On a certain hot summer's day, 344 people used the public swimming pool. The daily prices are $1.50 for children and $2.25 for adults. The receipts for admission total$715.50.How many children and how many adults swam at the public pool that day?

The area between z = -1.71 and z = 0.08 is 0.4883, in median .

What is median ?

The value of the middle observation found after the data are arranged in ascending order is referred to as the median of the data. In many cases, it is challenging to take all of the facts into account when representing something, and in these cases, median is helpful. The median is one of the simple to compute statistical summary metrics. The median is also known as the place average since it uses the data that is in the centre of a sequence to determine its value.

=> P(-1.71 < Z < 0.08 ) = P(Z < 0.08 ) - P(Z < -1.71)

= 0.5319 - [1 - P(Z < 1.71)]

= 0.5319- [1 - 0.9564]

= 0.4883

Hence, the area between z = -1.71 and z = 0.08 is 0.4883.

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Answer:

x=66 degrees so b

Step-by-step explanation:

132 and 2x are corresponding angles so 132=2x and x=66 degrees

Answer:

x = 66

Step-by-step explanation:

2x = 132

Divide by 2 on both sides

x = 66

Answer:

∴ x = 6.40 units (to 3sf)

Step-by-step explanation:

Following the Pythagoras' Theorem,

 x² = 4² + 5²

∴ x = 6.40 units (to 3sf)

The decimal equivalent of 13/10 is 1.3

Decimal:

Decimal is referred as the numbers, which consist of two parts namely, a whole number part and a fractional part separated by a decimal point. For example, 12.5 is a decimal number.

Give,

Here we have the fraction 13/10

And we need to find the decimal equivalent to the fraction.

In the given fraction, we have to use the long division method in order to get the decimal for,

The steps for the long division is as follows:

1) Set up the problem with long division bracket. Put dividend inside bracket and divisor on outside left.

2) 10 goes into 13 (1-times). Put a 1 in the next place of quotient and multiply 10 by 1 to get 10.

Subtract 10 from 13 to get remainder (13-10=3).

3) Now, bring down 0 from the dividend, to make 30. 10 goes into 30 (3-times). Put a 3 in the next place of quotient and multiply 10 by 3 to get 30.

Subtract 30 from 30 to get remainder (30-30=0).

Therefore, the decimal equivalent to 13/10 is 1.3

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The gradient of the function at the point (-1 -7) is 3a - 6.

What is the gradient of the function?

A function's gradient is also known as its slope, and the slope (of a tangent) at a given point on a function is known as its derivative.

We have,

g(x) = ax³ + 3x² + bx + c

Take the derivative of the function with respect to x, then substitute the x-coordinate of the point of interest in for the x values in the derivative to find the gradient.

to know the gradient of the function ax³ + 3x² + bx + c at the point (-1, -7) we would do the following:

Take the derivative with respect to x:

g'(x) = 3ax² + 6x + b

Substitute the x-coordinate (x = -1)in for x:

gradient = 3a(-1)² + 6(-1) + b = 3a - 6

So the gradient of the function at the point (-1 -7) is 3a - 6.

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Part (a)

To verify a point is on the line, we plug the coordinates into that equation. If we get a true result at the end, then we've confirmed the point is on that line.

For point A(-5,0) we have x = -5 and y = 0 pair up together. Let's plug these coordinates into x - 3y = -5 to get the following steps shown below.

x - 3y = -5

-5 - 3(0) = -5

-5 - 0 = -5

-5 = -5

We get a true statement at the end, which confirms x-3y = -5 is true when x = -5 and y = 0. Therefore, point A is definitely on the line x - 3y = -5

Repeat similar steps for B(4,3) to show that this point is also on x - 3y = -5

x - 3y = -5

4 - 3(3) = -5

4 - 9 = -5

-5 = -5

This confirms point B is also on this line.

We have confirmed line AB is x-3y = -5

I'll let you check to see if C(-3,9) and D(1,-3) are on the line 3x+y = 0. The steps will follow the same outline as shown above.

=========================================================

Part (b)

The standard form equation Ax+By = C has the slope m = -A/B, where B cannot be zero.

The equation x-3y = -5 has A = 1 and B = -3 to give us a slope of m = A/B = -1/(-3) = 1/3

Meanwhile the equation 3x+y = 0 gives the slope of A/B = -3/1 = -3

Notice that slopes 1/3 and -3 multiply to -1. This is one useful property of perpendicular lines. Their slopes always multiply to -1; this is assuming neither line is vertical, nor horizontal.

