a) Write - 2x² - 12x - 8 in the form a (x + b)² + c, where a, b
and c are numbers.
What are the values of a, b and c?
b) Hence, write down the coordinates of the turning point of the
curve y = -2x² - 12x - 8.

According to the given information, the equation of the ellipse is [tex](x+8)^2/256 + (y+1)^2/784 = 1.[/tex]

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that deals with the study of geometric shapes using algebraic principles. It involves the use of coordinates to represent points, lines, curves, and other geometric figures on a plane or in space.

we need to know the coordinates of its foci, co-vertices, and the center. We can start by finding the center of the ellipse, which is the midpoint of the line segment joining the foci:

Center = ( (-8 + (-8))/2 , (14 + (-16))/2 ) = (-8,-1)

Next, we can find the distance between the foci, which is given by:

[tex]distance between foci = 2c = sqrt[(14 - (-16))^2 + (-8 - (-8))^2] = 30[/tex]

where c is the distance from the center to either focus.

We also know that the distance between the co-vertices is given by:

distance between co-vertices = 2a = |-16 - 0| = 16

where a is the distance from the center to either co-vertex.

Finally, we can use the standard form equation for an ellipse centered at the origin:

[tex](x^2/a^2) + (y^2/b^2) = 1[/tex]

where b is the distance from the center to either vertex.

To find b, we can use the Pythagorean theorem:

[tex]b^2 = c^2 - a^2 \\b^2 = 30^2 - 16^2\\b^2 = 784\\b = 28[/tex]

Now we have all the information we need to write the equation of the ellipse:

[tex](x+8)^2/16^2 + (y+1)^2/28^2 = 1[/tex]

Therefore, according to the given information, the equation of the ellipse is [tex](x+8)^2/256 + (y+1)^2/784 = 1.[/tex]

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Answer:

(a) y = 0.80x + 50

(b) Plugging x = 5 into the equation from part (a), we get y = 0.80(5) + 50 = 54, so the ordered pair associated with x = 5 is (5, 54). This means that if the car is driven for 5 miles, the total charge to the renter is $54.

(c) Let y be the total charge and solve for x:

y = 0.80x + 50

187.60 = 0.80x + 50

137.60 = 0.80x

x ≈ 172

Therefore, the car must have been driven approximately 172 miles

Answer:

(a) y = 0.8x + 50

(b)  D. The ordered pair associated with the equation x = 5 is (5, 54) and it means that the charge for driving the car for 5 miles is $54.

(c) If the renter paid $187.60, the car must have been driven for 172 miles.

Step-by-step explanation:

In the given problem, we are told that the rental car costs a flat fee of $50 plus an additional charge of $0.80 per mile driven.

Let x be the number of miles driven.

Let y be the total charge to the renter (in dollars).

We know that the total charge y will depend on the number of miles driven x, and we can express this relationship using a linear equation in the form y = mx + b, where m is the slope of the line and b is the y-intercept (the value of y when x = 0).

In this case, we know that the cost per mile is $0.80 so the slope of the line is m = 0.8, and the flat fee is $50 so the y-intercept is b = 50.

Substitute these values into the equation to get:

[tex]y = 0.8x + 50[/tex]

[tex]\hrulefill[/tex]

To find the ordered pair associated with x = 5, substitute x = 5 into the equation:

[tex]\begin{aligned}x=5\implies y &= 0.8(5) + 50\\&=4+54\\&=54\end{aligned}[/tex]

The ordered pair associated with the equation x = 5 is (5, 54) and it means that the charge for driving the car for 5 miles is $54.

[tex]\hrulefill[/tex]

To find how many miles the car was driven if the renter paid $187.60, set y = 187.60 and solve for x:

[tex]\begin{aligned}\implies 0.8x + 50&=187.60\\0.8x + 50-50&=187.60-50\\0.8x&=137.60\\\dfrac{0.8x}{0.8}&=\dfrac{137.60}{0.8}\\x &= 172\end{aligned}[/tex]

Therefore, if the renter paid $187.60, the car must have been driven for 172 miles.

