The admission fee at the county fair is $2 for children and $4 for adults. on a certain day, 2000 people enter the fair and $5000 is collected. using c as the variable to represent the number of children and a as the variable for adults, write a system of equations to determine how many children and how many adults attended. note down this system of equations so that you can use it for the next question.

Solving question 1 : What is the sum or difference?

[tex] 4x^{10} - 9x^{10} \\\\
(4-9)x^{10} \\\\
-5x^{10} [/tex]

Hence, option A is correct i.e. [tex] -5x^{10} [/tex].

Solving question 2 : What is the sum or difference?

[tex] 6x^{5} - 9x^{5} \\\\
(6-9)x^{5} \\\\
-3x^{5} [/tex]

Hence, option D is correct i.e. [tex] -3x^{5} [/tex].

Solving question 3 : Write the Polynomial in standard form.

2 - 11x² - 8x + 6x²

We can combine like terms, and rewriting it in decreasing power of x's.

⇒ - 11x² + 6x² - 8x + 2

⇒ (-11 + 6)x² - 8x + 2

⇒ -5x² - 8x + 2

Hence, option A is correct i.e. -5x² - 8x + 2; quadratic trinomial.

Solving question 4 :

White-sided jackrabbits: 5.5x² - 9.2x + 6.9

Black-tailed jackrabbits: 5.5x² + 9.9x + 1.3

Total population = White-sided jackrabbits + Black-tailed jackrabbits

Total population = (5.5x² - 9.2x + 6.9 ) + (5.5x² + 9.9x + 1.3)

Total population = (5.5 + 5.5)x² + (9.9 - 9.2)x + (6.9 + 1.3 )

Total population = 11x² + 0.7x + 8.2

Hence, option A is correct i.e. 11x² + 0.7x + 8.2

The hypotenuse of the isosceles right triangle is [tex]\( 12 \)[/tex] units.

In an isosceles right triangle, the legs are congruent, and the hypotenuse can be found using the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse [tex]\( c \)[/tex] is equal to the sum of the squares of the lengths of the other two sides [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:

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

Given that each leg of the isosceles right triangle has a measure of [tex]\( 6\sqrt{2} \)[/tex], we can substitute this value into the formula:

[tex]\[ c^2 = (6\sqrt{2})^2 + (6\sqrt{2})^2 \]\[ c^2 = 36 \times 2 + 36 \times 2 \]\[ c^2 = 72 + 72 \]\[ c^2 = 144 \][/tex]

Now, we take the square root of both sides to find the length of the hypotenuse [tex]\( c \):[/tex]

[tex]\[ c = \sqrt{144} \][/tex]

[tex]\[ c = 12 \][/tex]

So, the hypotenuse of the isosceles right triangle is [tex]\( 12 \)[/tex] units.

The expression given to us is:

[tex] 9m^2-\frac{6}{5}m+c [/tex]

If the above expression is a perfect square then the middle term will have to be 2 times the square root of the first term times the square root of the last term. Thus:

[tex] -\frac{6}{5}m=2\times 3m\times \sqrt{c} [/tex]

[tex] \therefore \sqrt{c}=-\frac{1}{5} [/tex]

Thus, [tex] c=\frac{1}{25} [/tex]

Thus, for 9m^2-(6/5)m+c to be a perfect square, the value of c must be equal to [tex] \frac{1}{25} [/tex] or 1/25

The correct answer is:

$38.75

Explanation:

35 feet of trim at $1.75 per foot would give us a cost of

35(1.75) = 61.25.

Paying with a $100 bill, he would receive

100-61.25 = $38.75 in change.

Answer:

$144

Step-by-step explanation:

Just did test

Answer:

Look below

Step-by-step explanation:

The total cost of the trees must be added to the total cost of the pansies. The tree cost is the cost of one tree times eight. The pansy cost is the cost for 15 pansies multiplied by 8 trees, then divided by the number of pansies in a pack: 20.75(8) + 2.50(15)(8) ÷ 6.

Final answer:

The probability of drawing two green marbles consecutively without replacement from a basket of 4 green marbles and 8 blue marbles is 0.1, or 10%.

