A teacher asked her students how many pets they own. Here are the results:

Answer:

x = 25/4

Step-by-step explanation:

Because of the known similarity of the triangles, we know that

10       x

----- = ----

16      10

Cross-multiplying, we get 16x = 100, and thus x = 100/16 = 50/8 = 25/4

x = 25/4

For the given triangle the value of x is 25/4.

Hence the correct option is E.

The Pythagorean theorem states that,

For a right-angle triangle,

(Hypotenuse)²= (Perpendicular)² + (Base)²

Given that,

In ΔBAC

CB = 16

DB = x

AB = 10

Then CD = 16-x

Apply the Pythagorean theorem in ΔBAC,

Hypotenuse = CB

Perpendicular = AC

Base = AB

(Hypotenuse)²= (Perpendicular)² + (Base)²

(CB)²= (AC)² + (AB)²

(16)²= (AC)² + (10)²

(AC)² = 256 - 100

(AC)² = 156 ......(i)

Apply the Pythagorean theorem in ΔADB,

Hypotenuse = AB

Perpendicular = AD

Base = DB

Therefore,

(AB)²= (AD)² + (DB)²

(10)²= (AD)² + x²

(AD)²= 100 - x²   ......(ii)

Again apply the Pythagorean theorem in ΔADC,

Hypotenuse = AC

Perpendicular = AD

Base = CD

Therefore,

(AC)²= (AD)² + (CD)²

(AC)²= (100 - x²) + (16-x)²                          [ from (ii) ]

(AC)²= 100 - x² + 256 + x² - 32x              [Since (a-b)² = a² + b² -2ab ]

(AC)²= 356 - 32x  ....(iii)

Equating the equation (i) and (iii)

356 - 32x = 156

32x = 200

x = 200/32

x = 25/4

Hence, the value of x is 25/4.

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Answer:

168

Step-by-step explanation:

The first equation given as  [tex]10.50a+3.75b=2071.50[/tex]

Where a is the number of adults and b is the number of students

Since, the number of students are given as 82, we can plug 82 into b and then do algebra and solve for a (shown below):

[tex]10.50a+3.75b=2071.50\\10.50a+3.75(82)=2071.50\\10.50a+307.5=2071.50\\10.50a=2071.50-307.5\\10.50a=1764\\a=\frac{1764}{10.50}\\a=168[/tex]

Thus, 168 adult tickets were sold

Answer:

(-7,4)

Step-by-step explanation:

goal: (y-k)^2=4p(x-h)

y^2-8y=4x+12    Rearranged and added 4x and 12 on both sides

y^2-8y+(-8/2)^2=4x+12+(-8/2)^2 complete square time (add same thing on both sides)

y^2-8y+(-4)^2=4x+12+(-4)^2 (simplify inside the squares)

(y-4)^2=4x+12+16  (now write the left hand side as a square)

(y-4)^2=4x+28

(y-4)^2=4(x+7)    factored...

vertex is (-7,4)

Answer:

(-7,4)

Step-by-step explanation:

Answer:

  8r^4

Step-by-step explanation:

√(64r^8) = √((8r^4)^2) = 8r^4

_____

You can make use of either or both of these rules of exponents:

  (a^b)^c = a^(b·c) . . . . . used above

  [tex]\sqrt[n]{a}=a^{\frac{1}{n}}[/tex]

Using the second rule, you can write the expression as ...

[tex]\sqrt{64r^8}=\sqrt{64}\cdot r^{8\cdot\frac{1}{2}}=8r^4[/tex]

Answer:

B

Step-by-step explanation:

edg21

Answer:

B) 16 ft, 10.5 in

Step-by-step explanation:

There are a few different ways you can work this. Since we want to know the difference between length and heigh of the model and we are given skull length of the model, it makes a certain amount of sense to find the corresponding measurements of the actual skeleton.

The actual skeleton's length was 40 ft and its height was 13 ft, so the difference between these dimensions is ...

40 ft - 13 ft = 27 ft

The actual skull is 5 ft long, so the difference is ...

(27 ft)/(5 ft) = 5.4

times the length of the skull.

The same ratio will apply to the model, so the difference between the model height and model length is 5.4 times the length of the model skull:

desired difference = 5.4 × 3 ft 1.5 in = 16.2 ft + 8.1 in

= 16 ft 10.5 in

ANSWER

The vertex of this parabola is (-7,4)

EXPLANATION

The given parabola has equation:

[tex] {y}^{2} - 4x - 8y - 12 = 0[/tex]

[tex] {y}^{2} - 8y = 4x +12[/tex]

Complete the square for the quadratic equation in y.

