Determine the point(s), if any, at which the graph of the function has a tangent line with the given slope. Function y = x^2 + 2x Slope m = −8

The length of the curve will be given by the definite integral

[tex]\displaystyle \int_0^1 \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt[/tex]

From the given parametric equations, we get derivatives

[tex]x(t) = 3t^2 + 5 \implies \dfrac{dx}{dt} = 6t[/tex]

[tex]y(t) = 2t^3 + 5 \implies \dfrac{dy}{dt} = 6t^2[/tex]

Then the arc length integral becomes

[tex]\displaystyle \int_0^1 \sqrt{\left(6t\right)^2 + \left(6t^2\right)^2} \, dt = \int_0^1 \sqrt{36t^2 + 36t^4} \, dt \\\\ = \int_0^1 6|t| \sqrt{1 + t^2} \, dt[/tex]

Since 0 ≤ t ≤ 1, we have |t| = t, so

[tex]\displaystyle \int_0^1 6|t| \sqrt{1 + t^2} \, dt = 6 \int_0^1 t \sqrt{1 + t^2} \, dt[/tex]

For the remaining integral, substitute [tex]u = 1 + t^2[/tex] and [tex]du = 2t \, dt[/tex]. Then

[tex]\displaystyle 6 \int_0^1 t \sqrt{1 + t^2} \, dt = 3 \int_1^2 \sqrt{u} \, du \\\\ = 3\times \frac23 u^{3/2} \bigg|_{u=1}^2 \\\\ = 2 \left(2^{3/2} - 1^{3/2}\right) = 2^{5/2} - 2 = \boxed{4\sqrt2-2}[/tex]

An equation is formed of two equal expressions. The original height of the water in the pool in feet is 5.

An equation is formed when two equal expressions are equated together with the help of an equal sign '='.

Given the equation, 4.75 = x + (-0.25) can be used to find x, the original height of the water in a pool. Therefore, the height of the fool in the beginning is,

4.75 = x + (-0.25)

4.75 + 0.25 = x

x = 5

Hence, the original height of the water in the pool in feet is 5.

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The average rates of change of f(x) and their corresponding intervals are given as:

Interval          Rate of Change

[-5, -1]                 -8

[-4, -1]                 -7

[-3, 1]                  -4

[-2, 1]                  -3.

Step 1 - See Table Attached

Step 2 - State the formula for rate of change

The formula for rate of change is given as:
= [change in f(x)] / [change in x]

a) For interval [5, -1] ⇒

Rate of Change - [ f(1) - f(-5) ] / [-1 - (-5)]  

= [-1 - 35] / [-1+5]

= -36 / 4

= - 8

b) For interval [-4, -1] ⇒

rate of change = [ f(-1) - f(-4) ] / [ -1 - (-4) ]

= [3 - 24] / [-1 + 4]

= -21/3

= - 7

c) interval [-3,1] ⇒

rate of change = [ f(1) - f(-3) ] / [ 1 - (-3) ]

= [-1 - 15] / [1 + 3]

= -16/4

= - 4

d) interval [-2,1] ⇒

rate of change = [f (1) - f(-2)] / [1 - (-2)]

= [ -1 - 8] / [1 + 2]

= -9/3

= -3

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Answer + Step-by-step explanation:

Question 4 :

Construct the angle 0MN :

1) Draw a segment MO

2) Put the protractor so that it is in line with the segment MO

At the same time ,line up the points M with the little circle of the protractor.

3) Spot 40° on the protractor ,then draw a point.

4) Use the ruler to connect point M through that point

Construct the angle MON :

1) Put the protractor so that it is in line with the segment MO

At the same time ,line up the points O with the little circle of the protractor

2) Spot 70° on the protractor ,then draw a point.

3) Use the ruler to connect point O through that point.

Construct the point N :

N is the point of intersection of the lines that go (respectively) through M and O.

Question 6 :

1) Draw a segment XZ such that XZ = 2.

2) construct the angle ZXY = 25°.   (follow the steps described in the Previous question).