The value of height (h)  will be 6 cm.

What is Parallelogram?

A parallelogram is a four-sided polygon in which both pairs of opposite sides are parallel and equal in length. It is a special case of a quadrilateral, which means a polygon with four sides. The opposite angles in a parallelogram are also equal in measure, and the adjacent angles are supplementary, which means they add up to 180 degrees.

Given : height (H) = 5 cm

            base (B) = 12 cm

Similarly, height (h) = 5 cm

               base (h) = 12 cm

Now, we know that area of parallelogram will be same whether we use different method.

So, area of given parallelogram  =  base × height

                                         B × H    =  b × h

                                        12 × 5    =  10 × h

                                            60     =  10 h

So,   h = 60/10 = 6 cm.

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Answer:

To simplify F1 × F2 - F3 in terms of y, we need to first find the product of F1 and F2, and then subtract F3.

F1 × F2 can be expanded using the distributive property:

F1 × F2 = (4y - 6) × (9y + 3) = 4y × 9y + 4y × 3 - 6 × 9y - 6 × 3

= 36y^2 + 12y - 54y - 18

= 36y^2 - 42y - 18

Now we can subtract F3 from the result:

F1 × F2 - F3 = (36y^2 - 42y - 18) - (-y - 8)

= 36y^2 - 42y - 18 + y + 8

= 36y^2 - 41y - 10

Therefore, F1 × F2 - F3 in terms of y is 36y^2 - 41y - 10.

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The length distance of ab between the points  (-4,-6) and b (3,2) is   10.6 units.

Now to find the distance between points A(-4,6) and B(3,2), we  can use the distance formula which is :

d = sqrt((x₂ - x₁)² + (y₂ - y₁)²), where (x₁,y₁)=(-4,-6) and (x₂,y₂)=(3,2), now we substituting the  values into the formula  after which we get :

d = sqrt((3 - (-4))² + (2 - (-6))²) = sqrt((7)² + (8)²) = sqrt(49 + 64) = sqrt(113) ≈ 10.6

it comes out  that the distance between points A and B is approximately 10.6  units  which are rounded to the nearest tenth.

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The value of the x is 5√3 after we successfully do the application of the 30°-60°-90° Triangle theorem.

The 30°-60°-90° Triangle Theorem states that in such a triangle, the side opposite the 30° angle is half the length of the hypotenuse, and the side opposite the 60° angle is the product of the length of the hypotenuse and the square root of 3 divided by 2.

Using this theorem, we can write:

y = hypotenuse

Opposite of 30° angle = 5 = hypotenuse/2

Opposite of 60° angle = x = hypotenuse × (√(3)/2)

Solving for the hypotenuse in terms of y from the first equation, we get:

hypotenuse = 5×2 = 10

Substituting this value into the third equation, we get:

x = 10 × (√(3)/2) = 5 × √(3)

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By finding the derivative and evaluating it in x = -2, we will see that the slope is 12.

To find the slope of the tangent line at a given point, we need to take the derivative and evaluate it in the x-value of the given point.

Here we have the function:

f(x) = 3x² + 7

If we differentiate it with respect to x, we will get:

f'(x) = 2·3x

f'(x) = 6x

That is the derivative, now we want to find the slope at (-2, 19), to find the slope at that point we need to get:

f'(-2) = 6·-2

f'(-2) = -12

That is the slope of the graph.

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The height of the building in the scale drawing will be 5 inches.

The scale ratio must be used to determine the height of the building in the scale drawing.

The scale ratio is:

1 inch = 170 feet

We can devise a ratio:

1 inch / 170 feet = x inches / 850 feet

In the scale drawing, x represents the height of the building in inches.

To find x, we can cross-multiply and simplify:

1 inch * 850 feet = 170 feet * x inches

850 inches = 170 feet * x

Dividing both sides by 170 yields:

x = 850 inches / 170 = 5 inches

As a result, the building's height will be 5 inches in the scale drawing.

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Answer:

Step-by-step explanation:

To find the temperature corresponding to the 67th percentile, we need to find the z-score that has an area of 0.67 to the left of it in the standard normal distribution. We can use a table or a calculator to find this z-score.

