during the first years of growth the height of a tree can be modeled with the function h=-t^2 +12t+10 where t is the time in years since being planted and h is the height in inches. Enter the average rate cof change, in inches per year, from year 1 to year 5.

[tex]\dfrac{\partial f}{\partial x}=7+8xy^2[/tex]

[tex]\dfrac{\partial f}{\partial y}=8x^2y[/tex]

The first equation gives

[tex]f(x,y)=7x+4x^2y^2+g(y)[/tex]

Differentiating with respect to [tex]y[/tex] gives

[tex]8x^2=8x^2y+\dfrac{\mathrm dg}{\mathrm dy}\implies\dfrac{\mathrm dg}{\mathrm dy}=0\implies g(y)=K[/tex]

for some constant [tex]K[/tex]. So

[tex]f(x,y)=7x+4x^2y^2+K[/tex]

and by the fundamental theorem of calculus,

[tex]\displaystyle\int_C\nabla f\cdot\mathrm d\vec r=f\left(3,\frac13\right)-f(1,1)=25-11=\boxed{14}[/tex]

To find a function f such that f = ∇f, we need to find a function whose gradient is equal to itself. The function f(x, y) = 0 is such that ∇f = f. To evaluate c·f · dr along the curve c, we need to find the dot product of the tangent vector to c and the function f.

To find a function f such that f = ∇f, we need to find a function whose gradient is equal to itself. Let's start by finding the partial derivative of f with respect to x and y. ∂f/∂x = (7 + 8y2)i + 16xyj and ∂f/∂y = 16xyi + 8x2j. Equating both partial derivatives to f gives us the following equations:

7 + 8y2 = (7 + 8y2)i + 16xyj

8x2 = 16xyi + 8x2j

Simplifying these equations, we find:

i = 0

j = 0

Therefore, the function f(x, y) = 0 is such that ∇f = f.

In order to evaluate c·f · dr along the curve c, we need to find the dot product of the tangent vector to c and the function f. The tangent vector to c is given by dr/dt = (-1/t2, 1/t3). Evaluating f at (x, y) = (1, 1/t) gives us f(1, 1/t) = (7 + 8/t2)i + 8/t2j. Taking the dot product of this with (-1/t2, 1/t3) yields:

c·f · dr = ((7 + 8/t2)(-1/t2) + 8/t2/t3)dt = (-7/t4 + 8/t2/t3)dt = (-7/t4 + 8/t5)dt.

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Answer:

Part A) Option C. obtuse

Part B) Option B. acute

Step-by-step explanation:

we know that

Applying the Pythagoras Theorem

if [tex]c^{2}=a^{2}+b^{2}[/tex] ----> is a right triangle

if [tex]c^{2}>a^{2}+b^{2}[/tex] ----> is an obtuse triangle

if [tex]c^{2}<a^{2}+b^{2}[/tex] ----> is an acute triangle  

where

c is the greater side

Part A) A triangle has side lengths 32, 45 and 18. What type of triangle is it?

we have

[tex]c^{2}=45^{2}=2,025[/tex]

[tex]a^{2}+b^{2}=32^{2}+18^{2}=1,348[/tex]

therefore

[tex]c^{2}>a^{2}+b^{2}[/tex]

Is an obtuse triangle

Part B) A triangle has side lengths 44, 36 and 30. What type of triangle is it?

we have

[tex]c^{2}=44^{2}=1,936[/tex]

[tex]a^{2}+b^{2}=36^{2}+30^{2}=2,196[/tex]

therefore

[tex]c^{2}< a^{2}+b^{2}[/tex]

Is an acute triangle

ANSWER

The length is 10 inches and the width is 24 inches

EXPLANATION

The diagonal of the rectangular painting is d=26 inches.

Let l and w be the length and width of the painting respectively.

From Pythagoras Theorem,

[tex] {l}^{2} + {w}^{2} = {26}^{2} [/tex]

[tex]{l}^{2} + {w}^{2} = 676..(1)[/tex]

Its area is 240 square inches.

This implies that,

[tex]l \times w = 240[/tex]

[tex]l = \frac{240}{w} ...(2)[/tex]

Put equation (2) into (1).