[tex]{y}^{2} - 8y + {( - 4)}^{2} = 4x + 12 + {( - 4)}^{2} [/tex]

[tex]{y}^{2} - 8y + {( - 4)}^{2} = 4x + 12 + 16[/tex]

[tex]{( y- 4)}^{2} = 4x + 28[/tex]

[tex]{( y- 4)}^{2} = 4(x +7)[/tex]

The vertex of this parabola is (-7,4)

Answer:

(-7, 4)

Step-by-step explanation:

We are given the following equation for which we have to complete the square in order to find the vertex of this parabola:

[tex] y ^ 2 - 4 x - 8 y - 1 2 = 0 [/tex]

[tex]y^2-(\frac{8}{2} )^2-4x-12=(\frac{8}{2} )^2\\[/tex]

[tex] y ^ 2 - 1 6 - 4 x - 1 2 = 1 6 [/tex]

[tex] ( y - 4 ) ^ 2 - 4 x - 1 2-16=0[/tex]

[tex](y-4)^2=4x+28[/tex]

[tex](y-4)^2=4(x+7)[/tex]

[tex]x+7=0, y-4=0[/tex]

x = -7, y = 4

Therefore, the vertex of this parabola is (-7, 4).

Law of cosines.

Cos(angle) = Adjacent Leg / Hypotenuse

Cos(60) = 20 / x

X = 20/cos(60)

x = 40 ft.

Answer:

40

Step-by-step explanation:

You have a 30-60-90 right triangle. The hypotenuse is twice the length of the short leg.

BC = 2 * 20 = 40

Answer:

0.18 seconds

Step-by-step explanation:

Using the given function, it is found that the object will be 3 inches above the rest position after 0.18 seconds.

The function for an object's height after t seconds is given by:

[tex]h(t) = 8\sin{\left(\frac{2\pi}{3}t\right)}[/tex]

The height is of 3 inches when h(t) = 3, hence:

[tex]h(t) = 8\sin{\left(\frac{2\pi}{3}t\right)}[/tex]

[tex]3 = 8\sin{\left(\frac{2\pi}{3}t\right)}[/tex]

[tex]\sin{\left(\frac{2\pi}{3}t\right)} = \frac{3}{8}[/tex]

[tex]\sin^{-1}{\sin{\left(\frac{2\pi}{3}t\right)}} = \sin^{-1}{\left(\frac{3}{8}\right)}[/tex]

[tex]\frac{2\pi}{3}t = 0.3844[/tex]

[tex]t = \frac{3 \times 0.3844}{2\pi}[/tex]

[tex]t = 0.18[/tex]

The object will be 3 inches above the rest position after 0.18 seconds.

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Answer:

Step-by-step explanation:

98*x = 101.92

x = 101.92/98 = 1.04

The % increase is 1.04%

Final answer:

The percent increase of the company's rates from $98 to $101.92 is 4%. It is calculated by dividing the increase in rates by the original rate and then multiplying by 100.

Explanation:

To calculate the percent increase for a company's rate change from $98 a month to $101.92 a month, we first find the difference in rates. The increase is $101.92 - $98 = $3.92. To find the percentage, we divide the increase by the original amount and multiply by 100. Therefore, the percent increase is ($3.92/$98)  imes 100.

Calculating this gives us a percent increase of approximately 4%. So, the company's rates have increased by 4 percent. The percentage change, or growth rate, indicates how significantly the rates have increased in comparison to the starting rate.

The inequality that represents Sam's situation is: x <= 18.67

To determine the inequality that represents Sam's situation, we can use the following formula for compound interest:

A = P(1 + r/n)^(nt)

where:

A is the final amount

P is the principal amount

r is the interest rate

n is the number of compounding periods per year

t is the number of years

We know that Sam initially invested $4,500 (P = 4500) and that the interest rate is 3% (r = 0.03). We also know that Sam wants to determine the number of years, x (t = x), for which the account will have less than or equal to $7,020 (A = 7020).

Substituting these values into the formula, we get the following inequality:

7020 <= 4500(1 + 0.03/1)^(1x)

Solving for x, we get:

x <= log(7020/4500) / (0.03/1)

x <= 18.67

Therefore, the inequality that represents Sam's situation is:

x <= 18.67

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Answer:

A. Permutation; number of ways = 720

Step-by-step explanation:

For the first medal, we have 6 runners that can earn it.  

