100 points 2 questions


Question 1
There are 140 people in a theater watching a movie. The number of children present is 3 times the number of adults. How many children are watching the movie?

12 children

35 children

3 children

105 children

question 2
A catering company pays cooks $85 per shift and waiters $45 per shift. Yesterday, 10 people worked for the company at a special event. If the total amount paid to the employees was $570, how many cooks worked at the event?
3 cooks

7 cooks

10 cooks

12 cooks

The end behavior of a graph is defined as what is going on at the ends of each graph when the function approaches positive or negative infinity.

End behaviour of given graph is,  x⇒ -∞ then y⇒∞ and x ⇒∞ then y⇒ -∞

Since, here a  graph is attached.

From graph it is observed that, when x tends to negative side, graph is going to upward direction. and when x tends to positive side then graph is going to downward direction.

So, when  x tends negative infinity then y tends to positive infinity.

when x tends to positive infinity then y tends to negative infinity.

Therefore, end behaviour of given graph are,

                   x⇒ -∞ then y⇒∞ and x ⇒∞ then y⇒ -∞

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The possible values of X in the GP 72,x,18 is 36.

A geometric progression simply means a sequence that has a common ratio.

From the information, the first term is 72 and the third term is 18

First term a = 72

Third term ar² = 18

Therefore, 72r² = 18

r² = 18/72

r² = 1/4

r = ✓1/4

r = 1/2

Therefore second term will be:

= 72 × 1/2

= 36

Therefore, x is 36.

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Answer:

acute angle - UVW

obtuse angle - VWX

right angle - UYX

Answer:

  30 m

Step-by-step explanation:

Given a trapezium with area 756 m², height 28 m, and one base of length 24 m, you want to know the length of the other base.

The formula for the area of a trapezium is ...

  A = 1/2(b1 +b2)h

Using the given values, we have ...

  756 = 1/2(24 +b2)(28)

  54 = 24 +b2 . . . . . . . . . . . divide by 14

  30 = b2 . . . . . . . . . . . . subtract 24

The measure of the other parallel side is 30 m.

The fraction of 1 pizza to be shared with 4 friends are 3/8

Division is breaking a number up into an equal number of parts.

Given that, there are pizzas 1[tex]\frac{1}{2}[/tex] to be shared between 4 friends,

1[tex]\frac{1}{2}[/tex] = 3/2

For sharing = 3/2÷4 = 3/8

Hence, The fraction of 1 pizza to be shared with 4 friends are 3/8

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Answer: 14 is the ancer

Step-by-step explanation:

Answer:

its 14

Step-by-step explanation:

The distance of the satellite from tower A, using the law of sines, is given by:

11987.216 km.

Suppose we have a triangle in which the sides and angles can be seen as follows:

The lengths and the sine of the angles are related by the equality shown as follows:

[tex]\frac{\sin{A}}{a} = \frac{\sin{B}}{b} = \frac{\sin{C}}{c}[/tex]

The internal angles of the triangle in this problem are given as follows:

The distance of the satellite from the tower A is opposite to the angle of 85.8º, while the distance between the towers of 658 km is opposite from the angle of 3.1º, hence the following relation is established:

sin(85.8º)/d = sin(3.1º)/658.

Applying cross multiplication, the distance is obtained as follows:

d = 658 x sin(85.8º)/sin(3.1º)

d = 11987.216 km.

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The third term from the sequence should be 48 so that (a) the next term is 147 (b) the rule to find the nth term is 3×n² and (c) the 20th term is 1200.

From the question, 3, 12, 27, 64, 75, 108 the third term 64 is supposed to be 48. hence the terms are derived as follows:

1st term; 3 × 1² = 3

2nd term;3 × 2² = 12

3rd term; 3 × 3² = 27

4th term; 3 × 4² = 48

5th term; 3 × 5² = 75

6th term; 3 × 6² = 108

Therefore, the next term which the 7th term is 3 × 7² = 147, the rule for the sequence is 3×n² and the 20th term is 3 × 20² = 1200.

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Answer:

(0,-3)

Step-by-step explanation:

Answer:

120 inches by 80 inches

3 inches by 2 inches

Step-by-step explanation:

So the ratio of 12/8 is 1.5 so we divide all of the numbers to find ratios that equal that

120/80=1.5

3/2=1.5

but we dont put say 6 by 2 because 6/2 is 3

According to the information, it can be inferred that the turtle travels 0.5 miles in a quarter of an hour (1/4 hours).