NOTE : ∠Z = 180 - (∠X + ∠Y) = 180 - (25 + 45) = 110°

3) construct the angle XZY = 110° .  (follow the steps described in the Previous question)

4) Y is the point of intersection of the lines that go (respectively) through X and Z.

The conversion of 120 sq. inches is equivalent to 0.09264 sq. yd.

Conversion is the process of changing the value of one form to another for example inches to millimeters, or liters to gallons.

Here, we know that,

    1 square inches = 0.000772 sq yd

we have 120 sq. inches

so, 120 sq. inches = 120 X 0.000772 sq yd

                              = 0.09264 sq. yd.

Thus, the conversion of 120 sq. inches is equivalent to 0.09264 sq. yd.

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Answer:

z = 1/4

Step-by-step explanation:

See attached image

Answer:

z = 5

Step-by-step explanation:

simplify

5/(z-25) first

turning it into

((0 - 3/z) + 7/4z) - 5/(z - 25) = 0

then simplify 7/4z

((0 - 3/z) + 7/4z) - 5/(z-25) = 0

when the fractions denominator is 0 then the numerator must be 0

turning the equation into

-25  *  (z - 5)  = 0

solve

-25   =  0

something that is not zero cannot equal zero.

z - 5 = 0

5 - 5 = 0

z = 5

hope this helps:)

Answer: (8, -4)

Step-by-step explanation:

-5x - 3y = -28
  x + 2y = 0

1. Multiply both sides of the bottom system by 5 to cancel the x out

   5(x+2y=0)

   5x + 10y = 0

2. rewrite

   -5x - 3y = -28

    5x + 10y = 0

3. add

   0x + 7y = -28

4. divide by 7

   7y = -28

5. y = -4

6. plug in -4 for y in one of the original equations

   x + 2(-4) = 0

7. simplify

   x = 8

9. solution is

   (8, -4)

The measures of BC will be 100°. Option D is correct.

It is a point locus drawn equidistant from the center. The radius of the circle is the distance from the center to the circumference.

The angle at the center is twice as large as the angle at the perimeter.

BC = 2 ×∠CDB

BC=2×50°

BC= 100°

Hence option D is correct.

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Answer:

A and B are arithmetic sequences. they have a set and consistent increase/decrease in each term.

Step-by-step explanation:

Answer:

Q = 40.6°

Explanation:

Given three sides: 9.6, 8.1, 6.3

Use the cosine rule:

c² = a² + b² - 2ab cos(C)

Insert following variables:

6.3² = 9.6² + 8.1² - 2(9.6)(8.1) cos(Q)

39.69 = 157.77 - 155.52 cos(Q)

cos(Q) = -118.08/-155.52

cos(Q) = 41/54

Q = cos⁻¹(41/54) = 40.6°

Answer:

360 degrees

..............

[tex]\begin{array}{ccll} feet&seconds\\ \cline{1-2} 19 & 1\\ 600& x \end{array} \implies \cfrac{19}{600}~~=~~\cfrac{1}{x} \\\\\\ 19x=600\implies x=\cfrac{600}{19}\implies \stackrel{\textit{about half a minute}}{x\approx 31.58~seconds}[/tex]

Answer:

  10/19 min ≈ 0.5263 min

Step-by-step explanation:

The woman's sprinting speed is given in feet per second, and we are asked for it in minutes per 600 feet. To find the time, we can use the relation ...

  time = distance / speed

Using the given numbers will give a time in seconds, so we need to do a units conversion to find the answer in minutes.

__

  time = distance/speed

  time = (600 ft) / (19 ft/s) = (600/19) s

Converting the units gives ...

  time = (600/19) s × (1 min)/(60 s) = (600·1)/(19·60) min

The time it takes the woman to sprint 600 feet will be ...

  time = 600/(19·60) min = 10/19 min ≈ 0.5263 min

Answer:

[tex]0.95*10^8[/tex]

Step-by-step explanation:

Hello!

Rules for scientific notation format:

[tex]\bold{0.95*10^8}[/tex]

There is a multiplication to a power of 10, but the other factor is less than 1.