For the second medal, we have 5 runners because there's one who won the first one.

For the third, we have 4 runners.

And so on up to the 6th medal where we have just one runner left.

As this happens all at the same time, we have to multiply them.

Ways to award the medals = 6*5*4*3*2*1 = 6! = 720

Remember that a permutation is a combination where the order matters. So, in this case, is a permutation because each medal is different.

Answer:

a) Permutation; number of ways = 720

Step-by-step explanation:

[tex]\bf \qquad \textit{Amount for Exponential Decay} \\\\ A=P(1 - r)^t\qquad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{initial amount}\dotfill &8227\\ r=rate\to 7\%\to \frac{7}{100}\dotfill &0.07\\ t=\textit{elapsed time}\ \end{cases} \\\\\\ A=8227(1-0.07)^t\implies A=8227(0.93)^t[/tex]

Final answer:

An exponential decay model represents the car's value decreasing each year by 7%, with the equation V = 8227 x (1 - 0.07)^t, where V is the car's value and t is the time in years.

Explanation:

The student is dealing with a depreciation problem in which a car decreases in value by a fixed percentage each year. To express this situation mathematically, we can use an exponential decay model. With an initial value of $8227 and an annual decrease rate of 7%, the equation to represent the car's value V at any time t in years can be written as:

V = 8227 times (1 - 0.07)^t

This equation models the car's value as it depreciates 7% per year from its initial value. When t is 0 (at the time of purchase), V will be $8227, indicating the initial value.

Answer:

  Yes

Step-by-step explanation:

(3/8)·64 = 24 seats in the first class carriage are being used.

(7/13)·(78)·3 = 126 seats in the standard carriages are being used, for a total of ...

  24 + 126 = 150 . . . occupied seats

The number of available seats is ...

  64 +3·78 = 298

so half the seats on the train will be 298/2 = 149 seats.

  150 > 149, so more than half the seats on the train are being used.

Answer:

x= -5/7

Step-by-step explanation:

The equation involves only one variable x.

so, we have to isolate the variable to get the solution of the equation

Given

[tex]2(4 - 2x) - 3 = 5(2x + 3)\\8-4x -3 = 10x+15\\5-4x = 10x+15\\-4x = 10x +15 -5\\-4x-10x=10\\-14x = 10\\x = \frac{10}{-14}\\ x = -\frac{x=5}{7}[/tex]

Hence the value of x or solution is

x= -5/7

Answer:

53 miles per hour

Step-by-step explanation:

This is the correct answer

I hope this helps you!

Answer:

Domain is 2 and range is 4

Step-by-step explanation:

Answer:

0

1

Step-by-step explanation:

First question:

You are given a side, a, and its opposite angle, A. You are also given side b. Use that in the law of sines and solve for the other angle, B.

[tex] \dfrac{a}{\sin A} = \dfrac{b}{\sin B} [/tex]

[tex] \dfrac{10}{\sin 30^\circ} = \dfrac{40}{\sin B} [/tex]

[tex] \dfrac{1}{0.5} = \dfrac{4}{\sin B} [/tex]

[tex] \sin B = 2 [/tex]

The sine function can never equal 2, so there is no triangle in this case.

Answer: no triangle

Second question:

You are given a side, b, and its opposite angle, B. You are also given side c. Use that in the law of sines and solve for the other angle, C.

[tex] \dfrac{b}{\sin B} = \dfrac{c}{\sin C} [/tex]

[tex] \dfrac{10}{\sin 63^\circ} = \dfrac{}{\sin C} [/tex]

[tex] \sin C = \dfrac{8.9\sin 63^\circ}{10} [/tex]

[tex] C = \sin^{-1} \dfrac{8.9\sin 63^\circ}{10} [/tex]

[tex] C \approx 52.5^\circ [/tex]

One triangle exists for sure. Now we see if there is a second one.

Now we look at the supplement of angle C.

m<C = 52.5°

supplement of angle C: m<C' = 180° - 52.5° = 127.5°

We add the measures of angles B and the supplement of angle C:

m<B + m<C' = 63° + 127.5° = 190.5°

Since the sum of the measures of these two angles is already more than 180°, the supplement of angle C cannot be an angle of the triangle.

Answer: one triangle

In the first case with measures a=10, b=40, A=30°, no triangle can be formed as a is smaller than b sin(A). In the second case with measures b=10, c=8.9, B=63°, one triangle can be formed because b is greater than c.