Explanation:

The question involves calculating the probability of drawing two green marbles in succession without replacement from a basket containing 4 green marbles and 8 blue marbles. For the first draw, the probability of drawing a green marble is 4 out of 12, which reduces to 1/3 or about 0.3333. Once that marble is drawn, there are 3 green marbles left and 7 blue marbles, making a total of 10.

Therefore, the probability of drawing another green marble is 3 out of 10, or 0.3. To find the probability of both events happening consecutively, we multiply the two individual probabilities: (1/3) * (3/10) = 1/10 or 0.1. Hence, the probability that both marbles will be green is 0.1, or 10%.

Answer:

true

Step-by-step explanation:

Cost function: [tex]\( nc = 72 \)[/tex]. Cost per person for 12 students: $6. Answer: C.

To determine the function that models the data and to find the cost per person if 12 students go on the trip, we need to analyze the relationship between the number of students (n) and the cost per student (c).

Given the data:

- When [tex]\( n = 3 \), \( c = 24 \)[/tex]

- When [tex]\( n = 6 \), \( c = 12 \)[/tex]

- When [tex]\( n = 9 \), \( c = 8 \)[/tex]

- When [tex]\( n = 16 \), \( c = 4.5 \)[/tex]

We can observe that as the number of students increases, the cost per student decreases. This suggests an inverse relationship between the number of students and the cost per student. The form of an inverse relationship can be expressed as:

[tex]\[ c = \frac{k}{n} \][/tex]

where [tex]\( k \)[/tex] is a constant.

To find the constant [tex]\( k \)[/tex], we can use one of the data points. Let's use the first data point ([tex]\( n = 3 \), \( c = 24 \)[/tex]):

[tex]\[ 24 = \frac{k}{3} \][/tex]

Solving for [tex]\( k \)[/tex]:

[tex]\[ k = 24 \times 3 = 72 \][/tex]

So the function that models the data is:

[tex]\[ c = \frac{72}{n} \][/tex]

Now, we need to find the cost per person if 12 students go on the trip. We substitute [tex]\( n = 12 \)[/tex] into the function:

[tex]\[ c = \frac{72}{12} = 6 \][/tex]

Therefore, the cost per person if 12 students go on the trip is $6.

The correct answer is:

C. [tex]\( nc = 72 \)[/tex], $6

To confirm this, we can check that this function fits all the provided data points:

1. For [tex]\( n = 3 \)[/tex]:

[tex]\[ c = \frac{72}{3} = 24 \][/tex] (matches the given cost)

2. For [tex]\( n = 6 \)[/tex]:

[tex]\[ c = \frac{72}{6} = 12 \][/tex] (matches the given cost)

3. For [tex]\( n = 9 \)[/tex]:

[tex]\[ c = \frac{72}{9} = 8 \][/tex] (matches the given cost)

4. For [tex]\( n = 16 \)[/tex]:

[tex]\[ c = \frac{72}{16} = 4.5 \][/tex] (matches the given cost)

Hence, the function [tex]\( c = \frac{72}{n} \)[/tex] is validated by all the data points.

In isosceles trapezoids, the diagonals are not perpendicular. They are congruent, the bases are parallel, and the non-parallel sides are congruent.

An isosceles trapezoid is a type of quadrilateral that has a pair of parallel sides, known as the bases, and the other two sides, not parallel, are of equal length. The statement 'The diagonals are perpendicular' is NOT true for isosceles trapezoids. In isosceles trapezoids, the diagonals are congruent and not perpendicular. Just to put in context, the perpendicular diagonals are a characteristic of rhombuses and not of isosceles trapezoids.

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Answer:

540

Step-by-step explanation:

The answer should be 7092.822912 or 7092.82 when rounded to the nearest hundredth because the formula for volume is V= π times r^2 times height to get the answer. So: 3.14 x 9.8^2 x 23.52, and that's the answer.

QUESTION 1

We want to find the volume of a circular cylinder with a diameter of [tex]19.6yd[/tex] and a height of [tex]23.52yd[/tex].