To find the distance that the tortoise travels in a quarter of an hour we must perform the following operations and take into account the information provided.

To verify that the turtle takes 1/4 of an hour to travel 0.5 miles, we must divide the total number of miles it travels in one hour by 4, because we want to identify how much distance it travels in a quarter of the time, as shown below:

2 miles / 4 = 0.5 miles

Based on the above, we can infer that the tortoise travels 0.5 miles in 1/4 hour.

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Using the two points on the graph, Sandra can prove that her speed was constant using: Slope (m) = (18 - 6) / (3 - 1).

The slope of a graph is the ratio of the vertical distance along the line to the horizontal distance across the line, which is given as, m = change in y / change in x.

For a proportional graph, the slope is always constant, for whichever points we choose to use on the line.

Given two points on the line that Sandra chose, (3, 18) and (1, 6), therefore:

Slope (m) = change in y / change in x

Slope (m) = (18 - 6) / (3 - 1)

Slope (m) = 12/2

Slope (m) = 6

Therefore, to show her speed, which is the slope of the line, is constant, she can use the equation below:

Slope (m) = (18 - 6) / (3 - 1).

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Answer:

Step-by-step explanation:

You want the first 9 terms of the recursive sequence defined by ...

  [tex]\begin{cases}s_1=2\\s_2=3\\s_n=s_{n-2}(s_{n-1}-1)\end{cases}[/tex]

The attached spreadsheet uses the given formula for the next term. It shows the terms to be ...

  2, 3, 4, 9, 32, 279, 8896, 2481705, 22077238784

The attached spreadsheet sum function has been used to find the sum of the first 8 terms. That sum is ...

  2490930

The sequence of problem 1 has neither a common difference nor a common ratio between successive terms. It is neither arithmetic nor geometric.

The sum of the first n terms of an arithmetic sequence with first term a1 and common difference d is given by ...

  Sn = (2a1 +d(n -1))(n/2)

You have a sequence with a1 = 12 and d = (18-12) = 6. You want the sum of the first 200 terms.

  S200 = (2·12 +6(200 -1))(200/2) = 121,800

The sum is 121,800.

__

Additional comment

A spreadsheet is a nice tool for finding terms of a recursively-defined sequence. The formula can include as many terms as necessary, and it can be replicated thousands of times, if necessary. The limitation is that arithmetic is generally limited to 16 significant figures, or so.

Answer:

1.  

s₁ = 2

s₂ = 3

s₃ = 4

s₄ = 9

s₅ = 32

s₆ = 279

s₇ = 8,896

s₈ = 2,481,705

s₉ = 22,077,238,784

2. 2,490,930

3.  Neither.

4.  121,800

Step-by-step explanation:

A recursive rule for a sequence allows you to find the nth term of the sequence provided you know the value of the previous term in the sequence.

Given recursive rule:

[tex]\begin{cases}s_n=s_{n-2} \cdot (s_{n-1}-1)\\s_1=2\\s_2=3\end{cases}[/tex]

Therefore, the first 9 terms of the sequence are:

[tex]s_1=2[/tex]

[tex]s_2=3[/tex]

[tex]\begin{aligned}s_3&=s_{3-2} \cdot (s_{3-1}-1)\\&=s_{1} \cdot (s_{2}-1)\\&=2 \cdot (3-1)\\&=2 \cdot 2\\&=4 \end{aligned}[/tex]

[tex]\begin{aligned}s_4&=s_{4-2} \cdot (s_{4-1}-1)\\&=s_{2} \cdot (s_{3}-1)\\&=3 \cdot (4-1)\\&=3 \cdot 3\\&=9 \end{aligned}[/tex]

[tex]\begin{aligned}s_5&=s_{5-2} \cdot (s_{5-1}-1)\\&=s_{3} \cdot (s_{4}-1)\\&=4 \cdot (9-1)\\&=4 \cdot 8\\&=32 \end{aligned}[/tex]

[tex]\begin{aligned}s_6&=s_{6-2} \cdot (s_{6-1}-1)\\&=s_{4} \cdot (s_{5}-1)\\&=9 \cdot (32-1)\\&=9 \cdot 31\\&=279\end{aligned}[/tex]

[tex]\begin{aligned}s_7&=s_{7-2} \cdot (s_{7-1}-1)\\&=s_{5} \cdot (s_{6}-1)\\&=32 \cdot (279-1)\\&=32 \cdot 278\\&=8896\end{aligned}[/tex]

