The length of each side of a square is with the area of 1600 square inches is increased by 20 inches what is the resulting area

Answer: m = 1, b = 7.

Step-by-step explanation:

First, you must convert 2 + 3x + 5 - 2x = y to slope-intercept form. (y = mx + b)

y = mx + b
y = x + 7

The slope (m) is 1

The y-intercept (b) is 7

413 bricks are needed to build a wall whose area is equivalent to 10.32 m².

Given is that a brick wall of dimension 4.3 m by 2.4 m is to be built. There are 40 bricks per m².

The area of the wall will be -

[A] = 4.3 x 2.4 = 10.32 m²

[A] = 10.32 m²

Now, for the area of 1 m², 40 bricks are needed. So, for 10.32 m², we can say that -

10.32 x 40

412.8

413 bricks.

Therefore, for the area of 10.32 m², to only 413 bricks are needed to build it.

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The average speed in mile per hour is  5.455  miles per hour    .

In the question ,

it is given that ,

the 1 mile distance is covered in 11 minutes ,

So , in 1 minute the distance covered is = 1/11 miles

So , the speed in miles per minute is 1/11 miles per minute .

To convert speed in miles per minute to miles per hour ,

we multiply it by 60 ,

So , we get ,

Average Speed in miles per hour  is = 60 * (1/11)

= 60/11

= 5.45454

≈ 5.455

Therefore , the average speed is 5.455  miles per hour .

The given question is incomplete , the complete question is

What is your average speed in mile per hour and in feet per second if you travel a mile in 11 minute ?

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Answer: thus, 65% of the books were fictional.

Step-by-step explanation:

Consequently, we deduce that the typical output voltage is under 130.

In statistics, the standard deviation measures how widely distributed or how much a bunch of values can vary. While a big standard deviation shows that values are scattered throughout a wider range, a low standard deviation suggests that values are frequently close to the set's mean.

Here,

A defined output voltage of 130 is offered for an electrical circuit.

The voltage for this circuit was measured 40 times independently, with a sample mean of 128.6 and a standard deviation of 2.1.

Let's say the average output voltage.

Null Hypothesis, [tex]H_{0}[/tex]:  = 130 denotes that the average output voltage is 130.

A Different Hypothesis, [tex]H_{A}[/tex] :  < 130  ( it indicates that the average output voltage is below 130.)

One-sample t test statistics will be employed in this case because we are unsure about the population standard deviation;

            T.S.  =   ~ [tex]\frac{X^{'}-u }{\frac{s}{\sqrt{n} } }[/tex]

In this case, sample mean output voltage equals 128.6

Sample standard deviation equals 2.1, or s.

n = 40 for the sample of independent readers.

Thus, test statistics equal.

So, test statistics  =    ~[tex]\frac{128.6-130 }{\frac{2.1}{\sqrt{40} } }[/tex]

                              =  -4.216

Now, for a left-tailed test with a 0.05 level of significance, the t table provides a critical value of -1.685 at 39 degrees of freedom. We have enough data to reject our null hypothesis because it will fall in the rejection region because our test statistics are smaller than the crucial value of t, which is -4.216 -1.685.

Consequently, we deduce that the typical output voltage is under 130.

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The equation of the line perpendicular to y=-3x-1 that passes through the point (3,4) is y = (1/3)x + 3.

What do you mean by equation of line?

The slope of the line and a point on the line can be used to create the equation of a line. The slope of the line, which can be stated as a numeric integer, fraction, or the tangent of the angle it forms with the positive x-axis, is the line's inclination with respect to the positive x-axis.

The slope of the line given in the question is: -3,

The slope of perpendicular line is negative reciprocal of the slope of other line,

So,

The slope of perpendicular line is: 1/3.

The equation of the line is generally in the form:

y = mx + c

So, the equation of line becomes:

y = (1/3)x + c

We have given that the line passes through the point (3,4).

Putting the values of x and y in the above equation to get the value of c.

4 = (1/3)(3) + c

c = 3

Putting the value of c in the equation,

y = (1/3)x + 3

This is the required equation.

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Answer:

(-3,-4) and (3,4).

Step-by-step explanation:

The mean yield for the three temperatures are equal.

Temperature is a unit used to express how hot or cold something is. It can be expressed using the Celsius or Fahrenheit scales, among others. Temperature shows which way heat energy will naturally flow, i.e., from a hotter (body with a higher temperature) to a colder (body with a lower temperature) (one at a lower temperature).

Given:

Number of groups [tex]k=3[/tex]

Number of observation in group 50°C, [tex]n_{1} =5[/tex]

Number of observation in group 60°C, [tex]n_{2} =5[/tex]

Number of observation in group 70°C, [tex]n_{3} =5[/tex]

Total number of observation [tex]N=15[/tex]

Null hypothesis [tex]H_{0}[/tex] :

The mean yield for the three temperatures are equal.

Alternative hypothesis [tex]H_{1}[/tex] : The mean yield for the three temperatures are not equal.

