interior angles of a regular polygon, we use the formula: (n - 2) × 180°, where n is the number of sides of the polygon. This formula gives us the sum of all interior angles of the polygon.

To find the measure of one interior angle in a regular polygon (where all angles are equal), we divide this sum by the number of sides.

Interior Angle of a Regular Nonagon:

A nonagon has 9 sides. Using our formula:

Interior Angle of a Regular 100-gon:

A 100-gon has 100 sides. Following the same process:

Therefore, each interior angle of a regular nonagon is 140°, and each interior angle of a regular 100-gon is 176.4°.

Answer:

Step-by-step explanation:

The equation ...

  g(x) = a·f(x -h) +k

indicates a vertical stretch by a factor of "a", a horizontal translation to the right by "h" units, and a translation up by "k" units.

Matching the shapes of the curves, we see that the point of inflection of f(x) is (-2, -1). The corresponding point on g(x) is (2, -1). This is called the "turning point" in your question. It is where the graph turns from being concave downward to being concave upward.

The difference in x-values between g(x) and f(x) for the turning point is ...

  2-(-2) = 4

This is the amount by which the graph of f(x) is translated to the right: 4 units.

The vertical difference between the marked points on f(x) and the turning point is 1 unit. On g(x), those same marked points are 3 units away from the turning point vertically. Hence the vertical stretch factor is 3.

_____

Comment on the transformation of f(x)

Please note that the graph of g(x) is actually related to the graph of f(x) as ...

  g(x) = 3·f(x -4) +2

That is, for x=1 on g(x), the y-coordinate is ...

  g(1) = 3·f(1 -4) +2 = 3·(-2) +2 = -4 . . . . . . . point (1, -4) on g(x)

For x=3 on g(x), the y-coordinate is ...

  g(3) = 3·f(3 -4) +2 = 0 +2 = 2 . . . . . . . . . . point (3, 2) on g(x)

It may seem a little strange that there is a vertical translation of 2 units upward, when the point of inflection has the same vertical location. Actually, that is the clue that there is an upward translation.

The stretch factor operates about the origin, so stretching f(x) by a factor of 3 will make the turning point move from y=-1 to y=3·(-1) = -3. Since it shows on the graph of g(x) at location y=-1, it must have been translated 2 units upward from its stretched location.

Answer:

  KM = 20

Step-by-step explanation:

If V is the midpoint of KM, then ...

  KV = VM

  2.5z = 5z -10

  0 = 2.5z -10 . . . . . . subtract 2.5z

  0 = z - 4 . . . . . . . . . divide by 2.5

  4 = z . . . . . . . . . . . . add 4

We know that V bisects KM, so KV is half the overall length. That is ...

  KM = 2·KV = 2·2.5z = 5z

Using the value of z we found, ...

  KM = 5·4 = 20

Answer:

  564 ft²

Step-by-step explanation:

To account for the extra space between units, we can add 2" to every unit dimension and every box dimension to figure the number of units per box.

Doing that, we find the storage box dimensions (for calculating contents) to be ...

  3 ft 2 in × 4 ft 2 in × 2 ft 2 in = 38 in × 50 in × 26 in

and the unit dimensions to be ...

  (4+2)" = 6" × (6+2)" = 8" × (2+2)" = 4"

A spreadsheet can help with the arithmetic to figure how many units will fit in the box in the different ways they can be arranged. (See attached)

When we say the "packing" is "462", we mean the 4" (first) dimension of the unit is aligned with the 3' (first) dimension of the storage box; the 6" (second) dimension of the unit is aligned with the 4' (second) dimension of the storage box; and the 2" (third) dimension of the unit is aligned with the 2' (third) dimension of the storage box. The "packing" numbers identify the unit dimensions, and their order identifies the corresponding dimension of the storage box.

We can see that three of the four allowed packings result in 216 units being stored in a storage box.

If storage boxes are stacked 4 deep in a 9' space, the 2' dimension must be the vertical dimension, and the floor area of each stack of 4 boxes is 3' × 4' = 12 ft². There are 216×4 = 864 units stored in each 12 ft² area.