Put another way, the slopes are negative reciprocals of one another. We flip the fraction and flip the sign to go from 1/3 to -3, or vice versa.

I recommend using GeoGebra to plot out the points mentioned, and forming the lines AB and CD. This helps give a quick visual confirmation the lines are perpendicular.

----------

To summarize: We found the slopes of line AB and CD to be 1/3 and -3 respectively. The slopes multiply to -1 which is sufficient to conclude the lines are perpendicular.

=========================================================

Part (c)

We have this system of equations

x - 3y = -5

3x + y = 0

To represent lines AB and CD respectively.

Let's triple everything in line CD to go from 3x+y = 0 to 9x+3y = 0

Therefore, an equivalent system is this

x - 3y = -5

9x + 3y = 0

The first equation hasn't changed. After this point, we can see the y terms add to 0 since -3y+3y = 0y = 0. Therefore, the y terms cancel which lets us solve for x.

10x = -5

x = -5/10

x = -0.5

Then we use this to find y

x-3y = -5

-0.5-3y = -5

-3y = -5+0.5

-3y = -4.5

y = -4.5/(-3)

y = 1.5

The point of intersection is (-0.5, 1.5)

Notes:

-0.5 = -1/2

1.5 = 3/2

=========================================================

Part (d)

The x coordinates of points A and B are -5 and 4 in that order.

Add them up: -5+4 = -1

Divide the result in half: -1/2 = -0.5

This is the x coordinate of the midpoint of segment AB.

Repeat for the y coordinates

Add: 0+3 = 3

Divide in half: 3/2 = 1.5

The midpoint of segment AB is (-0.5, 1.5) which is exactly the result of part (c).

You should find that the midpoint of segment CD is also (-0.5, 1.5)

I'll let you do those steps.

=========================================================

Part (e)

Quadrilateral ACBD is a kite because of the perpendicular diagonals. This was confirmed in part (b).

The sides aren't all the same length (which you can confirm with the distance formula), which means we don't have a rhombus.

Answer:

a)  See below.

b)  See below.

[tex]\textsf{c)} \quad \left(-\dfrac{1}{2},\dfrac{3}{2}\right)[/tex]

[tex]\begin{aligned}\textsf{d)} \quad \textsf{Midpoint of $\overline{AB}$}&=\left(-\dfrac{1}{2},\dfrac{3}{2}\right)\\ \textsf{Midpoint of $\overline{CD}$}&=(-1,3)\end{aligned}[/tex]

e)  Kite

Step-by-step explanation:

Given endpoints:

Given equations of the lines:

To quickly check that the given equations are correct, input the x-value of a point on the line into the given equation of the line containing that point. If the y-value corresponds with the y-value of the point, the equation is correct.

Using point A to check equation AB:

[tex]\begin{aligned}x=-5 \implies -5-3y&=-5\\-3y&=0\\y&=0\end{aligned}[/tex]

As point A is (-5, 0), the equation is correct.

Using point C to check equation CD:

[tex]\begin{aligned}x=-3 \implies 3(-3)+y&=0\\-9+y&=0\\y&=9\end{aligned}[/tex]

As point C is (-3, 9), the equation is correct.

If two lines are perpendicular, their slopes are negative reciprocals.

The negative reciprocal of a number is -1 divided by the number.

Rearrange each equation to slope-intercept form, then compare slopes.

[tex]\begin{aligned}\textsf{Line}\;AB: \quad x-3y&=-5\\-3y&=-x-5\\3y&=x+5\\y&=\dfrac{1}{3}x+\dfrac{5}{3}\end{aligned}[/tex]

Therefore, the slope of line AB is ¹/₃.

[tex]\begin{aligned}\textsf{Line}\;CD: \quad 3x + y &= 0\\y&=-3x\end{aligned}[/tex]

Therefore, the slope of line CD is -3.

[tex]\textsf{As} \;\dfrac{-1}{-3}=\dfrac{1}{3}, \textsf{ the slopes are negative reciprocals}.[/tex]

Hence, the lines are perpendicular.

To find the point of intersection of AB and CD, solve the system of equations.

