A curve has the equation
y=8/x+2x
(i) Find the dy/dx and d^2y/dx^2
(ii) Find the coordinates of stationary points and state, with a reason, the nature of each stationary point.

The empirical probability that a speeding ticket will be issued in San Marcos is 5/8, or 5 out of 8.

The number of speeding tickets can be divided by the total number of tickets issued.

25 speeding citations were issued.

40 total tickets are available.

Empirical probability of a speeding ticket = Number of speeding tickets / Total number of tickets

Empirical probability of a speeding ticket = 25 / 40

Simplifying this fraction, we get

Empirical probability of a speeding ticket = 5/8

Therefore, the empirical probability that a speeding ticket will be issued in San Marcos is 5/8, or 5 out of 8.

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Answer:

Step-by-step explanation:

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The table of values for the function rule y = √(x + 6) is

x | f(x)

-15 | Not a real number

6 | 0

19 |  5

75 | 9

From the question, we have the following parameters that can be used in our computation:

y = √(x + 6)

To fill in a table using a function rule, we apply the function rule to each input to calculate the corresponding output, and write these values in the right-hand column of the table

We have the x values to be

x = -15, -6, 19 and 75

Substitute the known values in the above equation, so, we have the following representation

y = √(-15 + 6) = Not a real number

y = √(-6 + 6) = 0

y = √(19 + 6) = 5

y = √(75 + 6) = 9

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The rate at which the depth of the liquid is increasing when the depth of the liquid reaches one-third of the height of the bowl is 1.25 cm s⁻¹.

The volume of the liquid in the bowl is given by the following integral:

[tex]V = \int\limitsx_{0}^{h} \, \pi r^{2}(y) dy[/tex]

where r = radius of the bowl and y = height of the liquid.

The radius of the bowl is equal to the distance from the curve y = (4/(8-x)) - 1 to the y-axis. This can be found using the following equation:

r = √{(4/(8-x)) - 1}² + 1²

The height of the liquid is equal to the distance from the curve y = (4/(8-x)) - 1 to the x-axis. This can be found using the following equation:

h = (4/(8-x)) - 1

Substituting these equations into the volume integral:

[tex]V = \int\limitsx_{0}^{h } \, \pi {\sqrt{(4/(8-x)) - 1)^{2} + 1^{2} (4/(8-x))} - 1 dy[/tex]

Evaluate this integral using the following steps:

Expand the parentheses in the integrand.

Separate the integral into two parts, one for the integral of the square root term and one for the integral of the linear term.

Integrate each part separately.

The integral of the square root term can be evaluated using the following formula:

[tex]\int\limits^{b} _{a} \, dx \sqrt{x} dx = 2/3 (x^{3/2}) |^{b}_{a}[/tex]

The integral of the linear term can be evaluated using the following formula:

[tex]\int\limits^{b} _{a} \, {x} dx = (x^{2/2}) |^{b}_{a}[/tex]

Substituting these formulas into the integral:

V = π { 2/3 (4/(8-x))³ - 1/2 (4/(8-x))² } |_0^h

Evaluating this integral:

V = π { 16/27 (8-h)³ - 16/18 (8-h)² }

The rate of change of the volume of the liquid is given by:

dV/dt = π { 48/27 (8-h)² - 32/9 (8-h) }

The rate of change of the volume of the liquid is 7π cm³ s⁻¹. Also the depth of the liquid is one-third of the height of the bowl. This means that h = 2/3.

Substituting these values into the equation for dV/dt:

dV/dt = π { 48/27 (8-2/3)² - 32/9 (8-2/3) } = 7π

Solving this equation for the rate of change of the depth of the liquid:

dh/dt = 7/(48/27 (8 - 2/3)² - 32/9 (8 - 2/3)) = 1.25 cm s⁻¹

Therefore, the rate at which the depth of the liquid is increasing when the depth of the liquid reaches one-third of the height of the bowl is 1.25 cm s⁻¹.

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The area of each slice from the circle is 18π square units. Option C

The area of a circle is an enclosed region within a confined boundary. The area of a circle is measured as the product of π multiplied by the square of a radius.