If we assume that 2 weeks of production are 80 hours of production, then we need to store 80×500 = 40,000 units. At 864 units per 12 ft² of floor space, we need ceiling(40,000/864) = 47 spaces on the floor for storage boxes. That is ...

  47 × 12 ft² = 564 ft²

of warehouse floor space required for storage.

_____

The second attachment shows the top view and side view of units packed in a storage box.

Answer:

Part 1) Option A. h(2) = 86.00 means that after 2 seconds, the height of the ball is 86.00 ft

Step-by-step explanation:

we have

[tex]h(t)=-16t^{2}+150[/tex]

where

t ----> is the time in seconds after the ball is dropped

h(t) ----> he height in feet of a ball dropped from a 150 ft

Find h(2)

That means ----> Is the height of the ball  2 seconds after the ball is dropped

Substitute the value of t=2 sec in the equation

[tex]h(2)=-16(2)^{2}+150=86\ ft[/tex]

therefore

After 2 seconds, the height of the ball is 86.00 ft.

Answer:

Step-by-step explanation:

D is correct.  For any x in the domain of your function, there can be ONLY ONE corresponding y value.  If you find more than one y value associated, then this relationship is NOT a function.

Answer: D. A function is a set of ordered pairs in which each first component in the ordered pairs corresponds to exactly one second component.

Step-by-step explanation: In mathematics, Function is a relationship between two groups, in which, each element of the first group corresponds to the elements of the second group only once. They shows how a quantity depends on another. For example, the distance a person travels over time spent.

A function is a technique in which each element (x) of a set X is associated with an element of set Y or f(x). The set X is the domain of a function and set Y or f(x) is the codomain.

Each x is the input of a function or variable and each correspondent y is the value of the function or image of x by f.

Note that [tex]+\vee-[/tex] stands for plus or minus.

For the quadratic equation of the form [tex]ax^2+bx+c=0[/tex] the solutions are [tex]x_{1,2}=\dfrac{-b+\vee-\sqrt{b^2-4ac}}{2a}[/tex] that means that for [tex]a=4, b=-10, c=5\Longrightarrow x_{1,2}=\dfrac{-(-10)+\vee-\sqrt{(-10)^2-4\cdot4\cdot5}}{2\cdot4}[/tex] this simplifies to [tex]\boxed{x_1=\dfrac{5+\sqrt{5}}{4}}, \boxed{x_2=\dfrac{5-\sqrt{5}}{4}}[/tex]

Hope this helps.

Answer:

[tex]\large\boxed{x=\dfrac{5-\sqrt5}{4},\ x=\dfrac{5+\sqrt5}{4}}[/tex]

Step-by-step explanation:

[tex]\text{The quadratic formula for}\ ax^2+bx+c=0\\\\\text{if}\ b^2-4ac<0,\ \text{then the equation has no real solution}\\\\\text{if}\ b^2-4ac=0,\ \text{then the equation has one solution:}\ x=\dfrac{-b}{2a}\\\\\text{if}\ b^2-4ac,\ ,\ \text{then the equation has two solutions:}\ x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\\\\==========================================[/tex]

[tex]\text{We have the equation:}\ 4x^2-10x+5=0\\\\a=4,\ b=-10,\ c=5\\\\b^2-4ac=(-10)^2-4(4)(5)=100-80=20>0\\\\x=\dfrac{-(-10)\pm\sqrt{20}}{2(4)}=\dfrac{10\pm\sqrt{4\cdot5}}{8}=\dfrac{10\pm\sqrt4\cdot\sqrt5}{8}=\dfrac{10\pm2\sqrt5}{8}\\\\=\dfrac{2(5\pm\sqrt5)}{8}=\dfrac{5\pm\sqrt5}{4}[/tex]

Answer:

192.53= original bill

Step-by-step explanation:

The bill you receive is equal to the original bill minus the prepaid amount

bill = original - prepaid

82.53 = original - 110

Add 110 to each side

82.53+110 = original-100+100

192.53= original bill

ANSWER

The discriminant is 384. Because the discriminant is greater than 0 and is not a perfect square, the two roots are real and irrational.