Using a standard normal distribution table, we can look up the value that corresponds to an area of 0.67 to the left of the mean, which is 0.44. This means that P(Z ≤ 0.44) = 0.67, where Z is the standard normal random variable.

Next, we can use the formula for standardizing a normal random variable to convert this z-score to the corresponding temperature on the thermometer scale:

z = (x - μ) / σ

where μ is the mean, σ is the standard deviation, and x is the temperature we want to find.

Rearranging this formula, we get:

x = μ + z * σ

Plugging in the values, we get:

x = 0 + 0.44 * 1.00

x = 0.44

Therefore, the temperature corresponding to the 67th percentile is 0.44°C.

The player should be willing to pay up to $12 to play this game and not lose money in the long run.

First we need to multiply the probability of winning by the amount won and subtract the probability of losing by the amount lost.

The probability of rolling a 1 on a 6-sided die is 1/6, and the probability of rolling any other number is 5/6.

So, the expected value of the game is:

(1/6) x $4 - (5/6) x $2

= ($4/6) - ($10/6)

= -$1/3

This means that on average, for every game played, the player can expect to lose $1/3.

To find out how much the player should be willing to pay to play this game and not lose money in the long run, we can set the expected value equal to zero:

(1/6) x $4 - (5/6) x $2 = $0

Simplifying the equation, we get:

$4/6 = $10/6

Multiplying both sides by x, we get:

(1/6) x - $2 = 0

Solving for x, we get:

x = $12

Therefore, the player should be willing to pay up to $12 to play this game and not lose money in the long run.

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Answer:

76(/783/)-468

Step-by-step explanation:

The t-value such that the area under the t distribution to the right of the t-value is 0.10, assuming 15 degrees of freedom (df) is 1.753050356.

We have to determine the t-value.

The area in the right tail is 0.10 with 32 degrees of freedom(df).

The t-test is a test that is used as an alternative to the z-test in statistics. If the data are normally distributed but the sample size is small and the population standard deviation is unknown, the t-test is utilized.

A value that appears on the t distribution is the critical t value. The area under the curve and the degrees of freedom can be used to determine the t statistic value.

Using Excel Formula,

The t-value = (=TINV(0.1,15))  

The t-value = 1.753050356

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The angles for the given parallel lines are estimated.

m(∠DGF ) = 92; m(∠HGF ) = 88; m(∠AGD ) = 88; m(∠BDG ) = 92; m(∠BDC ) = 88; m(∠CDE ) = 92 ; m(∠EDG ) = 88.

A line that cuts over two parallel lines is referred to as a transversal line.

Each pair of internal angles located on the exact side of a transversal that meets two parallel lines is supplementary, or they add up to 180°.

The opposing angles created by the junction of two lines are known as vertical angles but rather vertically opposite angles.

m(∠AGH ) = m(∠BDG ) (corresponding angles)

3x +  2 = 2x + 32

x = 30

m(∠AGH ) = 3(30) +  2 = 92

m(∠BDG ) = 2(30) + 32 = 92

m(∠DGF ) = m(∠AGH ) = 92 (vertically opposite angles)

m(∠HGF ) = 180 - m(∠AGH ) = 180 - 92 = 88 (linear pair).

m(∠AGD ) = m(∠HGF ) = 88 (vertically opposite angles)

m(∠BDG ) = 92 (calculated earlier)

m(∠BDC ) = m(∠AGD ) =  88 (corresponding angles)

m(∠CDE ) = m(∠BDG ) = 92  (vertically opposite angles)

m(∠EDG ) = m(∠BDC ) =  88  (vertically opposite angles).

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Answer:

huh

Step-by-step explanation:

From the given probability distribution, the probability of x being 4 in the given probability distribution is 0.15,

According to the given probability distribution in Table 1, the probability of x being 4 is 0.15. This means that out of all the possible values of x (0 to 6), there is a 15% chance that x will be equal to 4.