From the information given:

The radius of the circle (r) = 12 units

The area of the circle = πr²

The area of the circle = π(12)²

The area of the circle = 144π

However, if the area is divided into 8 congruent(similar and equal) slices, the slice of each area will be:

=  144π/8

= 18π

The circumference of a circle = 2πr

If the radius = 12

The circumference of a circle = 2π(12)

The circumference of a circle = 24π

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The length of each side of the wood is 5 centimetres.

Doug has 8 square pieces of wood. Each piece of wood have a side length of s cm. The total area of all 8 pieces of wood is 200 cm².

Hence, using quadratic equation let's find the length of each side of the wood.

Therefore,

8s² = 200

Hence,

8s² = 200

divide both sides of the equation by 8

s² = 200 / 8

s² = 25

s = √25

s = 5 centimetres

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Answer:

Step-by-step explanation:

You want two numbers such that the larger is 2 less than 5 times the smaller, and their sum is 22 after it has been increased by 6.

Let x represent the smaller number. Then the larger is (5x-2) and the relation with regard to the sum is ...

  (x) +(5x -2) +6 = 22

  6x = 18 . . . . . . . . . . . . subtract 4 and simplify

  x = 3 . . . . . . . . . divide by 6

  (5x -2) = 5·3 -2 = 13

The two numbers are 3 and 13.

Check

The sum of the two numbers is 3+13 = 16. When that is increased by 6, the result is 16+6 = 22, as required.

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The speed of the radius changing at the instance the radius is 4 feet is 0. 72 ft per second.

Given that the balloon is inflating at the rate of 144 cubic feet per second :

dV  / dt = 144 cubic feet/second.

The volume equation of a sphere would need to be differentiated as a result to find the rate at which the radius of the balloon is changing:

dV/ dt = d / dt [4/3 x π x r ³ ]

dV/ dt = 4π x r²  x dr / dt

144 = 4 π x (4)² x dr / dt

144 = 64 π x dr / dt

dr / dt = 144 / ( 64 π )

dr / dt = 9/( 4 π ) feet /second.

dr / dt = 0. 72 ft per second

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Answer:

Blue

Step-by-step explanation:

The answer is blue because there is more blue marble than red.

Answer:

If two linear functions have the same coefficients of x, then the graphs of the two linear functions do not intersect at exactly one point.

Step-by-step explanation:

The statement that is logically equivalent to q→ p is:

If two linear functions have the same coefficients of x, then the graphs of the two linear functions do not intersect at exactly one point.

This is because the contrapositive of an implication is logically equivalent to the original implication. The contrapositive of q→ p is ¬p→ ¬q, which means:

If two linear functions do not have different coefficients of x, then the graphs of the two functions do not intersect at exactly one point.

Simplifying this statement, we get:

If two linear functions have the same coefficients of x, then the graphs of the two linear functions do not intersect at exactly one point.

Answer:

C, 3:7

Step-by-step explanation:

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Answer:

[tex] \huge{ \boxed{36 {x}^{3} - \frac{6 {x}^{2} }{ \sqrt{x} } - 12x \sqrt{x} + 2 \sqrt{x} }}[/tex]

Step-by-step explanation:

To find the derivative of the function [tex](3 {x}^{2} - \sqrt{x} )^{2} [/tex] , the chain rule can be applied.

According to the chain rule, the derivative of f(x) with respect to x is given by:

[tex] \dfrac{df}{dx} = \dfrac{df}{du} \times \: \dfrac{du}{dx} [/tex]

where

[tex] \dfrac{df}{du} [/tex] represents the derivative f(x) with respect to u.