[tex]\begin{aligned}s_8&=s_{8-2} \cdot (s_{8-1}-1)\\&=s_{6} \cdot (s_{5}-1)\\&=279 \cdot (8896-1)\\&=279\cdot 8895\\&=2481705\end{aligned}[/tex]

[tex]\begin{aligned}s_9&=s_{9-2} \cdot (s_{9-1}-1)\\&=s_{7} \cdot (s_{8}-1)\\&=8896\cdot (2481705-1)\\&=8896\cdot 2481704\\&=22077238784\end{aligned}[/tex]

Given series:

[tex]\displaystyle \sum^8_{k=1} s_k[/tex]

The sum notation asks to find the sum of the first 8 terms of the sequence from question 1.

Therefore:

[tex]\begin{aligned}\displaystyle \sum^8_{k=1} s_k&=s_1+s_2+s_3+s_4+s_5+s_6+s_7+s_8\\&=2+3+4+9+32+279+8896+2481705\\&=2490930\end{aligned}[/tex]

If a sequence is arithmetic, the difference between consecutive terms is the same (this is called the common difference).

If a sequence is geometric, the ratio between consecutive terms is the same (this is called the common ratio).

As the difference between consecutive terms it not the same, the sequence is not arithmetic.

As the ratio between consecutive terms is not the same, the sequence is not geometric.

Therefore, the sequence is neither arithmetic nor geometric.

[tex]\boxed{\begin{minipage}{7.3 cm}\underline{Sum of the first $n$ terms of an arithmetic series}\\\\$S_n=\dfrac{1}{2}n[2a+(n-1)d]$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the first term. \\ \phantom{ww}$\bullet$ $d$ is the common difference.\\ \phantom{ww}$\bullet$ $n$ is the position of the term.\\\end{minipage}}[/tex]

Given arithmetic sequence:

Therefore:

To find the sum of the first 200 terms, substitute the found values of a and d into the formula along with n = 200:

[tex]\begin{aligned}S_{200}&=\dfrac{1}{2}(200)[2(12)+(200-1)(6)]\\&=100[24+(199)(6)]\\&=100[24+1194]\\&=100[1218]\\&=121800\end{aligned}[/tex]

Answer: Break-even point in units= 20,000

Step-by-step explanation:

Break-even point in units= 20,000

Step-by-step explanation:

Giving the following information:

Selling price per unit= $29.99

Unitary variable cost= $14.25

Fixed costs= $314,800

To calculate the break-even point in units, we need to use the following formula:

Break-even point in units= fixed costs/ contribution margin per unit

Break-even point in units= 314,800 / (29.99 - 14.25)

Break-even point in units= 20,000

Answer:

20,000

Step-by-step explanation:

29.99-14.25 (15.74) it the money that they make per unit without accounting for the fixed cost.

314800 = 15.74x  Divide both sides by 15.74

20000 = x

Using Midsegment theorem, the value of x, y and DE in the triangle are 17, 2, and 20 respectively.

The Midsegment theorem states that the midsegment connecting the midpoints of two sides of a triangle is parallel to the third side of the triangle, and the length of this midsegment is half the length of the third side.

Therefore, DH is midsegment of  triangle DEF.

Let's find the value of x.

1 / 2 (28) = x - 3

14 = x - 3

x = 14 + 3

x = 17

Let's find y

x - 7 = y + 8

17 - 7 = y + 8

10 = y + 8

y = 10 - 8

y = 2

Let's find DE

DE =  y + 8 + x - 7

DE  = 2 + 8 + 17 - 7

DE = 20

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Answer:

Two correct answers: subtracting exponents and

3^ 2/9

Step-by-step explanation:

To divide, subtract exponents. To subtract fractions keep the bottom number the same and subtract the top number. See image.

The domain of function f using interval notation is (-3, 3).

The range of function f using interval notation is (-5, 4).

In Mathematics, a domain simply refers to the set of all real numbers for which a particular function is defined. This ultimately implies that, a domain is the set of all possible input numerical values (numbers) to a function and the domain of a graph comprises all the input numerical values which are shown on the x-axis.

Additionally, the vertical extent of a graph represents all range values and they are always read and written from smaller to larger numerical values, and from the bottom of a graph to the top.

By critically observing the graph of this function shown, we can reasonably infer and logically deduce the following:

Domain = (-3, 3).

Range = (-5, 4).

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x= 90°.