To understand the probability distribution better, we can visualize it using a graph. The x-axis represents the possible values of x, while the y-axis represents the probability of each value. We can plot the values from Table 1 to create a histogram or a bar graph.

From the graph, we can see that the probability distribution is skewed to the right, with the highest probability being at x=2. This means that there is a higher chance that x will be closer to 2 than to 0 or 6.

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In linear equation, 28  pieces of cake did Mrs Devi buy.

What in mathematics is a linear equation?

An algebraic equation with simply a constant and a first-order (linear) term, such as y=mx+b, where m is the slope and b is the y-intercept, is known as a linear equation. Sometimes, the aforementioned is referred to as a "linear equation of two variables," where x and y are the variables.  

                                   Equations with variables of power 1 are referred to as linear equations. One example with only one variable is where ax+b = 0, where a and b are real values and x is the variable.

Cost of M cake = $3.5

For every 3 M cakes she bought 7 B cakes.

Let 1 unit be 3 M cakes and 7 B cakes. 1 unit costs= 3(3.5) + 7(2.5)= $28

She bought $112 / $28 = 4 units of cakes.

Which means 12 M cakes and 28 B cakes.

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The probability of one event given the known outcome of a (possibly) related event is known as conditional probability.

Conditional probability is the measure of the probability of an event occurring given that the another event will already occurred. It is denoted by P(A | B), which represents the probability of event A given that event B has occurred. The conditional probability of A given B can be calculated using the formula:

P(A | B)=P(A and B)/P(B)

where P(A and B) represents the probability of both events A and B occurring, and P(B) represents the probability of event B occurring.

For example, consider a deck of 52 playing cards. The probability of drawing a king from the deck is 4/52, or 1/13. If one card is drawn from the deck and it is revealed to be a heart, then the probability of drawing a king from the remaining cards in the deck that are not hearts is:

P(king | heart) = P(king and heart) / P(heart)

P(king and heart) = 1/52 (there is only one king of hearts in the deck)

P(heart) = 13/52 (there are 13 hearts in the deck)

P(king | heart) = (1/52) / (13/52) = 1/13

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Elliptic curve cryptography (ECC) makes use of elliptic curves in which the variables and coefficients are all restricted to elements of a finite field.

ECC is a type of public-key cryptography that is based on the difficulty of solving the elliptic curve discrete logarithm problem (ECDLP), which is a variant of the discrete logarithm problem in which the group operation is performed on points on an elliptic curve.

ECC is particularly useful in settings where computational resources are limited, such as mobile devices and smart cards, as it provides the same level of security as other public-key cryptographic systems but with smaller key sizes.

ECC also offers other advantages over traditional public-key cryptography such as faster computation times, lower power consumption, and smaller message sizes.

ECC is widely used in a variety of applications, including digital signatures, encryption, and key exchange. It is implemented in many cryptographic standards, such as the Transport Layer Security (TLS) protocol used to secure internet communications, and is considered to be one of the most promising cryptographic techniques for the future.

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Complete question is:

___________ makes use of elliptic curves in which the variables and coefficients are all restricted to elements of a finite field.

Answer:

[tex](x-4)^2+(y+1)^2=162[/tex]

Step-by-step explanation:

Determine r² by using the equation of a circle and plugging in the center (h,k)->(4,-1) as well as (x,y)->(13,8):

[tex](x-h)^2+(y-k)^2=r^2\\(x-4)^2+(y-(-1))^2=r^2\\(13-4)^2+(8+1)^2=r^2\\9^2+9^2=r^2\\81+81=r^2\\162=r^2\\[/tex]

Hence, the equation of the circle that meets these criteria is[tex](x-4)^2+(y+1)^2=162[/tex]

Check the picture below.

so if the triangles are congruent, then CPCTC.

Answer:

m = -5/8

Step-by-step explanation:

Slope = rise/run or (y2 - y1) / (x2 - x1)

Points (-1, -7) (-9, -2)

We see the y increase by 5 and the x decrease by 8, so the slope is

m = -5/8

(a) The function of price is $1600

(b) The marginal revenue is $80

(c) The revenue and marginal revenue when price is $5 is $30

Marginal revenue is the increase in revenue from selling an additional unit of product. Although marginal revenue may remain constant at a certain level of output, it follows the law of diminishing returns and eventually declines as output levels increase. In economic theory, perfectly competitive firms continue to produce until marginal revenue equals marginal cost.