EXPLANATION

The given quadratic equation is:

[tex] - 3 {x}^{2} - 18x + 5 = 0[/tex]

We compare this equation to:

[tex]a {x}^{2} + bx + c = 0[/tex]

We have a=-3,b=-18, and c=5.

The discriminant of a quadratic equation is calculated using the formula:

[tex]D=b^{2} - 4ac[/tex]

We plug in the values to obtain:

[tex]D= {( - 18)}^{2} - 4( - 3)(5)[/tex]

[tex]D= 324 + 60[/tex]

Simplify:

[tex]D= 384[/tex]

The discriminant is greater than zero, hence there are two distinct real roots.

Since 384 is not a perfect square, the roots are irrational.

Answer:

I think the answer is 4.5 if you subtract 75 from 129 and divide by 12?

After deducting the cost of materials, the remaining amount spent on labor was $54. By dividing this amount by the friend's hourly rate of $12, it is determined that Jared's friend worked for 4.5 hours on constructing the treehouse.

Jared is trying to calculate how many hours his friend worked on building a treehouse based on the total cost of the project. Jared spent $75 on materials and paid his friend $12 per hour for labor. The total cost for building the treehouse was $129.

To find out how many hours Jared's friend worked, we start by subtracting the cost of materials from the total cost:

Total cost of treehouse = $129

Cost of materials = $75

Total cost minus materials cost = $129 - $75 = $54

This $54 represents the total amount paid for labor. We can then divide this amount by the hourly rate to find the number of hours worked:

Hourly wage = $12

Labor cost / hourly wage = $54 / $12 = 4.5 hours

Therefore, Jared's friend worked for 4.5 hours on the treehouse.

Answer:

years worked: 14

at least 10 years: 73%

Step-by-step explanation:

The mean is found by adding the years of service and dividing by the number of employees. The total years of service is 417, so the average is ...

average years worked = 417/30 = 13.9 ≈ 14 . . . years

__

The percentage of employees that have worked there at least 10 years is found by counting the number with 10 or more years of service and dividing that count by the total number of employees. The result is then expressed as a percentage.

(10 years or over)/(total number) = 22/30 = 0.73_3 (a repeating decimal) ≈ 73%

_____

Comment on the working

A spreadsheet can be helpful for this. It has a function that can calculate the mean for you. Sorting the years of service into order can make it trivially easy to count the number that are 10 or more, or you can write a function that will do the count for you. (Also, less than 10 means the years are a single digit. There are 8 single-digit numbers in your list.) The hard part is copying 30 numbers without error.

The answers are: 14 years and 50%. 14 years is the population mean for the number of years worked, and 50 % of the employees worked for the company for at least 10 years.

To determine the answers, steps:

1. Calculate the Population Mean: Sum all the years worked and divide by the number of employees (30).

   [tex]Total\ years\ worked: 8 + 13 + 15 + 3 + 13 + 28 + 4 + 12 + 4 + 26 + 29 + 3 + 10 + 3 + 17 + 13 + 15 + 15 + 23 + 13 + 12 + 1 + 14 + 14 + 17 + 16 + 7 + 27 + 18 + 24 = 412[/tex]

   [tex]Population\ mean =\frac{412}{30} \approx 14[/tex]

2. Calculate the Percentage of Employees who Worked At Least 10 Years: Count the employees who worked for 10 years or more, and divide by the total number of employees, then multiply by 100 to convert to percentage.

   [tex]Number\ of\ employees\ who\ worked\ at\ least\ 10\ years: 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 15[/tex]

   [tex]Percentage = \frac{15}{30} * 100 = 50\%[/tex]

Answer:

>,<,<,>,<

Step-by-step explanation:

The bigger side in the >/< Symbols mean that value is larger

For this case we have the following system of equations:

[tex]y = x ^ 2-4x + 3\\y = -x + 3[/tex]

Matching we have:

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

We solve by means of

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

Where:

[tex]a = 1\\b = -3\\c = 0[/tex]

Substituting:

[tex]x = \frac {- (- 3) \pm \sqrt {(- 3) ^ 2-4 (1) (0)}} {2 (1)}\\x = \frac {3 \pm \sqrt {9}} {2}\\x = \frac {3 \pm3} {2}[/tex]

Finally, the roots are:

[tex]x_ {1} = \frac {3-3} {2} = 0\\x_ {2} = \frac {3 + 3} {2} = \frac {6} {2} = 3[/tex]

Answer:

[tex]x_ {1} = 0\\x_ {2} = 3[/tex]

Answer:

Step-by-step explanation:

Dilation does not change the slope, so it remains ...

  m = ∆y/∆x = (8-2)/(3-2) = 6/1

  m = 6

The length is multiplied by the dilation factor. The original length (d) is given by ...

  d = √((∆x)^2 +(∆y)^2) = √(1^2 +6^2) = √37

Then the dilated length is ...