[tex]4 + 2 sin(x)= 14-8 sin(x)[/tex]

[tex]-4+4 + 2 sin(x)= 14-8 sin(x)-4\\ \\2 sin(x)= 10-8 sin(x)[/tex]

[tex]2 sin(x)+8 sin(x)= 10-8 sin(x)+8 sin(x)\\ \\10 sin(x)= 10[/tex]

[tex]\frac{10 sin(x)}{10} = \frac{10}{10} \\ \\sin(x)=1[/tex]

[tex]arcsin(sin(x))=arcsin(1)\\ \\x=90[/tex]

x= 90°.

The measures of Angle X, Y and Z are 160°,  15°,  5° respectively.

a figure which is formed by two rays or lines that shares a common endpoint is called an angle. The word “angle” is derived from the Latin word “angulus”, which means “corner”. The two rays are called the sides of an angle, and the common endpoint is called the vertex.

The sum of the three angles X, Y, Z is 180°

X + Y + Z = 180---------------1

if X is 8 times the sum of Y and Z, it means,

X = 8( Y + Z)--------------------2

Also

Y is three times Z, which means

Y = 3Z---------------------3

Substitute equation 3 and 2 in equation 1

8( 3Z + Z) + 3Z + Z = 180

32Z +4Z = 180

36Z = 180

Z = 180/36

Z = 5°

substitute the value of z in equations 1 and 3 to find X and Y

Y = 3Z,

at Z = 5

Y = 3 x 5

Y = 15°

X = 8(Y +Z)

X = 8( 15 + 5)

X = 8 x 20

X  = 160°

In conclusion, the Value of the angles X, Y and Z are 160°, 15° and 5° respectively.

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The linear function that models the puppy's growth is written as: y = 3x + 3.

The values we need to find before we can write he linear function that models a situation, are:

The linear function would be expressed as y = mx + b, which is in the slope-intercept form.

The table of the puppy's growth that Shayla is tracking is shown below. Using two points on the table, say, (0, 3) and (1, 6):

Slope (m) = change in y / change in x = (6 - 3)/(1 - 0)

Slope (m) = 3/1 = 3

Substitute m = 3 and (0, 3) into y = mx + b to find b:

3 = 3(0) + b

3 = 0 + b

3 = b

b = 3

To write the linear function, substitute m = 3 and b = 3 into the equation, y = mx + b:

y = 3x + 3

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Answer: A

Step-by-step explanation: FLVS question the answer is A

We have a translation of 7 units to the left, and it is:

F(c + 7) = 4c + 39

Here we have the linear function:

F(x) = 4x + 11

And we want to find:

F(c + 7)

So we just need to evaluate the function in x = c + 7, we will get:

F(c + 7) = 4*(c + 7) + 11

F(c + 7) = 4c + 28 + 11

F(c + 7) = 4c + 39

So basically we had a translation of 28 units up of our linear function (or 7 units to the left)

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Considering the figure, the line and the plane are

A plane extends infinitely in two dimensions, but a line is defined as anything that extends infinitely in one direction

The extension of the line and plane in the diagram are defined hence discrete

PQ represents the line on the plane PQS

While Plane covers more space and in two dimension, plane can say to be covering area or measured as area. Line measures just the length

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The value of the fraction −3 4/9÷(−2 1/3) is 1 10/21.

A fraction is simply a piece of a whole. The number is represented mathematically as a quotient where the numerator and denominator are split. In a simple fraction, the numerator as well as the denominator are both integers.

In this case, the fraction will be illustrated as:

= −3 4/9÷(−2 1/3)

Change to improper fraction

= -31 / 9 ÷ - 7/3

= 31/9 × 3/7

= 31 / 21

= 1 10/21

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The standard form of the quadratic function y=ax² + bx + c = a (x - h)² + k

By definition, quadratic function is a function in the form f(x) = ax² + bx + c, where a, b, and c are numbers with a not equal to zero.

ax² + bx + c = a (x² - 2xh + h²) + kax² + bx + c

= ax² - 2ah x + (ah² + k)

Comparing the coefficients of x on both sides,b = -2ah ⇒ h = -b/2a .............................................(i)

Comparing the constants on both sides,c = ah² + kc = a (-b/2a)² + k (From (1))

c = b²/(4a) + kk = c - (b²/4a)k = (4ac - b²) / (4a)

Therefore, we can use the formulas h = -b/2a and k = (4ac - b²) / (4a) to convert the standard form to its vertex form.

the vertex form of the parabola is  y = a (x - h)² + k

The standard form of the parabola is y = ax² + bx + c

The standard form of the quadratic function y=ax²+bx+c is x=-b± [tex]\frac{\sqrt{b^{2}- } 4ac}{2a}[/tex]

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The auxiliary function used in the composition between the two functions f(x) and g(x) is equal to linear function g(x) = x + 3.

