After a rain storm, Lily measures the depth of several puddles in her backyard.
How much deeper was the puddle under the swing than the puddle on the sidewalk?

(i) The coordinates of the point Q is [tex](x, y) = (21, 0)[/tex].

(ii) The equation of the curve is [tex]y = 6\cdot \ln (2\cdot x + 1)-1.592[/tex].

(iii) The rate of increase of the y-coordinate is 1.2 units per second.

(i) By analytical geometry, we know that the slope of a line normal of a curve ([tex]m_{\perp}[/tex]) by the following expression:

[tex]m_{\perp} = - \frac{1}{\frac{dy}{dx} }[/tex] (1)

Where [tex]\frac{dy}{dx}[/tex] is the slope of the line tangent to the curve.

If we know that [tex]x = 1[/tex], then the slope of the normal to the curve is:

[tex]\frac{dy}{dx} = \frac{12}{2\cdot (1)+1}[/tex]

[tex]\frac{dy}{dx} = 4[/tex]

And the slope of a line normal of a curve is:

[tex]m_{\perp} = -\frac{1}{4}[/tex]

By definition of secant curve we calculate the remaining coordinate of the point Q:

[tex]m_{\perp} = \frac{y_{Q}-y_{P}}{x_{Q}-x_{P}}[/tex] (2)

Where:

If we know that [tex](x_{P}, y_{P}) = (1, 5)[/tex], [tex](x_{Q}, y_{Q}) = (x, 0)[/tex] and [tex]m_{\perp} = -\frac{1}{4}[/tex], then the remaining coordinate of point Q is:

[tex]-\frac{1}{4} = \frac{0-5}{x-1}[/tex]

[tex]-\frac{1}{4}\cdot (x-1) = -5[/tex]

[tex]-\frac{1}{4}\cdot x +\frac{1}{4} = -5[/tex]

[tex]\frac{1}{4}\cdot x = \frac{21}{4}[/tex]

[tex]x = 21[/tex]

The coordinates of the point Q is [tex](x, y) = (21, 0)[/tex].

(ii) The equation of the curve is derived by separating the variables and integrating the resulting expression:

[tex]dy = \frac{12}{2\cdot x + 1}\,dx[/tex]

[tex]\int\, dy = 6\int {\frac{2\,dx}{2\cdot x + 1} }[/tex]

[tex]\int \, dy = 6\int {\frac{du}{u} }[/tex]

[tex]y = 6\cdot \ln u[/tex]

[tex]y = 6\cdot \ln (2\cdot x + 1)+C[/tex]

And the integration constant is found for [tex](x,y) = (1, 5)[/tex]:

[tex]5 = 6\cdot \ln(2\cdot 1 + 1) + C[/tex]

[tex]C = -1.592[/tex]

The equation of the curve is [tex]y = 6\cdot \ln (2\cdot x + 1)-1.592[/tex].

(iii) The rate of increase of the y-coordinate by using the differential equation presented in statement and the definition of rate of change:

[tex]\frac{dy}{dt} = \left(\frac{12}{2\cdot x +1}\right)\cdot \frac{dx}{dt}[/tex] (3)

If we know that [tex]\frac{dx}{dt} = 0.3[/tex] and [tex]x = 1[/tex], then the rate of increase of the y-coordinate is:

[tex]\frac{dy}{dt} = \left[\frac{12}{2\cdot (1)+1} \right]\cdot (0.3)[/tex]

[tex]\frac{dy}{dt} = 1.2[/tex]

The rate of increase of the y-coordinate is 1.2 units per second.

To learn more on differential equations, we kindly invite to check this verified question: https://brainly.com/question/25731911

i). The coordinates of Q is equal to:

[tex](x,y) = (21, 0)[/tex]

ii). The equation of the curve is:

[tex]y = 6. In(2.x + 1) - 1.592[/tex]

iii). The rate of increase for y-coordinate when x = 1:

[tex]1.2 units[/tex] per second

i). Using analytical geometry,

[tex]xp[/tex] and [tex]xq[/tex] are the P and Q's x-coordinates.