The volume of a cylinder is given by the formula

[tex]V=\pi r^2h[/tex]

where [tex]h=23.52yd[/tex] and [tex]r=9.8yd[/tex] is half the diameter of the cylinder and [tex]\pi=3.14[/tex].

We substitute all these values into the formula to obtain,

[tex]V=3.14\times 9.8^2\times 23.52[/tex]

[tex]V=7092.82[/tex] square yards to the nearest hundredth.

QUESTION 2

We want to find the volume of a right circular cylinder with a base diameter of [tex]18yd[/tex] and a height of [tex]3yd[/tex].

The volume of a cylinder is given by the formula

[tex]V=\pi r^2h[/tex]

where [tex]h=3yd[/tex] and [tex]r=9yd[/tex] is half the diameter of the cylinder.

We substitute all these values into the formula to obtain,

[tex]V=\pi \times 9^2\times 3[/tex]

[tex]V=243\pi[/tex] square yards.

Answer:

Midpoint of a segment in the complex plane with endpoints at 6 -2i and -4 + 6i is:

1+2i

Step-by-step explanation:

The midpoint of a segment in the complex plane with endpoints at 6 -2i and -4 + 6i is:

[tex]\dfrac{6-2i-4+6i}{2} \\\\=\dfrac{2+4i}{2}\\ \\=1+2i[/tex]

Hence, midpoint of a segment in the complex plane with endpoints at 6 -2i and -4 + 6i is:

1+2i

Answer:  2 + 4 i

Step-by-step explanation:

Hi, to solve this we have to apply the next expression:

(a1 +a2)/ 2 + (b1 +b2 )/2 i=

Where a is the real part, and b is the imaginary part (with i)

For example, for our case:

6 -2i , 6 is the real part (2) and -2 is the imaginary part (b)

Replacing with the values given

(6 -4) /2+ (-2 +6) /2 i = 2 + 4 i

Feel free to ask for more if needed or if you did not understand something.

To model the pressure function in the pipe that oscillates between 40 and 280 ten times an hour, we can use a sine function. The formula is P = f(t) = 120 sin(π/3 t) + 160.

The pressure in the pipe oscillates between 40 and 280 ten times an hour, this is a trigonometric function scenario. Assuming the oscillation is sinusoidal, we can use a sine function to model the pressure in the pipe. The oscillation's period is 6 minutes because the pressure changes happen 10 times per hour. Thus, the function modelling this pressure will be of the form

P = a sin(b(t - c)) + d.

Given that the middle value of the pressure (between the max of 280 and the minimum of 40) is 160, this makes

'd' = 160.

The amplitude 'a' is half the total swing of the pressure which is 120.

To find 'b', we use the fact that the period of a sinusoid in this form is (2π/b).

As our period is 6 minutes, that makes 'b' = π/3.

The pressure is at a minima at t=0 so the phase shift 'c' = 0

Hence the formula P = f(t) = 120 sin(π/3 t) + 160.

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Answer : 1 and 3 are the correct probabilities.

→According to the given Venn diagram.

Total number of elements  = 59.

1)P(C)=[tex]\frac{21}{59}[/tex] and [tex]P(A\cap C)=\frac{14}{59}[/tex] then

[tex]P(A|C)=\frac{P(A\cap C)}{P(C)}[/tex][tex]=\frac{\frac{14}{59}}{\frac{21}{59}}=\frac{14}{21}=\frac{2}{3}[/tex]

2)P(B)=[tex]\frac{27}{59}[/tex] and  [tex]P(C\cap B)=\frac{11}{59}[/tex] then

[tex]P(C|B)=\frac{P(C\cap B)}{P(B)}[/tex][tex]=\frac{\frac{11}{59}}{\frac{27}{59}}=\frac{11}{27}[/tex][tex]\neq \frac{8}{27}[/tex]

3) P(A) =[tex]\frac{number\ of\ elements\ in\ A}{Total\ elements}=\frac{31}{59}[/tex]

4) P(C) =[tex]\frac{number\ of\ elements\ in\ C}{Total\ elements}=\frac{21}{59}[/tex][tex]\neq \frac{3}{7}[/tex]

5) [tex]P(B|A)=\frac{P(B\cap A)}{P(A)}[/tex][tex]=\frac{\frac{13}{59}}{\frac{31}{59}}=\frac{13}{31}[/tex][tex]\neq \frac{13}{27}[/tex]

Therefore, option 1 and 3 are correct.