Assume that a certain product has the demand function given by:

9 = 1000e -0.02p

R(x) = 80x

P(x) = -0.25x² + 40x -100

R'(x) =80

P'(x) = -0.5x + 40

Because we have refurbished x = 20 iPad this month x = 20.

Thus,

R(20) = 80(20) = $1600

P(20) = -0.5(20)² + 40(20) - 1000

         = -$300

R'(20)= $80

And, P'(20) = -0.5 (20) + 40

                   = $30

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Answer:To simplify the expression, you need to combine the like terms, which are the terms that have the same variable and power. In this case, the like terms are -3a and 5a:

(-3a + 56) + (5a + 40)

= (-3a + 5a) + (56 + 40)

= 2a + 96

Therefore, the simplified expression is 2a + 96.

Enjoy (:

Step-by-step explanation:

Answer:

Let's start by expressing "Z increased by 16%" using algebra.

Let Z be the original value of some quantity.

To increase Z by 16%, we need to add 16% of Z to Z:

Z + 0.16Z

Simplifying this expression by factoring out Z, we get:

Z(1 + 0.16)

Combining like terms, we have:

Z(1.16)

Therefore, "Z increased by 16%" can be expressed algebraically as:

Z increased by 16% = Z(1.16)

Answer:

z(1.16)

Step-by-step explanation:

Answer:

Step-by-step explanation:

We can use the formula for compound interest:

A = P(1 + r/n)^(nt)

where A is the final amount, P is the principal (initial amount), r is the annual interest rate (as a decimal), n is the number of times the interest is compounded per year, and t is the time (in years).

In this case, we know that P = $5500, r = 4.5% = 0.045, and we want to find t when A = $6700. We also know that the interest is compounded annually, so n = 1.

Substituting these values into the formula, we get:

$6700 = $5500(1 + 0.045/1)^(1t)

Dividing both sides by $5500, we get:

1.218181818 = (1.045)^t

Taking the natural logarithm of both sides, we get:

ln(1.218181818) = ln(1.045)^t

Using the property of logarithms that ln(a^b) = b ln(a), we can rewrite the right side as:

ln(1.218181818) = t ln(1.045)

Dividing both sides by ln(1.045), we get:

t = ln(1.218181818)/ln(1.045) ≈ 4.2

Therefore, the person must leave the money in the bank for about 4.2 years to reach $6700. To the nearest tenth of a year, the answer is 4.2 years.

The area to the right of x is 0.877.

In a normal distribution, the entire area under the curve is identical to 1. The area to the left of a specific value of x represents the possibility of observing a value largely lesser than or same tox.

However, we're capable to discover the area to the right of x with the aid of abating the left area from 1, If the place to the left of x is given.

In this case, the area to the left of x is 0.123. thus, the place to the right of x is

1-0.123 = 0.877

Thus, the area is 0.877 to the right of x.

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The equation of the line passing through (-3, -1) with slope 3/5 is y = (3/5)x + 4/5.

What is point slope form?

The equation of a line is expressed in the point-slope form as follows: y - y1 = m(x - x1), where m is the slope of the line and (x1, y1) is a point on the line. When we know the slope of a line and a point on the line but not the intercepts, this version of the equation is helpful. It eliminates the need to independently compute the intercepts by allowing us to state the equation of the line in terms of the given point and slope.

Given that, line passes through (-3, -1) and has a slope of 3/5.

The points slope form is given as:

y - y1 = m(x - x1)

Substituting the values we have:

y - (-1) = (3/5)(x - (-3))

y + 1 = (3/5)x + 9/5

y = (3/5)x + 4/5

Therefore, the equation of the line passing through (-3, -1) with slope 3/5 is y = (3/5)x + 4/5.

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Answer:

Proofs attached to answer

Step-by-step explanation:

Proofs attached to answer