  3.5d = 3.5√37 ≈ 21.290

Answer:

[tex]P(B)=\frac{1}{3}[/tex]

Step-by-step explanation:

The number of red marbles in the bag is; [tex]n(R)=3[/tex].

The number of blue marbles in the bag is; [tex]n(B)=4[/tex].

The number of white marbles in the bag is; [tex]n(W)=5[/tex].

The total number of marbles in the bag is: [tex]n(S)=3+4+5=12[/tex].

The probability of selecting a blue marble is [tex]P(B)=\frac{n(B)}{n(S)}[/tex]

We substitute the given information to obtain:

[tex]P(B)=\frac{4}{12}[/tex]

We simplify to obtain:

[tex]P(B)=\frac{1}{3}[/tex]

Answer:

x = 29.2

Step-by-step explanation:

Given

Perpendicuar=10

Angle=20 degrees

Hypotenuse=x

We will use the trigonometric ratios of right angled triangle to solve this question. Since, we have to find x which is the hypotenuse of the given triangle. So we will use the ratio which involves perpendicular and hypotenuse.

sin⁡ 20=Perpendicular/Hypotenuse

sin⁡ 20=10/x

x=10/sin ⁡20  

x=10/0.3420

x=29.239

Rounding off to nearest tenth.

x=29.2

Answer:

there could be 8 trucks to 10 cars

16 trucks to 20 cars

or just continuously multiply by 2

Step-by-step explanation:

Answer:

there could be 8 trucks to 10 cars

16 trucks to 20 cars

or just continuously multiply by 2

Answer:

Strain Hardening as name implies, physical straining of metal is induced to increase strength and thus load carrying capacity of the specimen under consideration. The level of straining is dependent on the increased strength required. Strains are classified into two as 'Lateral Strain' which is decrease of cross sections and 'Linear Strains' which is increase in physical extensions (usually 'length') of the specimen.

Step-by-step explanation:

ANSWER

The curve is a parabola with a vertex at left parenthesis 3 comma negative 4 right parenthesis and is traced from left to right for increasing values of t.

EXPLANATION

The given curve is defined parametrically as;

[tex]x = 3+ t[/tex]

[tex]y = {t}^{2} - 4[/tex]

We need to eliminate the parameter by making t the subject in the first equation and substitute into the second equation.

[tex]t = x - 3[/tex]

[tex]y = {(x - 3)}^{2} - 4[/tex]

This is a parabola that has its vertex at (3,-4).

This parabola opens upwards.

The correct description is a parabola with a vertex at (-3, -4) and, because x increases linearly with t, it is traced from left to right as t increases. This corresponds to option 3.

The student's question involves finding the description of a curve represented by parametric equations x = -3 + t and y = t^2 - 4. Considering the equation for y which is a second-order polynomial in t, it indicates that the graph is a parabola. The presence of t^2 indicates the parabola opens upwards as the coefficient is positive. If we were to eliminate t from the parametric equations, the resultant equation would still describe this parabola.

To analyze the vertex, the standard form for a parabola's equation is y = ax^2 + bx + c. Given that y = t^2 - 4 has a constant term of -4, this suggests the vertex's y-coordinate is -4. For the x-coordinate of the vertex, we must consider the constant -3 in the x equation, which adjusts the x-coordinate of the vertex. Since the parametric equation for x alone does not yield an obvious vertex, we realize that the parabola is shifted from the origin, and the -3 signifies a leftward shift from the y-axis.

Therefore, The correct description is a parabola with a vertex at (-3, -4) and, because x increases linearly with t, it is traced from left to right as t increases. This corresponds to option 3.