In the context of the Ambiguous Case of the Law of Sines, we can find the number of triangles formed given the measures. For the first case, a = 10, b = 40, and A = 30°, no triangle can be formed because a is less than b sin(A), which means the given side (a) is too short to reach the other side (b).

For the second case, b = 10, c = 8.9, and B = 63°, one triangle can be formed. Here, b is greater than c and therefore capable of forming one valid triangle as per the Ambiguous Case of the Law of Sines.

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The value of x is 11/10.

To solve for x in the equation 5(2x - 1) = 6, we need to follow these steps:

Distribute the 5 into the parenthesis:

5 × 2x - 5 × 1

= 10x - 5

Set up the equation:

10x - 5 = 6

Add 5 to both sides of the equation to get

10x = 11

To solve for x, which gives us

x = 11/10

Answer: The height of the robot is 200 millimeters

Step-by-step explanation:

Answer:

It is 1,200

Step-by-step explanation:

Answer:

The third choice is the one you want.

Step-by-step explanation:

The formula for an arithmetic sequence is as follows:

[tex]a_{n}=a_{1}+d(n-1)[/tex]

Our first number is 8, so a1 = 8.  If the second term is 5, then d = -3.  Filling in our formula gives us this:

[tex]a_{n}=8-3(n-1)[/tex]

Now we need domain.  Our choices are n ≥ 1 and n ≥ 0 so let's try both.  Replace n in the formula with each one, one at a time, and see what the result is.

If n ≥ 0:

[tex]a_{0}=8-3(0-1)[/tex] so [tex]a_{0}=8-(-3)[/tex] which gives you that the first term, defined by [tex]a_{0}[/tex] is 11.  That's not correct.  Let's check n ≥ 1[tex]a_{1}=8-3(1-1)[/tex]

and [tex]a_{1}=8-0[/tex] which is 8, the first term.

Answer:

Step-by-step explanation:

1. There are several ways the area can be divided up so that formulas for common figures can be used to find the areas of the pieces. In the attached figure, we have identified an overall rectangle ABXE and a trapezoid BXDC that is subtracted from it.

The area of the rectangle is the product of length and width:

  area ABXE = (180 cm)(140 cm) = 25,200 cm²

The area of a trapezoid is the product of its height (DX = 70 cm) and the average of its base lengths ((BX +DC)/2 = 95 cm).

  area BXDC = (70 cm)(95 cm) = 6650 cm²

Then the area of figure ABCDE is the difference of these areas:

  area ABCDE = area ABXE - area BXDC = (25,200 - 6,650) cm²

  area ABCDE = 18,550 cm²

__

2. In order to find the area of the figure, we need to know the length DE. That length is one leg of right triangle DEA, so we can use the Pythagorean theorem. That theorem tells us ...

  DE² + EA² = AD²

  DE² + (8 ft)² = (10 ft)² . . . . . substitute the given values

  DE² = 36 ft² . . . . . . . . . . . . .subtract 64 ft²

  DE = 6 ft . . . . . . . . . . . . . . . take the square root

Now, we can choose to add the area of triangle DEA to that of square ABCE, or we can treat the whole figure as a trapezoid with bases AB=8 ft and DC=14 ft. In the latter case, the average base length is ...

  (8 ft + 14 ft)/2 = 11 ft

and the area is the product of this and the 8 ft height:

  area ABCD = (11 ft)(8 ft) = 88 ft²

Answer:

  $851.62

Step-by-step explanation:

The value multiplier wll be ...

  (1 +r/n)^(nt)

where r is the annual interest rate (3.8%), n is the number of compoundings per year (12), and t is the number of years (3). Filling in these numbers, we see the ending value will be ...

  A = $760(1 +.038/12)^(12·3) = $760(1.0031667^36) = $851.62

Answer:

$851.62

Step-by-step explanation:

Answer:

56.3 cm

Step-by-step explanation:

(sinA)/(27) = (sinC)/c

(sin28°)/(27) = (sin102°)/c

For this case we have that by definition, the sum of the internal angles of a triangle is 180 degrees.

Then we look for the measure of the third angle:

[tex]102 + 28 + x = 180\\x = 180-102-28\\x = 50[/tex]

According to the Law of sines:

[tex]\frac {sin (50)} {a} = \frac {Sin (28)} {27}\\a = \frac {27 * sin (50)} {sin (28)}\\a = \frac {0.76604444 * 27} {0.46947156}\\a = 44.06[/tex]

Answer:

[tex]a = 44.1[/tex]

Answer:

It's 224.