[tex]{( \frac{240}{w} )}^{2} + {w}^{2} = 676[/tex]

This implies that,

[tex] {w}^{4} - 676 {w}^{2} + 57600 = 0[/tex]

[tex]( {w}^{2} - 100)( {w}^{2} - 576) = 0[/tex]

[tex]{w}^{2} - 100 = 0 \: or \: ({w}^{2} - 576= 0[/tex]

[tex]{w}^{2} = 100 \: or \: {w}^{2} = 576[/tex]

Take positive square root to get,

[tex]{w} = 10\: or \: {w} = 24[/tex]

When w=24,

[tex]l = \frac{240}{24} = 10[/tex]

when w=10

[tex]l = \frac{240}{10} = 24[/tex]

Hence the length is 10 inches and the width is 24 inches.

The dimensions of the painting can be found by setting two equations, one for the area of the rectangle, and another based on the Pythagorean theorem, and solving for the length and the width. The key is to remember that these values must be positive, as they represent physical dimensions.

First, let's recall what we know. The area of a rectangle is given by the formula length x width = area. Here, we know that the area of the painting is 240 square inches.

Next, remembering the Pythagorean theorem, which states that for any right-angle triangle, the square of the length of the hypotenuse (the diagonal in this case) is equal to the sum of the squares of the other two sides. Here, we can write the Pythagorean theorem as length² + width² = diagonal² (26 inches).

Setting up these two equations, we would solve for length and width. Keep in mind that we're seeking positive and realistic solutions, as these represent the physical dimensions of the painting.

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Answer: TRUE

Step-by-step explanation: This is true, because there is an infinite amount of real numbers in both, and they are both countably infinite (so these infinities are equal). Hope this helps!

Answer: Second Option

[tex]\sigma=0.760[/tex]

Step-by-step explanation:

We have the average January surface water temperatures (°C) of Lake Michigan from 2000 to 2009.

5.07, 3.57, 5.32, 3.19, 3.49, 4.25, 4.76, 5.19, 3.94, 4.34

We know that the average [tex]{\displaystyle {\overline {x}}}[/tex] of the data is:

[tex]{\displaystyle {\overline {x}}}=\frac{\sum_{i=1}^{10}x_i}{10}=4.312[/tex]

We also know that the variance [tex]\sigma^2[/tex] is equal to 0.577.

So by definition the standard deviation [tex]\sigma[/tex] is:

[tex]\sigma=\sqrt{\sigma^2}\\\\\sigma=\sqrt{0.577}\\\\\sigma=0.760[/tex]

Answer:0.760

just right by chance

Answer: 4/20

Step-by-step explanation:

Option b is correct. The probability that you and your friend are both chose is 4/20.

The probability provides a means of getting an idea of likelihood of occurrence of different events resulting from a random experiment in terms of quantitative measures ranging between zero and one.

P(E) = number of favorable outcomes/Total number of outcomes

Where,

P(E) is the probability of an event.

An arrangement of objects where the order in which the objects are selected does not matter is called combination.

[tex]C_{n,k} = \frac{n!}{(n-k)!k!}[/tex]

Where,

n is total number of objects in set

[tex]C_{n, k}[/tex] is number of combinations

k is number of choosing objects from the set

According to the given question

We have

Total six people.

And,

There are total [tex]6C_{3}[/tex]  ways to choose the 3 contestants.

⇒ [tex]6C_{3} = \frac{6!}{(6-3)!3!}[/tex] = [tex]\frac{(6)(5)(4)(3!)}{(3!)(3!)}[/tex] = 20

⇒ total number of outcomes = 60

⇒ There are 20 ways to select 3 contestant among 20 people.

Now, the number of ways that, you and your friend are chosen, along with 1 person from a set of 4 is given by

[tex]C_{4,1} = \frac{4!}{1!3!}= 4[/tex]

⇒ total number of favorable outcomes = 4

Therefore,

The probability that you and your friend are both chose = [tex]\frac{4}{20}[/tex]

Hence, option b is correct.

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From the equation, you can see that each value in the equation is multiplied (x - 2)

To get it into factored form, you can just factor out (x - 2) from the equation, and you would be left with:

x²(x - 2) - 3(x - 2)

(x - 2)(x² - 3)      Your answer is A

Answer:

The graph of the given equation is letter A.

Step-by-step explanation:

I used the geogebra application to plot the graph of the equation.

You input the formula and the app will show you its graph.

Answer:

  124°

Step-by-step explanation:

Supplementary angles add to 180°.

  56° + second angle = 180°

Subtract 56° from both sides of the equation:

  second angle = 180° - 56° = 124°

A and c —————- Bc 20 +8 is 28

Answer:

The answer is (8,20) on APEX. just did the test. good luck!

Answer

m = 1

Explanation

Simple Method: Rise (go up) over Run (go left or right)

Start from a perfect point then rise and run.

Rise: 3, Run: 3 = 3/3 = 1

Alternatively, you could pick two points and use the formula y2 - y1/x2 - x1

Just as an example, I will choose (-2, -3) and (2, 1)

plug in:

1 + 3/2 + 2 = 4/4 = 1

ANSWER

1

EXPLANATION

The slope of the given line is obtained using the formula:

[tex]m = \frac{rise}{run} [/tex]

From the diagram the rise is 2 units and the run is 2 units.

This implies that,

[tex]m = \frac{2}{2} = 1[/tex]

Hence the slope of the given line is 1.

Answer:

Part 1) The width of the pad is [tex]1\ m[/tex]

Part 2) The solutions are

[tex]x=\frac{5(+)\sqrt{97}} {4}[/tex]  and [tex]x=\frac{5(-)\sqrt{97}} {4}[/tex]

Step-by-step explanation:

Part 1)

Let

x----> the width of the pad

we know that

[tex](15+2x)(18+2x)=340[/tex]

Solve for x

[tex]270+30x+36x+4x^{2} =340\\ \\4x^{2}+66x-70=0[/tex]

Solve the quadratic equation by graphing

The solution is [tex]x=1\ m[/tex]

see the attached figure

Part 2) we have

[tex]2x^{2} -5x-9=0[/tex]

we know that

The formula to solve a quadratic equation of the form [tex]ax^{2} +bx+c=0[/tex] is equal to

[tex]x=\frac{-b(+/-)\sqrt{b^{2}-4ac}} {2a}[/tex]

in this problem we have

[tex]2x^{2} -5x-9=0[/tex]  

so

[tex]a=2\\b=-5\\c=-9[/tex]

substitute in the formula

[tex]x=\frac{5(+/-)\sqrt{-5^{2}-4(2)(-9)}} {2(2)}[/tex]

[tex]x=\frac{5(+/-)\sqrt{97}} {4}[/tex]

[tex]x=\frac{5(+)\sqrt{97}} {4}[/tex]

[tex]x=\frac{5(-)\sqrt{97}} {4}[/tex]

Answer:

55.4 in³

Step-by-step explanation:

The volume (V) of a triangular prism is

V = area of triangular face × length

note the triangle is equilateral with area (A)

A = [tex]\frac{s^2\sqrt{3} }{4}[/tex] ( s is the length of the side )

   = [tex]\frac{16\sqrt{3} }{4}[/tex] = 4[tex]\sqrt{3}[/tex] in², hence

V = 4[tex]\sqrt{3}[/tex] × 8 = 32[tex]\sqrt{3}[/tex] ≈ 55.4 in³

I’m not quite sure what the function is but if it’s sine the period is pi/8

ANSWER

The period is

[tex] \frac{\pi}{8} [/tex]

EXPLANATION

The given function completes four cycles on the interval

[tex]

[ - \frac{\pi}{4} , \frac{\pi}{4} ][/tex]

Therefore the period is

[tex] = \frac{ \frac{\pi}{4} - \frac{\pi}{4} }{4} [/tex]

Simplify the numerator:

[tex] = \frac{ \frac{\pi}{2}}{4} [/tex]

Simplify further to get:

[tex]= \frac{\pi}{8} [/tex]

Answer:

a = 5.

Step-by-step explanation:

The graph here is a probability density graph. What will represent [tex]P(X\le a)[/tex] on the graph?

[tex]P(X\le a)[/tex] is the area

The area under the graph between 0 and 9 is a trapezoid. Consider the trapezoid in three slices from left to right:

The area of the leftmost triangle plus that of the rectangle is exactly 0.6. In other words, the area to the left of [tex]a = 5[/tex] between the graph and the x-axis is [tex]0.6[/tex]. [tex]P(X \le 5) = 0.6[/tex]. [tex]a = 5[/tex].