Now, Sum of the squares of between groups:

⇒∑  [tex]n_{i} (X_{i} -X)^{2}[/tex]

Here, [tex]X_{i}[/tex] is the mean of the jth group, [tex]X[/tex] is the overall mean and [tex]n_{1}[/tex] is the sample size of the jth group.

[tex]X=\frac{X_{1} +X_{2}+...+X_{i} }{i}[/tex]

[tex]X_{1} =34+24+36+39+32=165\\X_{2} =30=31+34+23+27=145\\X_{3} =23+28+28+30+31=140[/tex]

[tex]X_{i} =\frac{X_{i} }{n_{i} }[/tex]

⇒ [tex]X_{1} =\frac{165}{5} =33[/tex]

⇒ [tex]X_{2} =\frac{145}{5} =29[/tex]

⇒ [tex]X_{3} =\frac{140}{5} =30[/tex]

[tex]=n_{1} (X_{i} -X)^{2}+n_{2} (X_{2} -X)^{2}+n_{3} (X_{3} -X)^{2}[/tex]

SSB= 5 × [tex](33-30)^{2}[/tex]+ 5 × [tex](29-30)^{2}[/tex] + 5 × [tex](28-30)^{2}[/tex]

      = 5 × [tex]3^{2}[/tex] + 5 × [tex]1^{2}[/tex] + 5 × [tex]2^{2}[/tex]

      = 5 × 9 + 5 × 1 + 5 × 4

      = 45 + 5 + 20

      = 70

Now, Sum of difference between each element and grand mean squared (SST)

⇒ [tex](34-30)^{2} +(34-30)^{2}+(36-30)^{2} +...+(31-30)^{2}[/tex]

⇒ 306

Now, SSE = SST - SSB

= 306 - 70 = 236

Therefore,

Source of         Sum of                 Degrees of                   Mean

variation         squares             freedom                    square

Between      [tex]SSB=70[/tex]           [tex]df_{1} =k-1=2[/tex]            [tex]MSB=\frac{SSB}{df_{1} } =\frac{70}{2} =35[/tex]          

groups

Error    [tex]SSE=236[/tex]        [tex]df_{2} =N-k=12[/tex] [tex]MSE= \frac{SSE}{df_{2} } =\frac{236}{2} =19.67[/tex]                                          

Total                [tex]306[/tex]

Given .05 level of significance. i.e, [tex]F(0.05,2,12)=3.8853[/tex] tabulated value.

As [tex]1.7794 < 3.8853[/tex], tabulated value is higher than calculated value.

So, the null hypothesis is accepted.

Hence, the mean yield for the three temperatures is equal.

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The approximate total length of iron edging needed to create the square frame and the two diagonals is 43.5 inches. So, option A is correct.

When two vertices of a polygon or polyhedron are not on the same edge, a diagonal in geometry is a line segment that connects them. The term "diagonal" is a colloquial term for any sloping line.

The elements on the line connecting the top left corner and bottom right corner of a square matrix make up the diagonal of a matrix in matrix algebra.

Given, The diagonal of a square is 9 inches.

The side of the square be x as we have assumed.

We have to apply Pythagoras theorem for Δ ABC

AB²+BC²=AC²

x²+x²=9²

x=9/√2

We know that the diagonals in the square bisect each other.

OA = 4.5 inches=OC

OD = 4.5 inches=OB

Therefore, the total length of iron edge that is required for the window is:

=4×Length of square+2 diagonals

= 4×6.36+2×9=43.46 inches

The approximate total length of iron edging needed to create the square frame and the two diagonals is 43.5 inches.

Hence, option A is correct.

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Answer:

should be option 2, but im sorry if it isnt.

Step-by-step explanation:

Answer:

(c) 13 cm, 7 cm, 6 cm

Step-by-step explanation:

To determine which of the given sets of measurements cannot be the sides of a triangle, we can use the triangle inequality. The triangle inequality states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

In other words, if the sides of the triangle are a, b, and c, then a + b > c.

In answer choice c: 7 + 6 = 13, meaning that it does not satisfy the above inequality.

Therefore, the answer is (c) 13 cm, 7 cm, 6 cm

if this helps you, it would help me a lot if you could mark this as brainliest :)

The graph of Stewart's assignment added as an attachment

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

Number of assignments = 36

Rate of assignment per day = 4

This means that the equation of the scenario is

Number of assignments remaining = Initial number of assignments - Rate of assignment per day * Number of days

Represent the number of assignments remaining with y and the number of days with x

So, we have the following representation

y = Initial number of assignments - Rate of assignment per day * x

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

y = 36 - 4 * x

Evaluate the products

y = 36 - 4x

So, the equation is y = 36 - 4x and the graph is added as an attachment

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Answer:

x=- 5

Step-by-step explanation:

A vertical  line has an undefined slope (the rise is divided by zero, since there is no change in x.  x is always -5.

The required equation of the vertical line will be x = -5 which passes through the points (-5,-2).

The equation of a vertical line is of the form x = k, where k is the x-coordinate of the point through which the line passes.

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

We know that a vertical line's equation is x = k, where k is the x-coordinate of the point at which the line crosses.

As per the given question, the line passes through the point (-5, -2), so the equation of the line is x = -5.