Step-by-step explanation:

We use the power rule for a derivative.

If f(x) = ax^n then the derivative  f'(x) = anx^(n-1).

So the derivative of 12x^2 + 8x

= 2*12 x^(2-1) + 8x^(1-1)

= 24x + 8x^0

= 24x + 8.

When x = 9 the derivative = 24(9) + 8

=  224.

The value of first order derivative with x=9 is 224. Therefore, option D is the correct answer.

The process of finding derivatives of a function is called differentiation in calculus. A derivative is the rate of change of a function with respect to another quantity.

The given function is f(x)=12x²+8x at x=9.

Here, first order derivative is

f'(x)=24x+8

= 24×9+8

= 224

Therefore, option D is the correct answer.

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Answer:

  see below

Step-by-step explanation:

Eric will drive between 70 -4 = 66 mph and 70+4 = 74 mph. He will not drive less than 66 or more than 74 mph.

Answer:

Step-by-step explanation:

In general, solutions to absolute value inequalities, as in this case, take two forms:

If | x | <a, then x<a or x> -a.

If | x |> a, then x> a or x <-a.

In this case, you have |x − 70| ≤ 4. So, you have two cases:

x − 70 ≤ 4 and x − 70 ≥ -4

Solving both equations:

x − 70 ≤ 4      

x ≤ 4 + 70

x≤ 74

and

x - 70 ≥ -4

x ≥ -4+70

x ≥ 66

It is convenient to graph both solutions, as shown in the attached image .

The intersection between both conditions is the solution to the inequality (that is, in the image it is shown as the interval painted by both colors). In this case, the solution is 66≤x≤74

This indicates that Eric can drive within this speed range.

The range of speeds Eric would not drive at under the given conditions is x≤66 and x≥74,  as shown in the other image.

Answer:

Step-by-step explanation:

Real solutions equal

x=6,3,1

Hope that helps!

ANSWER

[tex]x=1,x=3,x=6[/tex]

EXPLANATION

To solve an equation using the x-intercept method, we must graph the corresponding function and locate x-values of the x-intercepts.

The given polynomial equation is

[tex] {x}^{3} - 10 {x}^{2} + 27x - 18 = 0[/tex]

The graph of the corresponding function

[tex]y = {x}^{3} - 10 {x}^{2} + 27x - 18 [/tex]

Is shown in the attachment.

The x-intercepts are:(1,0), (3,0), (6,0)

Therefore the solutions are,

[tex]x=1,x=3,x=6 [/tex]

Answer:

  $5308.79

Step-by-step explanation:

The future value can be computed from ...

  FV = P(1 +r/n)^(nt)

where P is the principal invested, r is the annual interest rate, n is the number of times per year it is compounded, and t is the number of years.

Filling in the given numbers, we have ...

  FV = $5000(1 +.03/12)^(12·2) ≈ $5308.79

Myron's withdrawal will be in the amount of $5308.79.

Answer:

  6.  quadratic

  7.  linear

Step-by-step explanation:

6. First differences are ...

Second differences are ...

These are constant at -2, so the data are quadratic. The data can be described by y = 6-(x-3)^2.

__

7. The x-values are evenly spaced (though decreasing). For our purpose, we can still look at the first differences of the table values in the order given. First differences are ...

These are constant at +4, so the data are linear. The data can be described by y = -4x +12.

Answer:

x = 13

Step-by-step explanation:

There are two ways to do this. You could use the Pythagorean Theorem because you are given that the triangles are right angled triangles.

Method One

a^2 + b^2 = c^2

a = 5

b = 12

c = ?

c^2 = 5^2 + 12^2

c^2 = 25 + 144

c^2 = 169

sqrt(c^2) = sqrt(169)

c = 13

x = 13

Method Two

You can do this problem with similar triangles.

48/12 = 52/x                    Cross multiply

48x = 12 * 52                   Combine the right  

48x = 624                        Divide by 48

48x/48 = 624/48              Do the division

x = 13  

Answer:

PD = 10.6

Step-by-step explanation:

Point P has coordinates (3,-1) and point Q has coordinates (-5,6)

(3 - (-5) ) = 8

-1 - 6 = -7

PD = √8^2 + (-7)^2

PD = √(64 + 49)

PD = √113

PD = 10.6