As a side note [tex]a = 5[/tex] shall be the only answer to this question since the area under the graph to the left of [tex]a[/tex] can only increase or stay constant but not decrease as the value of [tex]a[/tex] increases.

Answer:

d

Step-by-step explanation:

d is most likely

There are 3 trainers and 12 dogs. The correct answer would be an option (D).

The equation is defined as a mathematical statement that has a minimum of two terms containing variables or numbers that are equal.

Let's assume the number of trainers would be x and dogs would be y

As per the given information, we can write the system of equations as

x+y=15        .....(i)

2x+4y=54   .....(ii)

From equation (i),

x = 15 - y

From equation (ii), and solve for y

2(15-y)+4y=54

30-2y+4y=54

2y=24

y = 12

Substitute the value of y = 12 in equation (iii), and we get the value of x = 3

Hence, there are 3 trainers and 12 dogs.

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Answer:

[tex]\large\boxed{C.\ (2,\ -45^o)}[/tex]

Step-by-step explanation:

Regular coordinates (x, y)

Polar coordinates (r, φ)

[tex]r=\sqrt{x^2+y^2}\\\\\psi=\arctan\frac{y}{x}[/tex]

We have the point

[tex](\sqrt2,\ -\sqrt2)[/tex]

Substitute:

[tex]r=\sqrt{(\sqrt2)^2+(-\sqrt2)^2}=\sqrt{2+2}=\sqrt4=2\\\\\psi=\arctan\left(\frac{-\sqrt2}{\sqrt2}\right)=\arctan(-1)=-45^o[/tex]

Final answer:

The rectangular coordinates (√2, -√2) are converted to polar coordinates using formulas for radius and angle, resulting in polar coordinates (2, -45°), which matches Option C.

Explanation:

The question involves converting rectangular coordinates to polar coordinates. The given rectangular coordinates are (√2, -√2). To convert these to polar coordinates, we use the formulas r = √(x² + y²) and θ = arctan(y/x). First, calculate the radius r which is √((√2)² + (-√2)²) = √(2 + 2) = √4 = 2. Next, calculate the angle θ which is arctan((-√2)/(√2)) = arctan(-1). In degrees, arctan(-1) is -45°.

However, since polar coordinates represent angles positively in a counter-clockwise direction starting from the positive x-axis, we consider the equivalent positive angle which makes θ = 315°. However, to match the provided options more closely, we note that -45° is an equivalent way of expressing the direction taking the negative angle into consideration.

The correct polar coordinates are (2, -45°), which matches Option C.

Answer:

LOLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL

Step-by-step explanation:

Answer:

(n - 43)³

Step-by-step explanation:

Let the number be n.

Then the desired expression is (n - 43)³

Answer:

Algebraic expression = (x-43)^3

Step-by-step explanation:

Given , the verbal expression is the cube of the difference of a number and 43.

Let x be the number

Now, according to question

Difference means subtraction  in which a number is subtracted from the other number

Difference of a number and 43

It means 43 is subtracted from x

Now , the algebraic  expression =x-43

Cube of the difference of a number x and 43 =(x-43)(x-43)(x-43)

Cube of a number means multiply a number three times with self .

Therefore, Cube of the difference of a number x and 43=[tex](x-43)\times (x-43)\times(x-43)[/tex]

 Cube of the difference  of a number x and 43=(x-43)^3

Because in multiply  when base same then power of the number added.

I

Answer:

D) (4,5)

Step-by-step explanation:

Vertex E was dilated by 1/2 because 8/2 = 4. Thus, you will divide all the other coordinates by the same scale factor.