[tex]\begin{cases}x - 3y = -5\\3x + y = 0\end{cases}[/tex]

Rearrange the second equation to isolate y:

[tex]\implies y=-3x[/tex]

Substitute the found expression for y into the first equation and solve for x:

[tex]\implies x-3(-3x)=-5[/tex]

[tex]\implies x+9x=-5[/tex]

[tex]\implies 10x=-5[/tex]

[tex]\implies x=-\dfrac{5}{10}[/tex]

[tex]\implies x=-\dfrac{1}{2}[/tex]

Substitute the found value of x into the expression for y and solve for y:

[tex]\implies y=-3\left(-\dfrac{1}{2}\right)[/tex]

[tex]\implies y=\dfrac{3}{2}[/tex]

Therefore, the solution to the system of equations is:

[tex]\left(-\dfrac{1}{2},\dfrac{3}{2}\right)[/tex]

[tex]\boxed{\begin{minipage}{7.4 cm}\underline{Midpoint between two points}\\\\Midpoint $=\left(\dfrac{x_2+x_1}{2},\dfrac{y_2+y_1}{2}\right)$\\\\\\where $(x_1,y_1)$ and $(x_2,y_2)$ are the endpoints.\\\end{minipage}}[/tex]

[tex]\begin{aligned}\textsf{Midpoint of $\overline{AB}$}&=\left(\dfrac{x_B+x_A}{2},\dfrac{y_B+y_A}{2}\right)\\&=\left(\dfrac{4+(-5)}{2},\dfrac{3+0}{2}\right)\\&=\left(-\dfrac{1}{2},\dfrac{3}{2}\right)\end{aligned}[/tex]

[tex]\begin{aligned}\textsf{Midpoint of $\overline{CD}$}&=\left(\dfrac{x_D+x_C}{2},\dfrac{y_D+y_C}{2}\right)\\&=\left(\dfrac{1+(-3)}{2},\dfrac{-3+9}{2}\right)\\&=\left(-1,3\right)\end{aligned}[/tex]

The midpoint of line segment AB is the solution to the system of equations.

Quadrilateral ABCD is a kite:

The required suitable unique real numbers a, b, and C that represents the given graph are a = -1, b = 1, and c = 3.

A graph can be defined as a pictorial representation or a diagram that represents data or values.

The polynomial function of the graph which is given in the question:

f(x) = (x - a)²(x - b)(x - c)

This means that the polynomial's factors are as follows:

(x - a)², (x - b) and (x - c).

The polynomial solution will now be the values of x when the components equal zero.

This shows that a, b, and c are polynomial function solutions. This indicates that the x-intercepts are a, b, and C.

The x-intercepts on the graph are;

-1, 1, and 3.

Hence;

a = -1

b = 1

c = 3

Thus, the required suitable unique real numbers a, b, and C that represents the given graph are a = -1, b = 1, and c = 3.

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Answer:

a = 48°

Step-by-step explanation:

Forming the equation,

→ 42° + a = 90°

Now the value of a will be,

→ 42° + a = 90°

→ a = 90° - 42°

→ [ a = 48° ]

Hence, the value of a is 48°.

Answer:

Step-by-step explanation:

The probability that the region will be hit by a hurricane in 3 consecutive years is; 0.000343

The binomial probability is defined as the probability of exactly x successes on n repeated trials, with p probability. The general formula for binomial probability distribution is;

P(X = x) = ⁿCₓ * pˣ * (1 - p)^(n - x)

where;

P(X = x) is binomial probability

x is number of times for a specific outcome within n trials

ⁿCₓ is number of combinations

p is probability of success on a single trial

n is number of trials

Now, we are given that the probability of success for a particular region in Mexico being hit by hurricane is 0.07. Thus;

P(three consecutive years hurricane) = 0.07 * 0.07 * 0.07 = 0.000343

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The required implicit differentiation is given as dy/dx = -4x/3.

Given that,
To determine the implicit differentiation of the given function 4 = 2x² + 3y.

The method of determining the derivative of an implicit function by differentiating each term separately, expressing the derivative of the dependent variable as a symbol, and solving the resulting expression for the symbol

Here,
4 = 2x² + 3y
Differentiate with respect to x on both sides,
d4/dx = d[2x²]/dx + 3dy/dx
0 = 4x + 3dy/dx
dy/dx = -4x/3

Thus, the required implicit differentiation is given as dy/dx = -4x/3.

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Step-by-step explanation:

every marker marks obviously a unit of 6.

so, the values go up and down from 0 in multiples of 6.

for the blue dot we count down from 0 :

5 markers

each marker is worth 6. and we are going down into the negative realm.

so, the blue dot is

- 5×6 = -30

Answer:

g = -5

z = -4

z = 2.1

t = -36

Step-by-step explanation:

-4 = g +1  Subtract 1 from both sides

-5 = g

z - 4 = -8  Add 4 to both sides

z = -4

10.5 = 5z  Divide both sides by 5

2.1 = z

-9 = t/4  Multiply both side sby 4

-36 = t

Answer: (0,0),  (-5,0),  (2,0)

Step-by-step explanation:

c(t)=4t⁴+12t³-40t²

1) t=0

c(t)= 4(0⁴)+12(0³)-40(0²)

c(t)=4(0)+12(0)-40(0²)

c(t)=0+0+0

c(t)=0

Thus (0,0)

2) c(t)=0

4t⁴+12t³-40t²=0

Divide both parts of the equation by 4:

t⁴+3t³-10t²=0

t²(t²+3t-10)=0

t²=0

t=0

Thus (0,0)

t²+3t-10=0

t²+(5t-2t)-10=0

(t²+5t)-(2t+10)=0

t(t+5)-2(t+5)=0

(t+5)(t-2)=0

t+5=0

t=-5

Thus (-5,0)

t-2=0

t=2

Thus (2,0)

By the use of the distributive property, we have the result obtained as 6xy - 6xz.

The term distributive property has to do with the process by which we can remove the brackets that may sometimes occur in a mathematical expression. It is the way by which we can be able to rewrite the mathematical expression without adding the brackets thereby making mathematical operations much easier.

We have the expression; 6x (y - z)

Multiplying through by 6x we have;

6xy - 6xz

Using the distributive property, we obtain; 6xy - 6xz

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⇒The quadratic equation has already been factorized in such a way that the factors are equal to 0 ∴ they won`t be an need to multiply out, since you would still have to come back to the factorizing part.

⇒Let us solve for x now.

[tex]\sqrt{x} +2 =0\\\sqrt{x} =-2\\No\\ solution[/tex]

⇒There is no way that a square root of a number can result in a negative number... that is why the answer was a no solution.

[tex]\sqrt{x} -2=0\\\sqrt{x} =2\\(\sqrt{x} )^{2} =(2)^{2} \\x=4[/tex]

⇒The solution to the quadratic equation is only and only 4.

Answer:

12x + 9

Step-by-step explanation:

Combine the line terms

3x + 5x + 4x = 12x

8+ 2 -1 = 9

12x + 9

Answer:

----------------------------------

It is assumed boats leave from the same point.

Boats travel after 2 hours:

The angle formed between the two directions:

The line between the endpoints is opposite to this angle.

Use the law of cosines and find the distance:

Answer:

77.3 miles apart (nearest tenth)

Step-by-step explanation:

Given information:

Draw a diagram using the given information (see attached).

To find how far apart the boats are, model as a triangle and find the length of the missing side by using the cosine rule.

[tex]\boxed{\begin{minipage}{6 cm}\underline{Cosine Rule} \\\\$c^2=a^2+b^2-2ab \cos C$\\\\where:\\ \phantom{ww}$\bullet$ $a, b$ and $c$ are the sides.\\ \phantom{ww}$\bullet$ $C$ is the angle opposite side $c$. \\\end{minipage}}[/tex]

From inspection of the drawn diagram:

Substitute the values into the cosine rule and solve for c:

[tex]\implies c^2=40^2+56^2-2(40)(56) \cos 106^{\circ}[/tex]

[tex]\implies c^2=1600+3136-4480 \cos 106^{\circ}[/tex]

[tex]\implies c^2=4736-4480 \cos 106^{\circ}[/tex]

[tex]\implies c=\sqrt{4736-4480 \cos 106^{\circ}}[/tex]

[tex]\implies c=77.27131004[/tex]

Therefore, the boats are 77.3 miles apart (nearest tenth) after 2 hours.

