An adult ticket to an exhibition costs $5 while a child's ticket costs $2. If the number of children going to the exhibition was 3 more than the number of adults and a total amount of $20 was spent on the tickets. Find the number of adults going to the exhibition.

Hint:

Diameter(d)= 2 radius(r)

circumference= 2πr= πd

1. diameter= 19 × 2 = 38 inches

circumference= 3.18(38) = 119 inches (3 s.f.)

Note that I use 3.18 instead of π because the question states to use 3.14 for π.

Likewise, if you are given the diameter, divide it by 2 to find radius. Let's try a question which only gives you the diameter.

4. radius= 22 ÷ 2 = 11cm

circumference= 3.14(22) = 69.1cm (3 s.f.)

Answer: 12 1/3

Step-by-step explanation:

First, you need to add up 11/4 and

6 1/2

11/ 4 + 6 1/2 = 11/4 + 13/2 = 37 / 4

To find how many 3/4 we have in 37/4, we simply dividw 37/4 by 3/4

37/4 ÷ 3/4

= 37/4 × 4/3 (4 will cancel out 4)

= 37/3

=12 1/3

The number of groups of 3/4 that are in 11/4 and 6 1/2 is 3 2/3 and 8 2/3 respectively.

= 11/4 ÷ 3/4

= 11/4 × 4/3

= 11/3

= 3 2/3

= 6 1/2 ÷ 3/4

= 13/2 ÷ 3/4

= 13/2 × 4/3

= 8 2/3

Therefore, the number of groups of 3/4 that are in 11/4 and 6 1/2 are 3 2/3 and 8 2/3 respectively.

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Answer:

10,93,500.

Step-by-step explanation:

On first February total number of rabbits were 500.

Now it is given that after each month rabbit population will become triple of initial.

So, after one month rabbits will become 3×500.

After two months they will become [tex]3^{2}[/tex]×500.

Thus , after m months total number of rabbits will become [tex]3^{m}[/tex]×500.

Now,

On August 1 , 7 months will get passed from February 1 so putting m = 7 in the equation we get ,

Total number of rabbits = [tex]3^{7}[/tex]×500 = 10,93,500.

The function that represents the number of rabbits is f(m) = 500 * 3^m. Replacing m with 6, which represents the 6 months from February 1st to August 1st, we get that there would be 145800 rabbits in Lancaster on August 1st.

This question is about an exponential function, specifically a geometric sequence, where each term is tripled to get the next. The general form of an exponential function is f(m) = ab^m, where a is the initial amount, b is the rate of growth, and m is the time period.

In this case, the initial amount (a) of rabbits is 500, the rate of growth (b) is 3 (as the population triples every month), and m represents the months passed. Hence, the function relating to the rabbit population becomes f(m) = 500 * 3^m.

For the rabbit population on August 1st, we have to consider that February 1st to August 1st is 6 months. Thus, replacing m with 6 in our function: f(6) = 500 * 3^6, which equals 145800, so there would be 145800 rabbits in Lancaster on August 1st.

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Answer:

The Population Density is [tex]500\ People/mi^2[/tex].

Step-by-step explanation:

Given,

Total number of People = 900,000

Total Land Area = 1800 sq. mi.

Solution,

For calculating the population density, we have to divide the total number of people by the area of the land.

This can be framed in equation form'

[tex]Population\ Density=\frac{Total\ Number\ of\ People}{Land\ Area}[/tex]

Now putting the given values, we get;

[tex]Population\ Density=\frac{900,000}{1800\ mi^2}=500\ People/mi^2[/tex]

Hence The Population Density is [tex]500\ People/mi^2[/tex].

The number of hours the diesel train traveled  before the cattle train caught up is 15 hours

Solution:

Let t = travel time of the diesel train

Then  (t - 2) is the travel time of the cattle train (Left 2 hrs later)

Average speed of diesel train = 52 mph

Average speed of cattle train = 60 mph

To find: number of hours the diesel train traveled  before the cattle train caught up

Distance = speed x time

Distance traveled by diesel train:

Distance = 52 x t = 52t

Distance traveled by cattle train:

Distance = 60 x (t - 2) = 60t - 120

When the cattle train catches the diesel, they will have traveled the

same distance

Distance traveled by diesel train = Distance traveled by cattle train

52t = 60t - 120

60t - 52t = 120

8t = 120

t = 15

Thus the number of hours the diesel train traveled  before the cattle train caught up is 15 hours

Answer

Tina is 21, t-3

Step-by-step explanation:

Cole is 18, we don't really need that so just ignore it. Cole is 3 years younger than Tina. Therefore T which is Tina's age, minus 3 would equal Cole's age of 18.

Answer:

a

Step-by-step explanation:

Answer:

70

Step-by-step explanation:

We can rewrite the phrase for every '1 red light there are 9 blue lights' as there are 9 blue lights for every red, which may make it slightly clearer.

If there are 7 red lights, and 9 blues for every red, then there are 7*9 blue lights, or 63 blue lights. Now we can add the red and blue lights; 63+7=70, so there are 70 lights in all.

Answer:

70

Step-by-step explanation:

The expression [tex]\((-4x^2ya^3)^2\)[/tex] written as a monomial in standard form is [tex]\(16x^4y^2a^6\)[/tex].

To write the expression [tex]\((-4x^2ya^3)^2\)[/tex] as a monomial in standard form, you need to apply the exponent to each term inside the parentheses.

Remember that when raising a power to another power, you multiply the exponents.

[tex]\((-4x^2ya^3)^2\)[/tex] means you square each term inside:

[tex]\[ (-4)^2 \cdot (x^2)^2 \cdot (y)^2 \cdot (a^3)^2 \][/tex]

Now, perform the operations:

[tex]\[ 16 \cdot x^{2 \cdot 2} \cdot y^{2 \cdot 1} \cdot a^{3 \cdot 2} \][/tex]

Simplify the exponents:

[tex]\[ 16 \cdot x^4 \cdot y^2 \cdot a^6 \][/tex]

So, [tex]\((-4x^2ya^3)^2\)[/tex] written as a monomial in standard form is [tex]\(16x^4y^2a^6\)[/tex].

for a³, when squared, it becomes [tex]a^(3*2) = a^6.[/tex] Thus, the simplified expression is [tex]16x^4y^2a^6[/tex]

To simplify the expression[tex](-4x^2ya^3)^2[/tex], apply the power rule, squaring each term within the parentheses. First, square the coefficients: (-4)² = 16. Then, square the variables inside the parentheses. For x², when raised to the power of 2, it becomes[tex]x^(2*2) = x^4.[/tex] For y^1, when squared, it becomes [tex]y^(1*2) = y^2[/tex]. Finally, for [tex]a^3,[/tex] when squared, it becomes a^(3*2) = a^6. Thus, the simplified expression is [tex]16x^4y^2a^6[/tex]

To simplify the expression[tex](-4x^2ya^3)^2,[/tex]start by understanding the exponent rule when raising a power to another power. Applying this rule, square the entire expression inside the parentheses:[tex](-4x^2ya^3)^2.[/tex]Begin by squaring the coefficient[tex](-4)^2,[/tex] resulting in 16. Then, square each variable term. For[tex]x^2,[/tex] when squared, it becomes[tex]x^(2*2) = x^4.[/tex]The y term, which is effectively[tex]y^1,[/tex]squared yields[tex]y^(1*2) = y^2.[/tex]Lastly, a^3, when squared, becomes [tex]a^(3*2) = a^6.[/tex]Therefore, combining the simplified coefficients and variables, the final answer is[tex]16x^4y^2a^6.[/tex]

The point (3, 2) is the solution to given system of equations

Solution:

Given that system of equations are:

[tex]x^2 + y^2 = 13[/tex]    ------ eqn 1

[tex]2x - y = 4[/tex]    ------- eqn 2

From eqn 2,

y = 2x - 4

Substitute y = 2x - 4 in eqn 1

[tex]x^2 + (2x - 4)^2 = 13\\\\x^2 + 4x^2 + 16 - 16x = 13\\\\5x^2 -16x + 3 = 0[/tex]