(1) The measure of angle B is [tex]140^o.[/tex]

(2) The measure of angle A is [tex]65^o.[/tex]

(3) The measure of angle C is [tex]80.5^o.[/tex]

(1) The quadrilateral [tex]\(ABCD\)[/tex] is inscribed in a circle. For any quadrilateral inscribed in a circle, the opposite angles are supplementary (i.e., their sum is [tex]\(180^\circ\)).[/tex]

In the first image:

[tex]- \( \angle DAB = x^\circ \)\\ - \( \angle DCB = (4x - 20)^\circ \)[/tex]

Since these two angles are opposite angles of the inscribed quadrilateral, we have:

[tex]\[ x + (4x - 20) = 180 \][/tex]

Solving for [tex]\(x\):[/tex]

[tex]\[ 5x - 20 = 180 \][/tex]

[tex]\[ 5x = 200 \][/tex]

[tex]\[ x = 40 \][/tex]

Therefore, [tex]\( \angle B = (4x - 20) = 4(40) - 20 = 160 - 20 = 140^\circ \).[/tex]

(2) In the second image:

[tex]- \( \angle ADC = x^\circ \)\\ - \( \angle ABC = 148^\circ \)[/tex]

These are opposite angles of the inscribed quadrilateral. Thus:

[tex]\[ x + 148 = 180 \][/tex]

Solving for [tex]\(x\):[/tex]

[tex]\[ x = 180 - 148 = 32 \][/tex]

Therefore, [tex]\( \angle A = (2x + 1) = 2(32) + 1 = 64 + 1 = 65^\circ \).[/tex]

(3) In the third image:

[tex]- \( \angle DAB = (x + 15)^\circ \)\\ - \( \angle DCB = (x + 10)^\circ \)\\ - \( \angle BCD = (x + 24)^\circ \)[/tex]

Using the property that opposite angles are supplementary:

Opposite angles are [tex]\( (x + 15) \)[/tex] and [tex]\( (x + 24) \),[/tex] thus:

[tex]\[ (x + 15) + (x + 24) = 180 \][/tex]

Solving for [tex]\(x\):[/tex]

[tex]\[ 2x + 39 = 180 \][/tex]

[tex]\[ 2x = 141 \][/tex]

[tex]\[ x = 70.5 \][/tex]

Therefore, the measure of angle C is [tex]\( (x + 10) = 70.5 + 10 = 80.5^\circ \).[/tex]

To simplify the given rational expression, we factor both the numerator and the denominator, cancel out the common factor, and express the result as (n^2 - 6) / (n^2 - 2), with restrictions on n that it can't equate to the square roots of 5 or 2.

The question involves simplifying the rational expression n^4 - 11n^2 + 30 divided by n^4 - 7n^2 + 10, and stating any restrictions on the variable. To simplify this, we first factor both the numerator and the denominator.

Numerator: (n^2 - 5)(n^2 - 6)
Denominator: (n^2 - 5)(n^2 - 2)

After factoring, we can cancel out the common factor (n^2 - 5) from both the numerator and the denominator. This leaves us with (n^2 - 6) / (n^2 - 2).

However, we must state the restrictions on the variable n. The original denominator cannot be equal to zero, thus n^2 cannot be equal to 5 or 2, leading to restrictions of n != sqrt(5) and n != sqrt(2).

Final answer:

The solutions to the equation 2sin^2x = sinx in the interval [0, 2π) are x = 0, π, π/6, 5π/6 and x = π/6, 5π/6.

Explanation:

The given equation is: 2sin^2x = sinx.

To solve this equation, we can first factor out sinx:

sinx(2sinx - 1) = 0.

Setting each factor equal to zero, we get two equations:

Solving for x, we find: