Write an equation of the parabola in vertex form.
Amusement Park Ride
Height (feet)
YA(0, 180)
(1, 164)
160
80
0 2 4
Time (seconds)
Equation:

Answer:

168

Step-by-step explanation:

Let the area of the square be [tex]x[/tex].

[tex]\frac{4}{7}x-\frac{5}{6}\left(\frac{3}{7}x \right)=36 \\ \\ \frac{4}{7}x-\frac{5}{14}x=36 \\ \\ \frac{8}{14}x-\frac{5}{14}x=36 \\ \\ \frac{3}{14}x=36 \\ \\ \frac{1}{14}x=12 \\ \\ x=168[/tex]

0.065=x/5800

x=$377; that is the dollar value of the decrease

The account is now worth $5423.

A.The length of each path is 130 meters.

Given:

The park resembles an isosceles trapezoid.

Figure shows a trapezium . AC and BD are diagonals. The length of BD is 14x plus 4, and AC is 11x plus 31.

we know that:

Diagonals are equal in isosceles trapezoid.

14x + 4 = 11x + 31

14x - 11x = 31 - 4

3x = 27

divide by 3 on both sides

3x/3 = 27/3

x = 9

Diagonal length = 14x + 4.

= 14*9 + 4

= 126 + 4

= 130 meters.

Therefore The length of each path is 130 meters.

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The correct way to replace equation 1 into equation 2 using the substitution method is 2x+(-5x+10) = 7.

According to the question,

We have the following two equations:

Equation 1: y = -5x+10

Equation 2: 2x+y = 7

Now, we will substitute the value of y from equation 1 in equation 2.

So, we have the following expression:

2x+(-5x+10) = 7

(More to know: we can also solve this equation to find the value of x and then the obtained value of x can be put in equation 1 to find the value of y.)

Hence, the correct option is B.

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Answer:

C______________4_⁴________

Answer:

Step-by-step explanation:

Answer:

the change is in %age of decrease of 10%

Step-by-step explanation:

original value = 20

new value = 18

decrease = 2

%age decrease = decrease/original value * 100

                         =  2/20 * 100

                         =  10%

The compound interest is obtained as $91.81, rounded off to the nearest hundredth.

In order to compute compound interest, multiply the principle of the original loan by the annual interest rate multiplied by the number of compound periods minus one. You will then be left with the principal amount of the loan plus compound interest. The entire compound interest is then obtained by deducting the initial principal.

The equation that results will look like this:

Compound Interest = [tex]P(\left(1+r\right)^{ t}-1)[/tex]

Amount, rate of interest and time is given as $1500, 2% and 3 years respectively.

First, change r as a percent to r as a decimal.

r = R/100

r = 2/100

r = 0.02 rate per year,

Then solve the equation for compound interest.

[tex]\text{Compound Interest}=1500(\left(1+0.02\right)^{3}-1)\\=1500(1.061208-1)\\=15000\cdot 0.061208\\\\=91.81[/tex]

So, the compound interest is obtained as $91.81, rounded off to the nearest hundredth.

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Answer: its 1591.81

Step-by-step explanation:

I took the test and tried with the answer on top and got it wrong.

Step-by-step explanation:

the angle TSW is the total angle and is 73°.

from that total angle, we have one part : angle 1 = 36°.

as angle TSW = angle 1 + angle 2

we have

73 = 36 + angle 2

angle 2 = 73 - 36 = 37°

Answer:

a = 6 , b = - 44

Step-by-step explanation:

x² + 12x - 8

using the method of completing the square

add/subtract ( half the coefficient of the x- term)² from x² + 12x

x² + 2(6)x + 36 - 36 - 8

= (x + 6)² - 44 ← in the form (x + a)² + b

with a = 6 and b = - 44

The values of y and x are -7 and 2 respectively, completing the ordered pairs: (6,-7) and (2,1).

Given the equation in the question;

To find the y-value that completes the ordered pair ( 6,y ), replace all occurrence of x with 6 in the equation and solve for y.

y = −2x + 5

y = −2( 6 ) + 5

Multiply -2 and 6

y = -12 + 5

y = -7

Hence, the output value is -7, this forms an ordered pair of (6,-7).

To find the x-value that completes the ordered pair ( x,1 ), replace all occurrence of y with 1 in the equation and solve for x.

y = −2x + 5

1 = −2x + 5

Add 2x to both sides

1 + 2x = −2x + 2x + 5

1 + 2x = 5

2x = 5 - 1

2x = 4

x = 4/2

x = 2

Hence, the input value is 2, this forms an ordered pair of (2,1).

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Equation which is equivalent to the formula solved for r in term of A is   [tex]Or = \sqrt{\frac{A}{r}-2 }[/tex].

What is circle?

A circle is a special kind of ellipse in which the eccentricity is zero and the two foci are coincident.

A circle is also termed as the locus of the points drawn at an equidistant from the centre.

What is radius?
Radius of a circle is the distance from the center of the circle to any point on it's circumference. It is usually denoted by 'R' or 'r'.

What is circumference?

Circumference of the circle or perimeter of the circle is the measurement of the boundary of the circle. Whereas the area of circle defines the region occupied by it.

Given that;

[tex]A = \pi{({r+2})^{2}}[/tex]

[tex](r+2)^{2} =\frac{A}{\pi}[/tex]  (equivalent transformation)

[tex]r + 2 = \sqrt{\frac{A}{\pi} }[/tex]

[tex]r = \sqrt{\frac{A}{\pi} -2}[/tex]

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Answer:

c. 7/12

Step-by-step explanation:

There are 36 possible outcomes when you roll a die twice.

11, 12, 13, 14, 15, 16

21, 22, 23, 24, 25, 26

31, 32, 33, 34, 35, 36

41, 42, 43, 44, 45, 46

51, 52, 53, 54, 55, 56

61, 62, 63, 64, 65, 66

The desired outcomes are the ones whose first number is greater than or equal to the second number. They are shown in bold and underline above.

There are 21 desired outcomes.

p(first number ≥ second number) = 21/36 = 7/12

Answer: c. 7/12

The three inlet pipes, each working alone can fill the pool in 8 seconds, 12 seconds, and 16 seconds, respectively. it will take them 3.69 seconds  fill the pool

The problem requires to determine how long it takes to fill the pool when the three pipes are working simultaneously

let the time to fill the the pool when the three pipes are working simultaneously be y, such that

8 seconds pipe fills the pool in y/8 seconds

12 seconds pipe fills the pool in y/12 seconds

16 seconds pipe fills the pool in y/16 seconds

Adding the times together

y/8 + y/12 + y/16 = 1

13y/48 = 1

y = 48/13

y = 3.69 seconds

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The sin(x) = 0.88 and sin(y) = 32/68=0.47 in the Trigonometric Functions

What is Trigonometric Functions?

Trigonometry uses six fundamental trigonometric operations. Trigonometric ratios describe these operations. The sine function, cosine function, secant function, co-secant function, tangent function, and co-tangent function are the six fundamental trigonometric functions. The ratio of sides of a right-angled triangle is the basis for trigonometric functions and identities. Using trigonometric formulas, the sine, cosine, tangent, secant, and cotangent values are calculated for the perpendicular side, hypotenuse, and base of a right triangle.

The given triangle is a right-angled triangle

As Z² = X²+Y²

So sin (x) = 60/68=0.88

and sin(y) = 32/68=0.47

Hence the sin(x) = 0.88 and sin(y) = 32/68=0.47.

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Answer:

i think it is 15/20

Step-by-step explanation:

because 6/8,9/12,12/16

The 160 feet range and 70 feet height of the path of Gunter Money which can be described with a quadratic equation, gives;

(a) The path is; y = -1.09375×10²·x² + 1.75·x

(b) The vertex form of the equation is; y = -1.09375·(x - 80)² + 70

(c) The angle at the start is approximately 60.3°

A quadratic equation is a single variable polynomial equation of the second degree.

The points in the path which is a path of a projectile and has a shape of a parabola are;

(0, 0), (80, 70), and (160, 0)

The equations are;

y₁ = a·x₁² + b·x₁ + c

y₂ = a·x₂² + b·x₂ + c

y₃ = a·x₃² + b·x₃ + c

Plugging in the coordinates of the three points in the path gives;

The matrix becomes;

[tex]A = \begin{bmatrix}0 & 0 & 1\\80^2 &80 & 1\\160^2 &160 &1 \\\end{bmatrix}[/tex]

[tex]X = \begin{bmatrix}a \\b \\c\end{bmatrix}[/tex]

[tex]Y = \begin{bmatrix}0 \\70 \\0\end{bmatrix}[/tex]

We have;

A·X = Y, which gives;

[tex]\begin{bmatrix}0 & 0 & 1\\80^2 &80 & 1\\160^2 &160 &1 \\\end{bmatrix}\cdot \begin{bmatrix}a \\b \\c\end{bmatrix} = \begin{bmatrix}0 \\70 \\0\end{bmatrix}[/tex]

[tex]X = \dfrac{Y}{A} = A^{-1}\cdot Y[/tex]

Where;

A⁻¹ = The inverse of the matrix A

[tex]\begin{bmatrix}a \\b \\c\end{bmatrix} = \frac{ \begin{bmatrix}0 \\70 \\0\end{bmatrix}}{\begin{bmatrix}0 & 0 & 1\\80^2 &80 & 1\\160^2 &160 &1 \\\end{bmatrix}} = \begin{bmatrix}\dfrac{1}{12800} & -\dfrac{1}{6400} & \dfrac{1}{12800} \\\\-\dfrac{3}{160} &\dfrac{1}{40} & -\dfrac{1}{160} \\\\1 &0 &0 \\\end{bmatrix}\bullet \begin{bmatrix}0\\ \\70\\ \\0\end{bmatrix}=\begin{bmatrix}-\dfrac{7}{640} \\ \\\dfrac{7}{4} \\ \\0\end{bmatrix}[/tex]

[tex]\begin{bmatrix}a \\b \\c\end{bmatrix} = \begin{bmatrix}-\dfrac{7}{640} \\ \\\dfrac{7}{4} \\ \\0\end{bmatrix}[/tex]

[tex]a = -\dfrac{7}{640}[/tex]

[tex]b = \dfrac{7}{4}[/tex]

c = 0

The equation for Gunter Money's path is therefore;

[tex]y = -\dfrac{7}{640} \cdot x^2 + 1\dfrac{3}{4} \cdot x + 0 = -1.09375\times 10^2 \cdot x^2 + 1.75\cdot x[/tex]

(b) The vertex of the path is (h, k) = (80, 70)

The vertex form of a quadratic equation is; y = a·(x - h)² + k

The equation of Gunter Money's path in vertex form is therefore;

[tex]y = -\dfrac{7}{640} \cdot \left(x - 80)^2 + 70 = -1.09375\times 10^2\cdot (x - 80)^2+70[/tex]

(c) The angle of the parabolic path of Gunter Money's motion at a point is given by the value of the arctangent of the differentiation of the function for the motion at the point as follows;

[tex]y' = -\dfrac{7\times 2}{640} \cdot x + 1\dfrac{3}{4}[/tex]

At the point Gunter Money takes off,  (0, 0), we have;

[tex]tan(\theta) = y' = -\dfrac{7\times 2}{640} \times 0 + 1\dfrac{3}{4} = 1\dfrac{3}{4} = 1.75[/tex]

[tex]\theta = arctan(1.75) \approx 60.3^{\circ}[/tex]

The angle is approximately 60.3°

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a) The confidence interval is given as follows: (71.43, 77.17).

b) The manufacturer's claim is not correct, as 78 is not a part of the interval.

c) A sample of 55 employees is needed to achieve the desired margin of error.

The bounds of the t-distribution confidence interval, used when only the standard deviation for the sample is known, are given by the equation presented as follows:

[tex]\overline{x} \pm t\frac{s}{\sqrt{n}}[/tex]

The variables of the equation are presented as follows:

The critical value, considering the 95% confidence level and 12 - 1 = 11 df, is of:

t = 2.201.

The remaining parameters, namely sample mean, sample standard deviation and sample size are given as follows:

[tex]\overline{x} = 74.3, s = 4.52, n = 12[/tex]

(obtained using a calculator from the sample given in this problem).

The bounds of the interval are given as follows:

The interval does not contain the number 78, hence it is not a good estimate for the mean amount.

For item c, the z-distribution is used, as the population standard deviation is used, hence the minimum sample size is obtained as follows:

[tex]M = z\frac{\sigma}{\sqrt{n}}[/tex]

[tex]1.04 = 1.96\frac{3.91}{\sqrt{n}}[/tex]

[tex]1.04\sqrt{n} = 1.96 \times 3.91[/tex]

[tex]\sqrt{n} = \frac{1.96 \times 3.91}{1.04}[/tex]

[tex](\sqrt{n})^2 = \left(\frac{1.96 \times 3.91}{1.04}\right)^2[/tex]

n = 55. (rounded up, as a sample size of 54 would mean that the margin of error would be slightly above 1.04).

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The observation of the graphing speed versus time is given below.

The rate at which the speed is changing is known as acceleration.

From the graph, the line segment OA and DE have linear acceleration which increases the velocity.

Similarly, from the graph, the line segment BC and EF have linear deacceleration which decreases the velocity.

From the graph, line segment AB has constant acceleration which increases the velocity.

And from the graph, the line segment CD has zero acceleration which makes the velocity to be constant.

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Answer: I believe both the x and y- intercepts are “(0,0)”

Step-by-step explanation: there is only one intercept and it is at the origin (0,0) making both the x and y - intercepts (0,0) :)

The length of the line segment will be 10 units.

What is Distance?

The distance between two points (x₁ , y₁) and (x₂, y₂) is defined as;

D = √( y₂ - y₁)² + (x₂ - x₁)²

Given that;

Two endpoints of the line segment are (6, -5) and (0, 3).

Now,

The length of the line segment is distance between two endpoints (6, -5) and (0, 3).

So, The distance between two endpoints (6, -5) and (0, 3) is calculated as;

D = √(0 - 6)² + (3 - (-5))²

D = √36 + 64

D = √100

D = 10 units

Thus, The length of the line segment will be 10 units.

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The ordered pair could be removed to make this relation a function is (4,-2).

What is an ordered pair?

An ordered pair is made up of the ordinate and the abscissa of the x coordinate, with two values given in parenthesis in a certain sequence.

The numeric values in an ordered pair can be integers or fractions.

Here,

A function is when each input has one output. So, for every x there is one y. The x values can not repeat.

But in the given value x is repeating so the ordered pair (4, -2) can be removed.

Hence, the ordered pair could be removed to make this relation a function is (4,-2).

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A) The value of x in log₄(x² - 12x + 48) = 2, is x = 4 OR x = 8

B) The value of x in 27^(4x-6) =(1/9)^(2x-7), is x = 2

From the question, we are to solve the given equations

The given equation is log₄(x² - 12x + 48) = 2

From one of the laws of logarithm, we have that

logₐ x = n ⇒ x = aⁿ

Thus,

log₄(x² - 12x + 48) = 2 becomes

(x² - 12x + 48) = 4²

x² - 12x + 48 = 16

x² - 12x + 48 - 16 = 0

x² - 12x + 32 = 0

Solve the quadratic equation

x² - 12x + 32 = 0

x² - 8x - 4x + 32 = 0

x(x - 8) -4(x - 8) = 0

(x - 4)(x - 8) = 0

x - 4 = 0 OR x - 8 = 0

x = 4 OR x = 8

B) 27^(4x-6) =(1/9)^(2x-7)

Solve for x

27^(4x-6) =(1/9)^(2x-7)

Express the bases in index form

3^3(4x-6) =(3)^-2(2x-7)

Equate the powers

3(4x - 6) = -2(2x - 7)

12x - 18 = -4x + 14

12x + 4x = 14 + 18

16x = 32

Divide both sides by 16

16x/16 = 32/16

x = 2

Hence, the value of x is 2

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The value of the investment after 3.25 years is $434.93.

The amount a lender charges a borrower, expressed as a percentage of the principle, is known as the interest rate.

Given that, the initial investment is P₀ = $400, the interest rate is n = 2.6%, and the time is x = 3.25 years.

The formula for the future value in continuous compounding is:

P(x) = P₀eⁿˣ

Substitute P₀ = 400, n = 2.6%, and x = 3.25 into the equation:

[tex]P(3.25) = (400)e^{2.6\%(3.25)}\\[/tex]

= 434.93

Hence, the value of the investment after 3.25 years is $434.93.

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The perpendicular distance from R to PQ is ​346 feet.

What are the trigonometric functions?

The right-angled triangle's angle and the ratio of its two side lengths are related by the trigonometric functions, which are actual functions. They are extensively employed in all fields of geometry-related study, including geodesy, solid mechanics, celestial mechanics, and many others.

We have,

consider the sides of leaning tower are,

opposite side of the angle 56° (x) = 180.5 feet.

Adjacent side of the angle 56° (y) = 125 feet.

We know,

[tex]tan\theta =\frac{ opposite \ side}{ adjacent \ side} = \frac{x}{y}[/tex]

[tex]\theta = \ tan^{-1} \left(\frac{ opposite \ side}{ adjacent \ side} \right)[/tex]

The angle of elevation is  [tex]\theta = \ tan^{-1} \left(\frac{180.5 }{125} \right)[/tex]

θ = 56°

The perpendicular distance from R to PQ is RQ ​

[tex]sin\theta = \frac{opposite side}{hypotanous} = \frac{180.5}{y}[/tex]

sin(56) = 180.5/PQ

RQ = 180.5/sin(56)

RQ = - 346.08

RQ = 346 (distance never be negative),

Hence, the perpendicular distance from R to PQ is ​346 feet.

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The perimeter of the shape is 38 total unit.

The perimeter of a shape is a measure of the distance round the boundary or edge of the shape.

The given shape comprises of squares of 1 length each, hence to calculate for the perimeter of the shape, we add up each boundary of 1 unit length which will amount to 39 in total.

Therefore by careful observation, the perimeter of the plane shape is calculated to give us 39 total perimeter length.

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The system of equations has no solutions, meaning that the two lines never intersect, as shown by the image at the end of the answer.

A system of equations is a set of equations in the context of a problem, and these equations are solved to find the numeric value of each variable of the problem.

When the system involves linear equations, such as is the case in this problem, the solution is given by the intersection point between these two lines.

In this problem, the equations are given as follows:

From the second equation, it is found that:

x = 8 + 2y.

Replacing on the first equation, the solution for y can be obtained as follows:

2(8 + 2y) - 4y = 4

16 + 4y - 4y = 4

0y = -12

Which is a contradiction, hence the system has no solutions and the two lines do not intersect.

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Answer:

B. After the booking fee the party will cost $20 per person

Answer:

none of them

Step-by-step explanation:

Answer:

x+5

Step-by-step explanation:

there are no like terms