Answer:

The final temperature of the solution was 5°C

Step-by-step explanation:

we know that

1) At the start of an experiment, the temperature of a solution was -12°C

In this moment the temperature is -12°C

2) During the experiment, the temperature of the solution rose 5°C each hour during 4 hours

so

5°C*4=20°C

In this moment the temperature is -12°C+20°C=8°C

3) The temperature changed by -3°C

In this moment the temperature is 8°C-3°C=5°C

Answer:

The final temperature of the solution was 5°C

Step-by-step explanation:

Answer:

x = 88.2

Step-by-step explanation:

The angle at the top of the triangle = 90° - 10° = 80°

and the left side of the triangle is x ( opposite sides of a rectangle )

Using the tangent ratio in the right triangle

tan80° = [tex]\frac{opposite}{adjacent}[/tex] = [tex]\frac{500}{x}[/tex]

Multiply both sides by x

x × tan80° = 500 ( divide both sides by tan80° )

x = [tex]\frac{500}{tan80}[/tex] ≈ 88.2

Answer:

Third option.

Step-by-step explanation:

You need to remember that the formula used to calculate the arc lenght is:

[tex]arc\ length=r C[/tex]

Where "r" is the radius and "C" is the central angle in radians.

You need to solve for "C":

[tex]C=\frac{arc\ length}{r}[/tex]

You know the radius and the arc lenght, therefore, you can substitute values to calculate the central angle in radians. Therefore, this is:

[tex]C=\frac{8.0cm}{6.0cm}[/tex]

[tex]C=\frac{4}{3}radians[/tex]

Answer:

[tex]h=2.04\ miles[/tex]

Step-by-step explanation:

Let

h-----> the high of the balloon

we know that

In the right triangle of the figure

[tex]tan(63\°)=\frac{4}{h}[/tex]

[tex]h=\frac{4}{tan(63\°)}[/tex]

[tex]h=2.04\ miles[/tex]

Answer:

See explanation

Step-by-step explanation:

Zeroe of the function is such velue of x at which f(x)=0.

1. Consider the function [tex]f(x)=-x(3x-2)^2 (x+9)^5.[/tex]  

Zeros are:

[tex]-x(3x-2)^2(x+9)^5=0\\ \\x=0\text{ or }x=\dfrac{2}{3}\text{ or }x=-9.[/tex]

Zero [tex]x=0[/tex] has multiplicity of 1, zero [tex]x=\dfrac{2}{3}[/tex] has multiplicity of 2, zero [tex]x=-9[/tex] has multiplicity of 5.

At [tex]x=0[/tex] or [tex]x=-9[/tex] the graph of the function crosses the x-axis, at [tex]x=\dfrac{2}{3}[/tex] the graph of the function touches the x-axis.

2. Consider the function [tex]f(x)=x^3+10x^2+25x=x(x^2+10x+25)=x(x+5)^2.[/tex]  

Zeros are:

[tex]x(x+5)^2=0\\ \\x=0\text{ or }x=-5.[/tex]

Zero [tex]x=0[/tex] has multiplicity of 1, zero [tex]x=-5[/tex] has multiplicity of 2.

At [tex]x=0[/tex] the graph of the function crosses the x-axis, at [tex]x=-5[/tex] the graph of the function touches the x-axis.

Answer:

Step-by-step explanation:

The volume of a rect. prism is V = (length)(width)(height).

Here the volume is V = (4 in)(6 in)(5 in) = 120 in³ or 120 in^3

Note:  Please use " ^ " to indicate exponentiation:  120 in^3

Answer:

120 in 3

Step-by-step explanation:

The 3rd one. The number of 8 ounce glasses of milk in 1 gallon. If you read all the options carefully you can see that there’s no way that the other options could be “normally” distributed. The only on that could be “normal” is the 3rd one. Hope I helped!

Answer:the answer is 5

Answer:

answer is 5

Step-by-step explanation:

Given that the scatter plot shows the results of a survey in which 10 students were asked the number of hours they spent studying for a test and the score they earned.

The scatter plot shows

2 students got 100, 1 student between 90 and 100 and 2 students got 90.

Others got below 90

Hence no of students who scores 90 or above = 5

Answer: $72000

the answer is 256π option B

Answer:

[tex]V = 256\pi \, u^{3}[/tex]

Step-by-step explanation:

The volume of the cylinder is determined by the following formula:

[tex]V = \pi\cdot r^{2}\cdot h[/tex]

Where:

[tex]r[/tex] - Radius of the cylinder's base.

[tex]h[/tex] - Height of the cylinder.

The volume of the cylinder is:

[tex]V = \pi \cdot (8\,u)^{2}\cdot (4\,u)[/tex]

[tex]V = 256\pi \, u^{3}[/tex]