This is NOT in Scientific Notation.

All the other options have a multiplication operation to a power of 10, and all the other factors are above 1 and less than 10.

We are given the equation for the height of the arrow. If you graph it, you see that it's a parabola and that the arrow kinda peaks and then falls back down. Another way of thinking about this problem is that you're looking for the time when the height is 0. You can see on the graph that there are two times that h=0. The first is obviously at t=0, when the arrow hasn't left the ground yet. The second is what we're looking for, when the arrow reaches the ground.

To solve this, let's set h=0. So 0=40t-5t^2. If you factor this, you get 5t(8-t) = 0. Continuing that leads to 5t=0 where t=0 which we already knew, and 8-t=0 where t=8. So that second time is when the arrow is back on the ground. Therefore your answer is 8 sec.

The correct answer is 6 / 7 as it is closer to 1.

The fraction is defined as the division of the whole part into an equal number of parts.

The number near 1 will be calculated as:-

7/ 6 = 1.1666

6 / 7 = 0.8571

Difference of the first ratio from one.

1.1666 - 1  = 0.1666

Difference of the second ratio:-

1 - 0.8571 = 0.1429

So the difference of 6 / 7 is lower than the difference of the fraction 7 / 6 from one.

Therefore the correct answer is 6 / 7 as it is closer to 1.

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Step-by-step explanation:

x = number of bottles Paula sold

y = number of bottles Pedro sold

x + y = 72

x = 72 - y

x/y = 3/5

(72 - y)/y = 3/5

5(72 - y) = 3y

360 - 5y = 3y

360 = 8y

y = 45

x + y = 72

x + 45 = 72

x = 27

so,

Paula sold 27 water bottles, and Pedro sold 45.

Answer:

Angle 1 = 90 degree

Angle 2 = 55 degree

Angle 3 = 35 degree

Step-by-step explanation:

A rhombus has four equal sides. Since the intersection angle of a rhombus is always 90 degree, angle 1 is 90 degree

Angle 3 = 180 degree - 90 degree - 55 degree = 35 degree

Angle 2 = 180 degree - 90 degree - 35 degree = 55 degree

The number of each type of coin that Joan has are; 16 dimes and 3 quarters

We are told that Joan has a total of 19 coins.

Now, the worth of the coins is $2.35

Let dimes be d and let quarters be q. Thus;

10d + 25q = 235

d + q = 19

Substitute d = 19 - q in the 1st equation to get;

10(19 - q) + 25q = 235

190 - 10q + 25q = 235

15q = 45

q = 45/15

q = 3

Thus;

d = 19 - 3

d = 16

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Answer:

See below ~

Step-by-step explanation:

⇒ Number of years since = 0 ⇒ t = 0

⇒ Apply in the formula

⇒ n(0) = 73e^(0.02 × 0)

⇒ n(0) = 73e⁰

⇒ n(0) = 73,000,000 rats

=============================================================

⇒ Number of years since 1993 : 2009 - 1993 = 16 ⇒ t = 16

⇒ Applying in the formula

⇒ n(16) = 73e^(0.02 × 16)

⇒ n(16) = 73e^(0.32)

⇒ n(16) = 73 × 1.37712776 × 10⁶

⇒ n(16) = 100.530326 x 10⁶

⇒ n(16) = 100,530,326 rats

Answer:

Given function:  

[tex]\large \text{$ n(t)=73e^{0.02t} $}[/tex]

where:

To calculate the rat population in 1993, substitute [tex]t = 0[/tex] into the function:

[tex]\large \begin{aligned}\implies n(0) & =73e^{0.02(0)}\\& = 73 \cdot 1\\& = 73 \sf \: million\end{aligned}[/tex]

To calculate a prediction of the rat population in 2009, first determine the value of t by subtracting the initial year of 1993 from the given year of 2009:

[tex]\large \text{$ \implies t=2009-1993=16 $}[/tex]

Substituting the found value of t into the function to find the predicted number of rats in 2009:

[tex]\large \begin{aligned}\implies n(16) & =73e^{0.02(16)}\\& = 73e^{0.32}\\& = 100.5303268...\\ & = 100.5 \sf \: million\:(1\:dp)\end{aligned}[/tex]

Answer:

Bonds

Step-by-step explanation:

A bond is a fixed-income instrument that represents a loan made by an investor to a borrower (typically corporate or governmental).