[tex] \dfrac{df}{du} = 2u(x)[/tex]

[tex] \frac{du}{dx} = \frac{d}{dx} ( {3x}^{2} ) - \frac{d}{dx} ( \sqrt{x} ) \\ \\ \frac{du}{dx} = {6x} - \frac{1}{2 \sqrt{x} } [/tex]

[tex] \frac{df}{dx} = 2u(x) \times (6x - \frac{1}{2 \sqrt{x} } ) \\ [/tex]

[tex] \frac{df}{dx} = 2(3 {x}^{2} - \sqrt{x} ) \times (6x - \frac{1}{2 \sqrt{x} } ) \\ = (6 {x}^{2} - 2 \sqrt{x} )( 6x - \frac{1}{2 \sqrt{x} } ) \\ = 36 {x}^{3} - \frac{6 {x}^{2} }{ \sqrt{x} } - 12x \sqrt{x} + 2 \sqrt{x} [/tex]

We have the final answer as

[tex]36 {x}^{3} - \frac{6 {x}^{2} }{ \sqrt{x} } - 12x \sqrt{x} + 2 \sqrt{x} [/tex]

The exponential function [tex]y = 2 + 3^x[/tex] is a exponential growth function with these following features:

An exponential function has the definition presented as follows:

[tex]y = ab^x[/tex]

In which the parameters are given as follows:

The function for this problem is given as follows:

[tex]y = 2 + 3^x[/tex]

The addition by 2 means that the asymptote of the function is given as follows:

y = 2.

Hence the y-coordinate of the y-intercept is given as follows:

[tex]y = 2 + 3^0[/tex]

y = 3.

The parameter b is given as follows:

b = 3.

As |b| > 1, we have an exponential growth function.

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It should be noted that triangle ABC is an isosceles triangle.

An isosceles triangle has two sides of equal length, and the third side is longer than the other two sides. In triangle ABC, AB = DCE, which means that sides AB and DCE are equal in length. The third side, AC, is longer than AB and DCE, because ABC = 80, which is greater than 50.

The two angles opposite the equal sides are also equal. In triangle ABC, angles A and C are equal because sides AB and AC are equal.

The altitude (a line drawn from a vertex perpendicular to the opposite side) bisects the opposite side. In triangle ABC, the altitude from vertex A bisects side BC.

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The speed of the butterfly travelling is 30 miles per hour as given on the graph G.

For F, the slope is 150/25 = 6 so this is false

For G, the slope is 150/5 = 30, this is true.

For H, the slope is 25/150 = 1/6, This is false

For J, th slope is 5/150 = 1/30, this is also false.

Thus, the correct answer is option G.

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Full Question:

See attached image

The cuboid has the following dimensions (A = 11 in, B = 4 in, C = 8 in, D = 4 in) and the following surface areas: Lateral: S = 264 in², Total: S' = 328 in².

In this problem we find a cuboid, whose lateral and total surface area must be found. The lateral surface area is the sum is the sum of the areas of all faces except bases and the total surface area is the sum of the areas of the six faces. The area formula needed for the surface area is the rectangle:

S = w · h

Where:

Sides

A = 11 in, B = 4 in, C = 8 in, D = 4 in

Lateral surface area

S = 2 · (11 in) · (4 in) + 2 · (11 in) · (8 in)

S = 88 in² + 176 in²

S = 264 in²

Total surface area

S' = 2 · (4 in) · (8 in) + 264 in²

S' = 328 in²

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Answer:

x = 12, y = -1

Step-by-step explanation:

x - 2y = 14            (call this equation '1.')

x + 3y = 9             (call this '2')

'2' - '1' :

0x  + (3 - -2)y = 9 -14

(3 + 2)y = -5

5y = -5

y = -1.

substitute than into '1':

x - 2(-1) = 14

x + 2 = 14

x = 12.

check we are correct by subbing into '2'.

x + 3y = 9

12 + 3(-1) = 12 - 3 =9

so, x= 12 and y = -1

Answer:

X=14+2y

14+2y+3y=9

14+5y=9

5y=9-14

5y=–5

Y=–1

The surface area of the prism is 62 in²

The formula for surface area of a right triangular prism is expressed as;

Surface area = (Perimeter of the base × Length of the prism) + (2 × Base Area)

Such that the parameters are;

Substitute the values

Perimeter= 2(12 + 4) = 2(16) = 32 in

Base area = 1/2 base ×height

Base area = 1/2 × 12 × 5

Multiply the values, we get;

Base area = 30 in

Surface area = 30 + 32

Add the values

Surface area = 62 in²

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The matrix is 3 × 3 square matrix . Its values are:

[tex]a_{11} = 50 \\ a_{12} = 44 \\a_{13} = -43 \\a_{21} = 31 \\a_{22} = 16 \\a_{23} = 7 \\a_{31} = 37 \\a_{32} = 119 \\a_{33} = 94[/tex]

Given,

Matrix A and Matrix B

Now,

According to matrix multiplication rule first row of one matrix will be multiplied with first column of second matrix if the columns in first matrix is equal to the rows of second matrix.

