a=9y+3yx solve for y

After time  [tex]t = 9[/tex] seconds the driver applies the brakes does the car come to a stop.

" Time is defined as the measurable slot of period in which required action is done."

Formula used

[tex]y = x^{n} \\\\\implies \frac{dy}{dx} = nx^{n-1}[/tex]

According to the question,

Position of the car [tex]'s' = 54t - 3t^{2}[/tex]

[tex]'s'[/tex] represents the distance

[tex]'t'[/tex] is the time in seconds

When driver applies break [tex]v= 0[/tex],

[tex]v = \frac{ds}{dt}[/tex]

Calculate the first derivative with respect to time we get,

[tex]'s' = 54t - 3t^{2}\\\\\implies \frac{ds}{dt} = 54- 6t[/tex]

As [tex]\frac{ds}{dt} =0[/tex] we get the required time as per given condition,

[tex]54-6t =0\\\\\implies 6t =54\\\\\implies t = 9[/tex]

Hence, after time  [tex]t = 9[/tex] seconds the driver applies the brakes does the car come to a stop.

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In 22 gross of nails, there are 264 dozens as one gross is equal to 12 dozen.

To find out how many dozens are there in 22 gross of nails, we need to determine the number of dozens in one gross and then multiply it by 22.

One gross is equal to 12 dozen.

So, to find the number of dozens in 22 gross of nails.

we can multiply 12 (number of dozens in one gross) by 22:

12 x 22

= 264

Therefore, there are 264 dozens in 22 gross of nails.

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The requried 475% of 9.2 is 43.7, as of the given percentage situation.

The percentage is the ratio of the composition of matter to the overall composition of matter multiplied by 100.

Here,
In the question, we are asked to determine the 43.7 is what percent of 9.2.

So, let the percent be x,
x % of  9.2 = 43.7
x/100 × 9.2 = 43.7
x = 43.7/9.2 × 100%
x = 475%

Thus, the requried 475% of 9.2 is 43.7, as of the given percentage situation.

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By subtracting the number of blue pens from the number of black pens, we find that there are 183,891 more black pens than blue pens.

To find out how many more black pens there are than blue pens, we need to subtract the number of blue pens from the number of black pens. The student provided the numbers as 365,493 blue pens and 549,384 black pens.

Here is the calculation:

There are 183,891 more black pens than blue pens.

The extreme values of a function are the minimum and the maximum values of the function.

The extreme values are: -8 and 131.91

The given parameters are:

[tex]\mathbf{F(x,y) = x^2 + y^2 + 4x - 4y}[/tex]

[tex]\mathbf{x^2 + y^2 \le 81}[/tex]

Find the gradient of F(x,y)

[tex]\mathbf{f_x(x) = 2x + 4}[/tex]

[tex]\mathbf{f_y(y) = 2y - 4}[/tex]

Set to 0, to solve for x and y

[tex]\mathbf{2x + 4 = 0}[/tex]

[tex]\mathbf{2x = -4}[/tex]

[tex]\mathbf{x = -2}[/tex]

[tex]\mathbf{2y - 4 = 0}[/tex]

[tex]\mathbf{2y = 4}[/tex]

[tex]\mathbf{y = 2}[/tex]

So, the critical point is:

[tex]\mathbf{(x,y) = (-2,2)}[/tex]

Also, we have: [tex]\mathbf{x^2 + y^2 \le 81}[/tex]

Calculate the gradients

[tex]\mathbf{g_x = 2x}[/tex]

[tex]\mathbf{g_y = 2y}[/tex]

Equate the gradients as follows:

[tex]\mathbf{f_x = \lambda \cdot g_x}[/tex]

[tex]\mathbf{f_y = \lambda \cdot g_y}[/tex]

So, we have:

[tex]\mathbf{2x + 4 = \lambda \cdot 2x}[/tex]

[tex]\mathbf{2y - 4 = \lambda \cdot 2y}[/tex]

Make [tex]\mathbf{\lambda}[/tex] the subject, in the above equations

[tex]\mathbf{\lambda = \frac{x + 2}{x}}[/tex]

[tex]\mathbf{\lambda = \frac{y - 2}{y}}[/tex]

Equate the above equations, so we have:

[tex]\mathbf{\frac{x + 2}{x} = \frac{y - 2}{y} }[/tex]

Cross multiply

[tex]\mathbf{xy + 2y = xy - 2x}[/tex]