Let us solve the above equation by quadratic formula,

[tex]\text {For a quadratic equation } a x^{2}+b x+c=0, \text { where } a \neq 0\\\\x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}[/tex]

Using the Quadratic Formula for [tex]5x^2 -16x + 3 = 0[/tex] where  a = 5, b = -16, and c = 3

[tex]\begin{aligned}&x=\frac{-(-16) \pm \sqrt{(-16)^{2}-4(5)(3)}}{2 \times 5}\\\\&x=\frac{16 \pm \sqrt{256-60}}{10}\\\\&x=\frac{16 \pm \sqrt{196}}{10}\end{aligned}[/tex]

The discriminant [tex]b^2 - 4ac>0[/tex] so, there are two real roots.

[tex]\begin{aligned}&x=\frac{16 \pm \sqrt{196}}{10}=\frac{16 \pm 14}{10}\\\\&x=\frac{16+14}{10} \text { or } \frac{16-14}{10}\\\\&x=\frac{30}{10} \text { or } x=\frac{2}{10}\\\\&x=3 \text { or } x=0.2\end{aligned}[/tex]

Substitute for x = 0.2 and x = 3 in 2x - y = 4

when x = 3

2(3) - y = 4

6 - y = 4

y = 2

when x = 0.2

2(0.2) - y = 4

0.4 - y = 4

y = 0.4 - 4

y = -3.6

Thus Option D is correct The point is (3, 2)

Answer:

48  chicken wings in 12 minutes.

Step-by-step explanation:

Answer:

4 chicken wings per minute

Step-by-step explanation:

12/3=4

Answer:

[tex](-4,32)[/tex]

Step-by-step explanation:

Given points are [tex](-1,-8)[/tex]

And given transformation is [tex]D_4[/tex] [tex]r_{x-axis}[/tex]

We will start from left to right.

First transformation is reflection about x-axis.

When we reflect about x-axis [tex](x,y)\ became\ (x,-y)[/tex]

So, [tex](-1,-8)=[-1,-(-8)]=(-1,8)[/tex]

Now next transformation is dilation with a factor 4.

If we do dilation with a factor [tex]'k'[/tex] to the point [tex](x,y)[/tex]

New co-ordinates after dilation became [tex](kx,ky)[/tex]

So, [tex](-1,8)\ became\ (-4,32)[/tex]

Answer:

Here's what I get.

Step-by-step explanation:

When you dilate an object by a scale factor, you multiply its coordinates by the same number.

If the scale factor is 4, the rule is (x, y) ⟶ (4x, 4y)

Your table then becomes

[tex]\begin{array}{lcl}\textbf{Vertices of}& \, & \textbf{Vertices of}\\\textbf{VWXY}& \, & \textbf{V'W'X'Y'}\\V(-1 ,1) & \quad & V'(-4, 4)\\W(-1, 2) & \quad & W'(-4, 8)\\X(2, -1) & \quad & X'(8 ,-4)\\Y(2, 2) & \quad & Y'(8, 8)\\\end{array}[/tex]

The diagram below shows figure VVXY as a green bow-tie and its image V'W'X'Y' in orange.

The scale factor is greater than one, so the dilation is an enlargement.

The cost per ticket is $0.75 if Clayton purchases the package.

Step-by-step explanation:

Given,

Cost of package = $18.75

Tickets in package = 25 tickets

To determine the cost of one ticket, we will divide the cost of package with number of tickets in package.

Cost per ticket = [tex]\frac{18.75}{25}[/tex]

Cost per ticket = $0.75

The cost per ticket is $0.75 if Clayton purchases the package.