[tex]yp[/tex] and [tex]yq[/tex] are the P and Q's y-coordinates.

Since ([tex]xp[/tex], [tex]yp[/tex]) = (1,5) and ( [tex]xq[/tex], [tex]yq[/tex]) = [tex](x, 0)[/tex] and m⊥ = - 1/4, the left coordinate of Q's point:-

-1/4 = (0-5)/(x - 1)

∵[tex]x = 21[/tex]

Thus, the coordinates of y = (21, 0)

ii). Through isolating the variables and employing integration:-

[tex]dy = \frac{12}{(2.x + 1)} dx[/tex]

by integration and integration constant (1,5), the equation is:

[tex]y = 6. In(2.x + 1) - 1.592[/tex]

iii). The rate of increase for y-coordinate:

[tex]dy/dt[/tex] = [tex]\frac{(12)}{(2.1 + 1)} .0.3[/tex]

∵ [tex]dy/dt[/tex] [tex]= 1.2[/tex]

Thus, the rate of increase for y-coordinate when x = 1 is 1.2 units per second.

Learn more about "Co-ordinates" here:

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Answer: 81x + 31y

Step-by-step explanation:

Answer:

are you missing a option because for me one option that has the same exact numbers that you mentioned is the cost of 8 hats is 60

Step-by-step explanation:

Answer:

−72,−2.8,−54, 43, 1.3

Step-by-step explanation:

Answer:

3,094,240

Step-by-step explanation:

(9.32 x 10^3) (3.32 x 10^2)

(9.32 x 1000) (3.32 x 100)

(9320) (332) = 3094240

Answer: the variable is p=6 .

Step-by-step explanation:

Isolate the variable by dividing each side by factors that don't contain the variable

Answer:

p = 6

Step-by-step explanation:

Step 1: Simplify both sides.

3 + 2p − 4 = 11

3 + 2p + −4 = 11

(2p) + (3+−4) = 11 (Combine Like Terms)

2p + −1 = 11

2p − 1 = 11

Step 2: Add 1 to both sides.

2p − 1 + 1 = 11 + 1

2p = 12

Step 3: Divide both sides by 2.

2p/2 = 12/2

p=6

Answer:

Just use a calcultor...

Step-by-step explanation:

Answer:

quince

Step-by-step explanation:

Answer:

4.5

Step-by-step explanation:

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

Factorise x

[tex]x(x - 5) = 0[/tex]

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Hint :

[tex]a \times b = 0[/tex]

[tex]a = 0 \: \: \: \: or \: \: \: \: b = 0[/tex]

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Thus ;

[tex]x = 0 \: \: \: \: or \: \: \: \: x - 5 = 0[/tex]

So :

[tex] x = 0 \: \: \: \: or \: \: \: \: x = 5[/tex]

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CHECK :

[tex]x = 0[/tex]

[tex] {0}^{2} - 5(0) = 0[/tex]

[tex]0 - 0 = 0[/tex]

[tex]0 = 0[/tex]

The solution is correct.

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[tex]x = 5[/tex]

[tex] {5}^{2} - 5(5) = 0[/tex]

[tex]25 - 25 = 0[/tex]

[tex]0 = 0[/tex]

The solution is correct.

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Answer:

m= e= mc square

Step-by-step explanation:

Answer:

C- -24x - 2.75

Step-by-step explanation:

Answer:

B answer is correct hiiii

Answer: 50

Step-by-step explanation:

Answer:

A defensive way is use all your might to jump when the ball is coming at you, because as you might know is it takes a lot of effert to jump in a pool, the gravity is very strong in the pool! Or if you get the ball you can throw and swing your arm as strong as you can a HIT that ball! You wanna hit it hard so the players on the other side of the pool don’t get the ball,instead it will go over them and you WIN!

Step-by-step explanation:

Hope this helped :)✨

Answer:

$568.16

Step-by-step explanation:

Answer: The answer to your problem is $568.16

Answer:

Step-by-step explanation:

Step 1: Write the equation in the correct form. In this case, we need to set the equation equal to zero with the terms written in descending order.