Subtract xy from both sides

[tex]\mathbf{2y =- 2x}[/tex]

Divide both sides by 2

[tex]\mathbf{y =- x}[/tex]

Substitute [tex]\mathbf{y =- x}[/tex] in [tex]\mathbf{x^2 + y^2 = 81}[/tex]

[tex]\mathbf{x^2 + (-x)^2 = 81}[/tex]

[tex]\mathbf{x^2 + x^2 = 81}[/tex]

[tex]\mathbf{2x^2 = 81}[/tex]

Divide both sides by 2

[tex]\mathbf{x^2 = \frac{81}{2}}[/tex]

Take square roots

[tex]\mathbf{x = \±\frac{9}{\sqrt2}}[/tex]

Rationalize

[tex]\mathbf{x = \±\frac{9\sqrt2}{2}}[/tex]

Recall that: [tex]\mathbf{y =- x}[/tex]

So, we have:

[tex]\mathbf{y = \±\frac{9\sqrt2}{2}}[/tex]

The ordered pairs are:

[tex]\mathbf{(x,y) = \{(\frac{9\sqrt2}{2},-\frac{9\sqrt2}{2}),(-\frac{9\sqrt2}{2},\frac{9\sqrt2}{2})\}}[/tex]

Hence, the points are:

[tex]\mathbf{(x,y) = \{(-2,2),(\frac{9\sqrt2}{2},-\frac{9\sqrt2}{2}),(-\frac{9\sqrt2}{2},\frac{9\sqrt2}{2})\}}[/tex]

Substitute these values in [tex]\mathbf{F(x,y) = x^2 + y^2 + 4x - 4y}[/tex]

[tex]\mathbf{F(2,2) = (-2)^2 + 2^2 + 4(-2) - 4(2) =-8}[/tex]

[tex]\mathbf{F(\frac{9\sqrt2}{2},-\frac{9\sqrt2}{2}) = (\frac{9\sqrt2}{2})^2 + (-\frac{9\sqrt2}{2})^2 + 4(\frac{9\sqrt2}{2}) - 4(-\frac{9\sqrt2}{2}) =131.91}[/tex]

[tex]\mathbf{F(-\frac{9\sqrt2}{2},\frac{9\sqrt2}{2}) = (-\frac{9\sqrt2}{2})^2 + (\frac{9\sqrt2}{2})^2 + 4(-\frac{9\sqrt2}{2}) - 4(\frac{9\sqrt2}{2}) =30.09}[/tex]

Hence, the extreme values of the function are: -8 and 131.91

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There are a total of 26 letters that can be made to follow. There are 4 independent events here, namely:

First: Starts with K and followed by two additional letters = 1 * 26 * 26 = 676

Second: Starts with K and followed by three additional letters = 1 * 26 * 26 * 26 = 17,576

Third: Starts with W and followed by two additional letters = 1 * 26 * 26 = 676

Fourth: Starts with W and followed by three additional letters = 1 * 26 * 26 * 26 = 17,576

The total possibilities that can be made from here is:

Total possibilities = 676 + 17,576 + 676 + 17,576

Total possibilities = 36,504

Answer:

Option A: Exponential growth

Step-by-step explanation:

If  for values of x values of y increases and forms a geometric sequence then it would be an exponential growth function because geometric sequence is an exponential function and since, it is increasing hence, an exponential growth.

Option B is incorrect because  because y is not decreasing

Option C and D are incorrect because geometric sequence can never be linear since, it gives common ratio.

Final answer:

The function described, where y-values form an increasing geometric sequence as x-values increase, is characterized as Exponential growth. This is consistent with an exponential curve where the rate of growth is proportional to the current value, which aligns with the behavior of a geometric sequence.

Explanation:

For the given scenario where the y-values of a function form a geometric sequence that increases for the x-values 1, 2, 3, and so on, the type of function is described as Exponential growth. This is because in a geometric sequence each term after the first is found by multiplying the previous term by a constant called the common ratio, which is analogous to the exponential function where the growth rate of the value is proportional to its current value. The more the x-value increases, the higher and faster the y-value grows, following the pattern of an exponential growth curve.

Answer A. Exponential growth fits the description as it involves an increasing geometric sequence. Answer B. Decreasing linear refers to a linear function that decreases with each increment in x, which does not describe the function in question. Similarly, Answer C. Increasing linear describes a function that increases at a constant rate, not geometrically. Answer D. Exponential decay would imply the y-values decrease as x increases, which is also not the case described.