Answer: 674 ft

Step-by-step explanation:

[tex]\tan 12^{\circ}=\frac{y}{1315} \\ \\ 1315\tan 12^{\circ}=y[/tex]

[tex]\tan 8^{\circ}=\frac{y}{x+1315} \\ \\ (x+1315)\tan 8^{\circ}=y \\ \\ (x+1315)\tan 8^{\circ}=1315 \tan12^{\circ} \\ \\ x \tan 8^{\circ}+1315 \tan 8^{\circ}=1315 \tan 12^{\circ} \\ \\ x \tan 8^{\circ}=1315 \tan 12^{\circ}-1315 \tan 8^{\circ} \\ \\ x=\frac{1315 \tan 12^{\circ}-1315 \tan 8^{\circ}}{\tan 8^{\circ}} \approx \boxed{674 \text{ ft}}[/tex]

The perimeter of Rosalie's new rectangular mural is given by P = 21 feet

The perimeter P of a rectangle is given by the formula, P=2 ( L + W) , where L is the length and W is the width of the rectangle.

Perimeter P of rectangle = 2 ( Length + Width )

Given data ,

Let the perimeter of the rectangular mural be represented as P

Now , the equation will be

Let the initial length of the rectangular mural be L = 3.5 feet

Let the initial width of the rectangular mural be W = 5.5 feet

Now , the new mural is having an extra length of 1.5 feet

So , the new length of the rectangular mural is L₁ = 3.5 + 1.5 = 5 feet

Now , Perimeter P of rectangle = 2 ( Length + Width )

Substituting the values in the equation , we get

Perimeter of rectangular mural = 2 ( 5 + 5.5 )

On simplifying the equation , we get

Perimeter of rectangular mural P = 2 x 10.5

Perimeter of rectangular mural P = 21 feet

Hence , the perimeter of rectangular mural is 21 feet

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Answer:

1400

Step-by-step explanation:

∆ADB is ~ ∆BEC because <D and <E are right angles which are equal to 90°.

A right angle is defined as an angle that is formed when two straight line are perpendicular at an intersection creating a 90° angle when measured.

From the diagram, ∆ADB is ~ ∆BEC because <D and <E are right angles which are equal to 90°.

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Answer:

  1 plane

Step-by-step explanation:

In Euclidean geometry, three non-collinear points will define exactly one plane.

__

Two points will define a line. That line can exist in an infinity of different planes.

A third point not on the line can only lie in exactly one plane with that line.

Three non-collinear points in Euclidean geometry can lie on one unique plane.

In Euclidean geometry, any three non-collinear points (points not lying on the same line) uniquely determine a plane.

This means that there is only one plane that contains all three of those points.

So, given three non-collinear points, you can find exactly one plane that passes through all of them.

Hence, if the three points not on the same line can lie on one plane.

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Answer:

See below ~

Step-by-step explanation:

Classifying the triangles by sides and angles :

Triangle ABC

⇒ By Sides : Scalene (All sides are unequal)

⇒ By Angles : Right (There is a right angle = 90°)

============================================================

Triangle GHI

⇒ By Sides : Isosceles (2 sides are equal)

⇒ By Angles : Obtuse (One angle is greater than 90°)

============================================================

Triangle DEF

⇒ By Sides : Equilateral (All sides are equal)

⇒ By Angles : Acute (All angles are less than 90°)

Answer:

6 cm²

Step-by-step explanation:

The formula to find the area of the parallelogram is :

Area = Base × Height

Given that,

Base ⇒ 3cm

Height ⇒ 2 cm

Let us find now.

Area = Base × Height

Area = 3cm × 2cm

Area = 6 cm²