Keywords: division, unit rate

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Answer:

The maximum value of P is 32

Step-by-step explanation:

we have following constraints

[tex]x\geq 0[/tex] ----> constraint A

[tex]y\geq 0[/tex] ----> constraint B  

[tex]2x+2y\geq 4[/tex] ----> constraint C

[tex]x+y\leq 8[/tex] ----> constraint D

Solve the feasible region by graphing

using a graphing tool

The vertices of the feasible region are

(0,2),(0,8),(8,0),(2,0)

see the attached figure

To find out the maximum value of the objective function P, substitute the value of x and the value of y of each vertex of the feasible region in the objective function P and then compare the results

we have

[tex]P=3x+4y[/tex]

so

For (0,2) ---> [tex]P=3(0)+4(2)=8[/tex]

For (0,8) ---> [tex]P=3(0)+4(8)=32[/tex]

For (8,0) ---> [tex]P=3(8)+4(0)=24[/tex]

For (2,0) ---> [tex]P=3(2)+4(0)=6[/tex]

therefore

The maximum value of P is 32

Answer:

Graph the inequalities given by the set of constraints. Find points where the boundary lines intersect to form a polygon. Substitute the coordinates of each point into the objective function and find the one that results in the largest value.

Step-by-step explanation:

the sample answer

Answer: t = 6

Step-by-step explanation: First we solve for -9(t-2) and that comes out to be -9t + 18. Then we solve for 4(t-15) which comes out to be 4t - 60. So the new equation we have is -9t + 18 = 4t - 60. In order to solve for t, we need to get t on one side of the problem by itself. To do this we will first add 9t to both sides and it comes out to be 18 = 13t - 60. t is still not by itself so now we add 60 to both sides and that gives us 78 = 13t. t is still not by itself so now we need to divide each side by 13 so that variable t is by itself. When we divide both sides by 13 we get 6 = t.

Answer:

6

Step-by-step explanation:

-9t + 18 = 4(t-15)

-9 +18 = 4t - 60

-9t = 4t-60-18

-9t = 4t - 78

-9 - 4 = -78

-13 = -78t =

t = -78/-13

t = 6

The correct function to express the area of the rectangular corral as a function of its width, given a fixed amount of fencing, is A(x) = x(70 - x).

The student is asking about expressing the area of a rectangular corral as a function of its width when a fixed amount of fencing is used. Given that Farmer Jones has 140 feet of fencing to construct the corral and the width is represented by x, the perimeter of the rectangle is twice the sum of its width and length.

Therefore, if x is the width, the length will be (140 - 2x)/2 or 70 - x. We can then express the area A as a function of x by multiplying the width by the length, leading to the function A(x) = x(70 - x).

The cost of package is $14.20

Step-by-step explanation:

Given,

Cost per pound of groundless beef = $5.99

Weight of package bought = 2.37 lbs

Cost of package = Cost per pound of beef * Weight of package

Cost of package = 5.99 * 2.37

Cost of package = $14.1963

Rounding off to nearest cent

Cost of package = $14.20

The cost of package is $14.20

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Answer: 30m - 7

Step-by-step explanation:

combine the two m's (:

Answer:

30m-7

Step-by-step explanation:

18m-7+12m=30m-7

Answer:

91.6

Step-by-step explanation:

Answer:91.6

Step-by-step explanation:

Answer:

1. See the graph attached

2. 13,400 thousands people will attend the new movie on the 8th day, if the pattern shown in the graph continues.

Explanation:

The table that shows the number of people who attend a new movie ofver teh course of a week is:

Day   Attendance (thousands)

1          12,200

3         12,600

5         13,000

7         13,400

8              x

The graph showing that pattern is attached.

It is a discrete graph because days can take only positive integer values.

You can see that the relation is linear and can calculate the change in the number of people every two days by subracting any two consecutive pairs of data:

Hence, every two days the increase in the number of people is 400 thousands.

For one day the increase is: 400 thousands / 2 days = 200 thousands/day.

Since you know the attendance for the day 7, you can calculate the attendance for the day 8 adding 200 thousands to 13,400:

Answer:

1- false; 2- true; 3- true; 4-false

Final answer:

The statements about drawing a card higher than 6 and certain to draw an even or odd number are true. It's false that drawing a multiple of 3 is unlikely and that drawing a card less than 8 is unlikely.

Explanation:

The question involves determining the likelihood of drawing a specific card from a shuffled deck of 10 cards numbered 1-10. Let's assess each of the four statements provided.