The simplest radical numbers according algebra properties are listed below:

In this question we find two cases of radical numbers that must be simplified in accordance with algebra properties, especially those related to square roots, and factor decomposition. Now we proceed to simplify each expression:

Case A: √175

Step 1 - Use factor decomposition

√175 = √(5² × 7)

Step 2 - Apply algebra properties to simplify the expression

√175 = √5² × √7

√175 = 5 × √7

√175 = 5√7

Case B: √(5 / 12)

Step 1 - Use factor decomposition

√(5 / 12) = √[5 / (3 × 2²)]

Step 2 - Apply algebra properties to simplify the expression

√(5 / 12) = √5 / √(3 × 2²)

√(5 / 12) = √5 / [√3 × √2²]

√(5 / 12) = √5 / (2√3)

Step 3 - Rationalize the resulting expression

√(5 / 12) = (√5 × √3) / 6

√(5 / 12) = √15 / 6

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The fraction that Martha has left to make matching throw pillows is 3 1/6 yards.

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. On the other hand, a fraction appears in the numerator or the denominator of a complex fraction.

From the information given, Martha had 12 2/3 yards of drapery material. She used 3/4 of it to make the drapes for her family room.

Since 3/4 has been used, the fraction left will be gotten by subtracting the fraction that had already been used from the total fraction that she has. In this case, it should be noted that this will be illustrated as:

Total fraction that Martha has - The fraction that has been used

= 1 - 3/4

= 1/4

Therefore, the fraction will be the fraction that is left multiplied by the yards of drapery material that Martha had at the beginning. This will be:

= Fraction unused × Total yards of material

= 1/4 × 12 2/3

= 1/4 × 38/3

= 19/6

= 3 1/6

The fraction is 3 1/6 yards.

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From the given details of Manuel wooden crates , where x represents the number of crates and y represents the cost then equation represents the given situation is given by y = 10x + 5.

As given in the question,

Given details of Manuel wooden crates :

Number of crates  : 2       4       6         8

Cost in dollars       : 25     45      65      85

From the given table where x represents the number of crates and y represents the cost :

(x₁ , y₁) = ( 2, 25)

(x₂ , y₂) = ( 4, 45)

For the given situation equation is given by :

( y - y₁) / (x- x₁) = (y₂ - y₁) / (x₂ - x₁)

⇒( y - 25) / (x -2) = (45 -25)/ (4 -2)

⇒(y -25) / (x -2) = 10

⇒y - 25 = 10(x-2)

⇒ y = 10x +5

Therefore, from the given details of Manuel wooden crates , where x represents the number of crates and y represents the cost then equation represents the given situation is given by y = 10x + 5.

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Over 48.6 hours are worked by 2.28 percent of faculty members each week.

Given that,

In a college, a full-time faculty member typically puts in 43 hours each week.

We have to find how many faculty members work more than 48.6 hours each week, assuming the standard deviation is 2.8 hours.

The Z-score determines the measure's standard deviations away from the mean. After calculating the Z-score, we look at the p-value associated with it in the z-score table. This p-value is the probability that the measure's value is smaller than X, or the percentile of X. By subtracting 1 from the p-value, one can determine the likelihood that the value of the measure is greater than X.

In a college, a full-time faculty member typically puts in 43 hours each week. 2.8 hour standard deviation.

This means μ=43 and σ=2.8

What proportion of academic staff employees put in more than 58.6 hours each week.

When X = 48.6, the proportion is calculated by deducting 1 from the p-value of Z.

So,

Z=X-μ/σ

Z=48.6-43/2.8

Z=5.6/2.8

Z=2

Z=2 has a p-value of 0.9772

1 - 0.9772 = 0.0228

0.0228*100% = 2.28%

Therefore, Over 48.6 hours are worked by 2.28 percent of faculty members each week.

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The solution of the equation 3x(x - 3) = 12 will be 4 and negative 1.

The allocation of weights to the important variables that produce the calculation's optimum is referred to as a direct consequence.

The definition of simplicity is making something simpler to achieve or grasp while also making it a little less difficult.

The equation is given below.

3x(x - 3) = 12

Simplify the equation, then we have

3x(x - 3) = 12

x(x - 3) = 4

x² - 3x - 4 = 0

Factorize the equation, then we have

x² - 3x - 4 = 0

x² - 4x + x - 4 = 0

x(x - 4) + 1(x - 4) = 0

(x - 4)(x + 1) = 0

x = 4, -1

The solution of the equation 3x(x - 3) = 12 will be 4 and negative 1.

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