Step 2: Use a factoring strategies to factor the problem.

Step 3: Use the Zero Product Property and set each factor containing a variable equal to zero.

Answer:

Standard Form:

720.058

Expanded forms can be written like a sentence or stacked for readability as they are here.

Expanded Notation Form:

 720.058 =

 700  

+ 20  

+ 0  

+   0.0

+   0.05

+   0.008

Expanded Factors Form:

   720.058 =

 7 × 100  

+ 2 × 10  

+ 0 × 1  

+ 0 ×   0.1

+ 5 ×   0.01

+ 8 ×   0.001

Expanded Exponential Form:

 720.058 =

7 × 102

+ 2 × 101

+ 0 × 100

+ 0 × 10-1

+ 5 × 10-2

+ 8 × 10-3

Word Form:

720.058 =

seven hundred twenty and fifty-eight thousandths

Step-by-step explanation:

Answer:

700 + 20 + 0 + 0.050 + 0.008

Step-by-step explanation:

Answer:

D

Step-by-step explanation:

it goes the next and last way and its the next colour

Answer:

y=6x-2

Step-by-step explanation:

The equation of the line is y=mx+b

FINDING SLOPE: To find the slope of a given line use the equation [tex]\frac{y2-y1}{x2-x1}[/tex]

So x1=-3, y1=-20 and x2=1, y2=4

Plug those numbers into the equation: [tex]\frac{4-(-20)}{1-(-3)}[/tex]  and we get the slope which is 6.

FINDING Y-INTERCEPT: Then we find the y intercept by plugging in our points (you can choose) I will use (1,4) and solve.

y=6x+b

4=6(1)+b

b=-2

Answer:

The equation of Nolan's line is [tex]y=2x+3[/tex]

Step-by-step explanation:

The slope-intercept form of a line is given by:

[tex]y=mx+b[/tex]

Here

m is the slope of the line

and b is the y-intercept.

Given

y-intercept = b = 3

slope = m = 2

The values of m and b can be put into the equation to find the line's equation

So,

Putting the value

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

Hence,

The equation of Nolan's line is [tex]y=2x+3[/tex]

Answer:

The answer is y 2x + 3.

Step by step:

Because i know

Answer:

1. $539

2. $300

3. C

Step-by-step explanation:

May I have brainliest?

Answer:

bruh 4 days duh

Step-by-step explanation:

Answer:

Then he has 15 months?

Step-by-step explanation:

you might need to explain your question

Answer: the answers are the second one and the last one

Step-by-step explanation:

Answer:

PQ=30

Step-by-step explanation:

Knowing that the segment has 3 parts that are 3:1:1 and sum up to 50, you could find how much 1 would equal by finding the total in relation to the ratio (doing 3+1+1) which gets you 5 parts. Then you can divide the total (50) by 5 to figure out how much one part would equal (10). Now, since you know that 1 part is equal to 10 and that PQ is 3 parts, you can find that PQ is equal to 3x10, or 30.

Answer:

The answer to your question is 24cm.

Step-by-step explanation:

6 multiplyed by 8 equals 48, and 48 divided by 2 equals 24.

Hope I helped. Have a spectacular day! ;)

Answer:

C. There is one solution, because the coefficients of the variables on both sides of the equation are different.

Step-by-step explanation:

The solution is m = 1

~theLocoCoco

the answer is the same

The value of x is (2w - t)/5.

The value of w is (5x + t)/2.

The value of t is  2w - 5x.

An expression is a way of writing a statement with more than two variables or numbers with operations such as addition, subtraction, multiplication, and division.

Example: 2 + 3x + 4y = 7 is an expression.

We have,

5x - 2w + t = 0

Solve for x.

5x - 2w + t = 0

5x = 2w - t

x = (2w - t)/5

Solve for w.

5x - 2w + t = 0

5x + t = 2w

w = (5x + t)/2

Solve for t.

5x - 2w + t = 0

t = 2w - 5x

Thus,

The value of x, w, and t is given above.

Learn more about expressions here:

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