Answer:

Collin: about $401 thousand

Cameron: about $689 thousand

Step-by-step explanation:

A situation in which doubling time is constant is a situation that can be modeled by an exponential function. Here, you're given an exponential function, though you're not told what the variables mean. That function is ...

[tex]P(t)=P_0(2^{t/d})[/tex]

In this context, P0 is the initial salary, t is years, and d is the doubling time in years. The function gives P(t), the salary after t years. In this problem, the value of t we're concerned with is the difference between age 22 and age 65, that is, 43 years.

In Collin's case, we have ...

P0 = 55,000, t = 43, d = 15

so his salary at retirement is ...

P(43) = $55,000(2^(43/15)) ≈ $401,157.89

In Cameron's case, we have ...

P0 = 35,000, t = 43, d = 10

so his salary at retirement is ...

P(43) = $35,000(2^(43/10)) ≈ $689,440.87

___

Sometimes we like to see these equations in a form with "e" as the base of the exponential. That form is ...

[tex]P(t)=P_{0}e^{kt}[/tex]

If we compare this equation to the one above, we find the growth factors to be ...

2^(t/d) = e^(kt)

Factoring out the exponent of t, we find ...

(2^(1/d))^t = (e^k)^t

That is, ...

2^(1/d) = e^k . . . . . match the bases of the exponential terms

(1/d)ln(2) = k . . . . . take the natural log of both sides

So, in Collin's case, the equation for his salary growth is

k = ln(2)/15 ≈ 0.046210

P(t) = 55,000e^(0.046210t)

and in Cameron's case, ...

k = ln(2)/10 ≈ 0.069315

P(t) = 35,000e^(0.069315t)

Answer:

c The square of a number is equal to the product of 32 and that number.

Step-by-step explanation:

Let the number be w.

The square of the number is [tex]w^2[/tex]

The product (multiplication) of the number and 32 is [tex]32w[/tex]

The symbol = is read "is equal to"

[tex]w^2=32w[/tex]

The square of a number = The product of 32 and that number

The square of a number equals to The product of 32 and that number

We can conclude that the correct answer is c The square of a number is equal to the product of 32 and that number.

The answer is:

The x-intercept or roots of the parabola are:

[tex]x_{1}=-5\\x_{2}=-1[/tex]

To solve the problem, we need to find the roots or zeroes of the parabola.

We can find the zeroes of the quadratic equation (parabola) by factoring its equation.

So,we are given the function:

[tex]f(x)=x^{2} +6x+5[/tex]

To factorize the equation, we need to find two numbers which product gives as result the number 5, and its addition gives as result the number "6", these numbers are 5 and 1.

So, rewriting the equation, we have:

[tex]f(x)=x^{2} +6x+5=(x+5)(x+1)=0[/tex]

Therefore, we have that the x-intercept or roots of the parabola are:

[tex]x_{1}=-5\\x_{2}=-1[/tex]

Have a nice day!

Note: I have attached a picture for better understanding.

Answer:

[tex]x = -5\\x = -1[/tex]

Step-by-step explanation:

To find the intercept with the x axis, you must do [tex]f(x) = 0[/tex]

So:

[tex]f(x) = x^2 + 6x + 5 = 0[/tex]

Now you must factor the expression.

To factor the expression you must find two numbers such that when you add them, you obtain 6 and multiplying it will result in 5.

You can verify that these numbers are 5 and 1.

So

[tex]f(x) = (x + 5)(x + 1) = 0[/tex]

Therefore the solutions are

[tex]x = -5\\x = -1[/tex]

Answer:

For a. [tex]x=7[/tex]

For b. [tex]x=2[/tex]

Step-by-step explanation:

To solve this, we are using the intersecting secants theorem. The theorem says that if two secant segments intersect a circle from an exterior point, then the product of the measures of the exterior segment and the whole secant  is equal to the product of the measures of the other exterior segment and its whole secant.

Applying this to our circles:

For a.

[tex]5(5+x)=6(6+4)[/tex]

[tex]25+5x=6(10)[/tex]

[tex]25+5x=60[/tex]

[tex]5x=60-25[/tex]

[tex]5x=35[/tex]

[tex]x=\frac{35}{5}[/tex]

[tex]x=7[/tex]

For b.