Card greater than 6: There are 4 cards greater than 6 (7, 8, 9, 10), so the probability is 4 out of 10 or 0.4. This means it is likely, making the statement true.

Multiple of 3: There are 3 multiples of 3 (3, 6, 9), so the probability is 3 out of 10 or 0.3. This is less than half, but not very small, so the statement is subjective; however, we would not generally describe a 30% chance as 'unlikely', hence the statement might be considered false.

Even or odd number: Every card is either even or odd, so it is certain that an even or odd card will be drawn, making the statement true.

Number less than 8: There are 7 cards less than 8 (1 through 7), so the probability is 7 out of 10 or 0.7. This means it's more likely to draw a card less than 8, which means the statement is false.

The domain of a relation is the set of all the x-terms of the relation.

Let's look at an example.

In the image provided I have attached a relation and we want to list the domain.

So, I will list all the x-terms. Notice however that I listed 7 once even though it appears twice in the relation. When listing the domain, you don't repeat the x-terms.

Answer:

The value of a = ±(√3)/(6)

Step-by-step explanation:

Points A and D belong to the x−axis.

All vertices on the parabola y = a (x+1)(x−5) = a (x² - 4x - 5)

So, points A and D represents the x-intercept of the parabola y

To find x-intercept, put y = 0

∴ a (x+1)(x−5) = 0  ⇒ divide both sides by a

∴ (x+1)(x−5) = 0 ⇒ x = -1 or x = 5

so, the x-coordinate of Point A is -1 or 5

And given that: m∠BAD=60°

So, the tangential line of the parabola at point A has a slope of 60°

∴ y' = tan 60° = √3

∴ y' = a (2x-4)

∴ a (2x-4) = √3

∴ a = (√3)/(2x-4)

Substitute with x = -1 ⇒ a = (√3)/(-6)

Substitute with x = 5 ⇒ a = (√3)/(6)

So, The value of a = ±(√3)/(6)

Also, see the attached figure, it represents the problem in case of a = (√3)/(-6)

Answer:

Step-by-step explanation:

[tex]a=+(3+9\sqrt{3})/52\\ a=-(3+9\sqrt{3})/52\\[/tex]

Answer:

The quantity of cherries bought is 1.67 lb

The quantity of grapes bought is 3.33 lb

Step-by-step explanation:

Given as :

The cost of cherries = $4 per lb

The cost of grapes = $2.50 per lb

Total money spend on fruits = $15

The quantity of both fruits to bought = 5  lb

Let The quantity of cherries bought = c  lb

Let The quantity of grapes bought = g    lb

Now, According to question

quantity of both fruits to bought = quantity of cherries bought + quantity of grapes bought

i.e c + g = 5 lb              ........1

And

Total money spend on fruits = cost of cherries × quantity of cherries bought + cost of grapes × quantity of grapes bought

Or , c   lb ×  $4 per lb +  g  lb  ×  $2.50 per lb = $15

Or, 4 c + 2.50 g = 15           .......2

Now, Solving equation 1 and 2

So, (4 c + 2.50 g) - 2.50 × (c + g) = 15 - 5 × 2.50

Or, (4 c - 2.50 c) + (2.50 g - 2.50 g) = 15 - 12.50

Or, 1.5 c + 0 = 2.5

∴ c = [tex]\dfrac{2.5}{1.5}[/tex]

I.e c = 1.67  lb

So, The quantity of cherries bought = c  = 1.67 lb

Putting the value of c in eq 1

Since , c + g = 5  lb

Or, g = 5  lb - c

Or, g = 5  lb - 1.67  lb

i.e  g = 3.33  lb

So, The quantity of grapes bought = g  = 3.33 lb

Hence, The quantity of cherries bought is 1.67 lb

and The quantity of grapes bought is 3.33 lb   Answer

Answer:

Use a spreadsheet.

Step-by-step explanation:

If you mean a household budget you could use a spreadsheet.

The columns would be  months and the rows would be  items of expenditure and income.

You use  formulae ( provided by the spreadsheet)  to  do additions, subtractions and so on.