So,

Apply multiplication rule,

[tex]a_{11} = 30 + 14 + 6 = 50 \\ a_{12} = 55 + 7 - 18 = 44 \\a_{13} = -20 -35 + 12 = -43 \\a_{21} = 24 - 2 + 9 = 31 \\a_{22} = 44 -1 -27 = 16 \\a_{23} = -16 + 5 + 18 = 7 \\a_{31} = 36 + 16 - 15 = 37 \\a_{32} = 66 + 8 + 45 = 119 \\a_{33} = -24 -40 -30 = -94[/tex]

Thus the values of the matrix can be calculated.

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Answer:

Step-by-step explanation:

The area of a circle is calculated using the formula A = πr², where A is the area, π is approximately 3.14 and r is the radius of the circle.

For the first circle with a radius of 3 ft, the area would be A = 3.14 * 3² = 28.3 ft². So the correct answer is C. 28.3 ft².

For the second circle with a radius of 7 cm, the area would be A = 3.14 * 7² = 153.9 cm². So the correct answer is H. 153.9 cm².

Based on the dot plots, it appears that students generally did more push-ups this year than last year.

The range of the push-ups completed this year is 12, while the range of the push-ups completed last year is only 10.

This means that there is a greater spread of values in the data set for this year, with some students completing many more push-ups than others. In the data set for last year, the values are more evenly distributed, with most students completing between 5 and 10 push-ups.

The range of a data set affects the shape of the dot plot by determining how spread out the values are. A wide range will result in a dot plot that is more spread out, while a narrow range will result in a dot plot that is more compact. In the case of the push-up data, the wider range of values for this year results in a dot plot that is more spread out than the dot plot for last year.

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The value of each variable is:

x = 11 units

y = 11√2 units

Trigonometry deals with the relationship between the ratios of the sides of a right-angled triangle with its angles.

We can find the value of each variable using trigonometric ratios:

tan 45° = x/11 (tan = opposite/adjacent)

1 = x/11

x = 11 units

sin 45° =  9/y (sine = opposite/hypotenuse)

(√2)/2 = 11/y

y = 11/ (√2)/2

y = 11√2 units

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perimeter: 28

area: 45

Charlotte's reading rate per month is 2/3 books per month

To find the rate we use the principle of division to evaluate the amount of books read over a certain period of time .

Here ;

Number of books read = 8

Number of months over which book was read = 12

Reading rate = Number of books read / Number of months

Reading rate = 8 / 12

Reading rate = 2/3

Here, Charlotte's reading rate is 2/3 books per month

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Another way to write the mixed number 2 4/12 is 2 1/3.

Another way to represent the shaded portions that model the mixed number 2 4/12 is by simplifying the fraction to its simplest form.

In this case, 4/12 can be simplified by dividing both the numerator and denominator by their greatest common divisor, which is 4. By simplifying, we get 1/3. Thus, the mixed number 2 4/12 can be expressed as 2 1/3.

This representation implies that there are two whole units (2) and an additional fractional part equivalent to one-third (1/3) of a unit. It provides a clearer and more concise representation of the same quantity, avoiding unnecessary complexity.

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The surface area of the cylinder is 905 square centimeters

From the question, we have the following parameters that can be used in our computation:

Diameter, d = 12 cm

Height, h = 18 m

This means that

Radius, r = 12/2 = 6 cm

Using the above as a guide, we have the following:

Surface area = 2πr(r + h)

Substitute the known values in the above equation, so, we have the following representation

Surface area = 2π * 6 * (6 + 18)

Evaluate

Surface area = 905

Hence, the surface area is 905 square centimeters

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