[tex]4(4+x)=3(3+5)[/tex]

[tex]16+4x=3(8)[/tex]

[tex]16+4x=24[/tex]

[tex]4x=8[/tex]

[tex]x=\frac{8}{4}[/tex]

[tex]x=2[/tex]

We can conclude that the value of x in (a) is 7, and the value of x in (b) is 2

Answer:

  3.5 in

Step-by-step explanation:

Put the given numbers in the formula for the volume of a cylinder, then solve for the unknown.

  V = πr²h

  73.5π = πr²·6 . . . . . fill in V and h

  12.25 = r² . . . . . . . . divide by the coefficient of r²

  3.5 = r . . . . . . . . . . . take the square root

The radius of the can is 3.5 inches.

Answer:

The domain of g(x) is {xI x > -5} ⇒ first answer

Step-by-step explanation:

* Lets talk about the transformation at first

- If the function f(x) translated horizontally to the right  

 by h units, then the new function g(x) = f(x - h)

- If the function f(x) translated horizontally to the left  

 by h units, then the new function g(x) = f(x + h)

- If the function f(x) translated vertically up  

 by k units, then the new function g(x) = f(x) + k

- If the function f(x) translated vertically down  

 by k units, then the new function g(x) = f(x) – k

* lets revise the meaning of the domain

- The domain is all values of x that make the function defined

- Find the values of x which make the function undefined

- The domain will be all the real numbers except those values

* Now we can solve the problem

∵ f(x) = √x

- f(x) translated 5 units to the left, then add x by 5

∴ f(x) ⇒ f(x + 5)

- f(x) translated 3 units up, then add f(x) by 3

∴ f(x) ⇒ f(x) + 3

- The function g(x) is created after the transformation

∴ g(x) = f(x + 5) + 3

∵ f(x) = √x

∴ g(x) = √(x + 5) + 3

- The function will be defined if the value under the square root

  is positive (means greater than 0)

∵ The expression under the square root is x + 5

∴ x + 5 > 0 ⇒ subtract 5 from both sides

∴ x > -5

- The domain will be all the real numbers greater than -5

∴ The domain of g(x) is {xI x > -5}

The domain of the function g(x), which is the function f(x) = square root of x translated left 5 units and up 3 units, is {x | x > -5}.

The original function f(x) has a domain of {x | x ≥ 0} because we cannot take the square root of a negative number. When the function is translated left 5 units, the new function, g(x), will start at x = -5 instead of x = 0, reflecting the domain shift due to translation. Therefore, the domain of g(x) is {x | x > -5}.

Explanation:

The sum of the measures of the interior angles of a triangle is 180°. Then the sum of the given angles is 180°:

m∠P +m∠Q +m∠R = 180°

(5x)° +(5x)° +(8x)° = 180°

18x = 180 . . . . . . . . . . . . . . . collect terms, divide by °

x = 10 . . . . . . . . . . . . . . . . . . divide by 18. This is your desired result.

Explanation:

The sum of the measures of the interior angles of a triangle is 180°. Then the sum of the given angles is 180°:

m∠P +m∠Q +m∠R = 180°

(5x)° +(5x)° +(8x)° = 180°

18x = 180 . . . . . . . . . . . . . . . collect terms, divide by °

x = 10 . . . . . . . . . . . . . . . . . . divide by 18. This is your desired result.

Is there a formula that I could use to solve it?

The area of the shaded region is,

⇒ A = 41.87 cm²

The circle is a closed two dimensional figure , in which the set of all points is equidistance from the center.

We have to given that;

Radius of small circle = 3 cm

And, Radius of large circle = 3 + 4 = 7 cm

Hence, Area of sector for large circle,

A = 120/360 × 3.14 × 7²

A = 51.29 cm²

And, Area of sector for small circle is,

A = 120/360 × 3.14 × 3²

A = 9.42 cm²

Hence, The area of the shaded region is,

⇒ A = 51.29 - 9.42

⇒ A = 41.87 cm²

Learn more about the circle visit:

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