Herein we find the result of a composition between two functions, where the variable x of the parent function f(x) is substituted by the auxiliary function g(x). The composition between the two functions is defined below:

h(x) = f ° g (x) = f [g (x)]

If we know that the parent function f(x) = x⁵, then the auxiliary function within the composition is the linear function g(x) = x + 3.

The statement is poorly formatted, the correct form is shown below: The function h(x) = (x + 3)⁵ can be expressed in the form f[g(x)] where f(x) = x⁵ and g(x) is defined below.

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Answer:

D, (5,-8)

Step-by-step explanation:

On the graph in the image, you can see option D's only point on the line.

Answer: D

Step-by-step explanation:

You will plug every value into the equation; if they are true, that is the correct option.

A.

5+8=3/7(3/7-5)

13=3/7(-32/7)

13=(-96/49)

A is INCORRECT

B.

8+8=3/7(-5-5)

16=3/7(-10)

16=-30/7

B is INCORRECT

C.

-5+8=3/7(8-5)

3=3/7(3)

3=9/7

C is INCORRECT

D.

-8+8=3/7(5-5)

0=3/7(0)

0=0

D IS CORRECT

Quadratic function whose graph has its vertex

at (−1, 2) and its y-intercept at y = 3 is f(x) = [tex](x+1)^{2}[/tex] + 2

What do you mean by quadratic equation?

A quadratic equation in algebra is any equation that can be transformed in standard form like the following:

[tex]ax^{2} +bx+c[/tex]

Where x stands for an unknown and a, b, and c are known numbers. The coefficients of the equation, denoted by the numbers a, b, and c, are the quadratic coefficient

It is given that graph of the quadratic funcion has vertex(-1,2) and y-intercept at y=3

Now, using formula, f(x) = a[tex](x-h)^{2}[/tex] + k, where (h,k) is the vertex of the parabola.

Here, vertex of the quadratic function is (-1,2)

So, f(x) = a[tex](x+1)^2[/tex] + 2

Now, we need to solve for a.

y-intercept is given as y=3, which means f(x)= 3 when x=0

Using this we get ,
f(0) = 3

a(1) + 2 = 3

a + 2 = 3

a = 3 - 2

a = 1

Quadratic function become:

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

Therefore, quadratic function whose graph has its vertex

at (−1, 2) and its y-intercept at y = 3 is f(x) = [tex](x+1)^{2}[/tex] + 2

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The total of the areas under the standard normal curve to the left of (z₁ = -2.575) and to the right of (z₂ = 2.575) is equal to 0.0098.

In Statistics, the standard normal distribution table is designed and developed to provide only the area to the left of a specified z-score. Additionally, since z-score (z₁) and z-score (z₂) are generally symmetric about z = 0 and are negatives of one another, then, by symmetry, the area to the right of z-score (z₂) is always equal to the area to the left of z-score (z₁).

This ultimately implies that, the total areas under a standard normal curve in two tails can be determined by calculating the area to the left of z-score (z₁) and multiplying the value by two (2).

From the z-score table, the area to the left of z-score (z₁ = -2.575) is given by:

The area to the left of z-score (z₁ = -2.575) = 0.0049

For the total areas, we have:

Total areas under a standard normal curve = 2 × 0.0049

Total areas under a standard normal curve = 0.0098.

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The cost of the 80 square inches metal sheet is $160

The length of the rectangular sheet = 8 inches

The width of the rectangular sheet = 10 inches

The area of  the rectangular sheet = The length of the rectangular sheet  × The width of the rectangular sheet

Substitute the values in the equation

= 8 × 10

= 80 square inches

The cost of metal per square inch = $2

Then the cost of metal sheet = The area of  the rectangular sheet × The cost of metal per square inch

Substitute the values in the equation

= 80 × 2

= $160

Hence, the cost of the 80 square inches metal sheet is $160

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Answer: The slope is 7/5

Step-by-step explanation:

(-2,2) and (3,9)

The slope formula is [tex]m=\frac{y_{2}- y_{1}}{x_{2}- x_{1}}[/tex]

Lets insert our numbers into there!

[tex]m=\frac{9-2}{3- -2}[/tex]

m= 7/5

Hope this helps! <33