The instantaneous velocity of the ball after 4 seconds is -0.4 m/s

Step-by-step explanation:

If f(x) is the function which represents the distance that a particle moves after x seconds, with velocity v and acceleration a, then

∵ A ball is thrown vertically from the top of a building

∵ The height of the ball after t seconds can be given by the

   function s(t)= -0.1(t -2)² + 10

- To find the function of the velocity differentiate s(t)

∵ s(t) = -0.1(t - 2)² + 10

- Solve the bracket

∵ (t - 2)² = t² - 4t + 4

∴ s(t) = -0.1(t² - 4t + 4) + 10

- Multiply the bracket by -0.1

∴ s(t) = -0.1t² + 0.4t - 0.4 + 10

- Add the like terms

∴ s(t) = -0.1 t² + 0.4t + 9.6

Now let us differentiate s(t)

∵ s'(t) = -0.1(2)t + 0.4(1)

∴ s'(t) = -0.2t + 0.4

- s'(t) is the function of velocity after time t seconds

∵ s'(t) = v(t)

∴ v(t) = -0.2t + 0.4

We need to find the instantaneous velocity of the ball after 4 seconds

Substitute t by 4

∴ v(4) = -0.2(4) + 0.4

∴ v(4) = -0.8 + 0.4

∴ v(4) = -0.4

∴ The v is -0.4 m/s ⇒ -ve means the velocity is downward

The instantaneous velocity of the ball after 4 seconds is -0.4 m/s

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Final answer:

To determine the estimated instantaneous velocity of the ball after 4 seconds, calculate the derivative of the height function s(t) and evaluate it at t = 4 seconds to find the velocity of -0.2 m/s.

Explanation:

The estimated instantaneous velocity of the ball after 4 seconds can be found by calculating the derivative of the height function s(t) with respect to time t, which gives the velocity function v(t) = -0.2(t - 2). Evaluating this at t = 4 seconds, we get a velocity of -0.2 m/s.

The function which is created by shifting the graph of function f up 5 units is  [tex]f(x)=4^x-1[/tex]

So, Option A is correct.

Step-by-step explanation:

We are given function: [tex]f(x)=4^x-6[/tex]

We need to determine the function which is created by shifting the graph of function f up 5 units.

The translation is vertical

If g(x)=f(x)+h then the graph is shifted up h units.

So, Applying translation:

[tex]f(x)=4^x-6[/tex]

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

Simplifying:

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

[tex]f(x)=4^x-1[/tex]

The function which is created by shifting the graph of function f up 5 units is  [tex]f(x)=4^x-1[/tex]

So, Option A is correct.

Keywords: Transformation

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Answer with Step-by-step explanation:

Jamel' spend on apples

= Jamel' spend on Green apples + on Red apples

= Cost per pound of apples *( Pounds of green apples + Pounds of red apples)

= 1.65*(2+3.2)

= 1.65*5.2

= $8.58

Answer: $8.58 is Jamel' total spend on apples.

Option A: 6x^2+75x+200 is the correct answer

Step-by-step explanation:

The profit is obtained by subtracting the cost from the revenue.

Given

[tex]R(x) = 6x^2+100x+300\\C(x) = 25x+100[/tex]

The Profit function will be obtained by subtracting the cost function from the revenue function

So,

[tex]P(x) = R(x) - C(x)\\= (6x^2+100x+300)-(25x+100)\\=6x^2+100x+300-25x-100\\=6x^2+100x-25x+300-100\\P(x)=6x^2+75x+200[/tex]

Hence,

Option A: 6x^2+75x+200 is the correct answer

Keywords: Functions, function operations

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Answer:

-1/2.

Step-by-step explanation:

( - 3 + 2) / 2

= -1 / 2.

The midpoint of a line segment with endpoints -3 and 2 is -0.5. This is calculated by adding the endpoints and dividing by 2.

The midpoint of a line segment is the average of its endpoints. We can find it by adding the two endpoints and dividing by 2. So, for the line segment with endpoints -3 and 2, we would calculate

(-3 + 2) / 2

The answer to this calculation is -0.5. Therefore, the midpoint of the line segment with endpoints -3 and 2 is -0.5.

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