A vertical line has an undefined slope (since there is no change in x, the rise is divided by zero). x is always a negative five number.

Therefore, the required equation of the vertical line will be x = -5.

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Answer:

x^2 - x +3, aka the last option

Step-by-step explanation: y intercept is positive in the table, and leaves us with the first option and the last option. plugging in 3 gets the right answer of 9 for the last choice and gives 15 for the first choice

Answer:

0.692

Step-by-step explanation:

Given: The computer that controls a bank's automatic teller machine crashes a mean of 0.5 times per day.

Mean per day = 0.5

Mean per 7 day = 7*0.5 = 3.5

X = Number of times the computer will crash

We have to find:

P( the computer will crash more than 1 time in any seven-day week) =?

P(x >1) = ?

P(X > 1) =1- P(X < or = 1)

P(X < or = 1) = 0.320847

P(X < or = 1) = 0.3208

P(x > 1) = 0.6792

So, the answer is 0.6792

In the given diagram, the measure of diagonal EG is 80

From the question, we are to determine the measure of diagonal EG

From the given diagram,

We have a parallelogram.

Since the diagonals of a parallelogram bisect each other, we can write that

EJ = JG

Thus,

4w + 4 = 2w + 22

Solve for w

4w - 2w = 22 - 4

2w = 18

w = 18/2

w = 9

But

EG = EJ + JG

EG = 4w + 4 + 2w + 22

Substitute the value of w

EG = 4(9) + 4 + 2(9) + 22

EG = 36 + 4 + 18 + 22

EG = 80

Hence, the measure of EG is 80

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A dealer bought some 9-volt batteries for $2,400. She sold them for $5,600, making a profit of $1.60 on each battery. 2,500 batteries were involved in the deal.

The dealer bought some 9-volt batteries for $2,400 and she sold them for $5,600. To find out how many batteries were involved, we need to calculate how much she made from each battery. We can do this by subtracting the cost of the batteries from the selling price and then dividing it by the number of batteries. Profit per battery = $5,600 - $2,400 = $3,200. Number of batteries = $3,200/$1.60 = 2,500. Therefore, the dealer bought 2,500 batteries.

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Answer:

infinite number of solutions

Step-by-step explanation:

5x - 4y = 10 → (1)

10x - 8y = 20 → (2)

multiply (1) by - 2

- 10x + 8y = - 20 → (3)

ad (2) and (3) term by term

0 + 0 = 0

0 = 0 ← true statement

this indicates the system has an infinite number of solutions

The area of the pyramid = 340 sq. inches

The area of a square pyramid= b^2+ 2.b.s

                                                  = 10^2 + 2*10*12

                                                  = 340 sq. inches

We can also find out the are by finding area of each sides

Area of square = 10*10 = 100 sq. inches

Area of triangle= 1/2( 12*10)

                           = 60

Area of four triangles = 4*60

                                    = 240

Total surface area = 240+100

                                = 340 sq. inches

Hence the total surface area is 340 sq. inches

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Answer:

around 8,000 or 100,000 is the answer

The 95% confidence interval estimate of the population mean is 98.10< μ < 98.30.

Given that

sample mean = ~x = 95.2

sample standard deviation = s = 0.7

sample size = n = 175

Degrees of freedom = d.f = n - 1 = 175 - 1 = 174

αt /2,d.f = 1.974.

The explanation for step 1

At 95% confidence level

 α= 1-0.95% =1-0.95 =0.05

α /2=0.05/ 2= 0.025

αt /2,d.f = t0.025,174 = 1.974

αt /2,d.f = 1.974

The explanation of Step 2

Margin of error = E = t/2,df x (s /Vn )    =  1.974 x ( 0.7/ V175)   = 0.10

Margin of error = E = 0.10

The 95% confidence interval estimate of the population mean is,

~x- E < μ < ~x + E

98.2 - 0.10< μ < 98.2 + 0.10

98.10< μ < 98.30

Margin of error = E = 0.10

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As a set of inputs with one output for each, a function is defined as a relationship between them.

An equation, rule, or law in mathematics that establishes the link between the independent variable and the dependent variable (the dependent variable). Mathematics is rife with functions, and the sciences depend on them for constructing physical relationships.

We have given, f(0) = 3 for all parts.

a) f(n+1) = -2 f(n)

For f(1) put f(n) = f(0), f(0+1) = -2 f(0)

f(1) = -2(3) = -6

For f(2) put f(n = 1), f(1+1) = -2 f(1)

f(2) = -2(-6) = 12

For f(3) put f(n) = f(2),  f(2+1) = -2 f(2)

f(3) = -2(12) = -24

For f(4) put n = 3, f(3+1) = -2 f(3)

f(4) = -2(-24) = 48

For f(5) put n = 4, f(4+1) = -2 f(4)

f(5) = -2(48) = -96

b) f(n+1) = 3, f(n) + 7

For f(1) put n = 0, f(0+1) = 3 f(0)+7

f(1) = 3(3)+7 = 16

For f(2) put n = 1, f(1+1) = 3 f(1)+7

f(2) = 3(16)+7 = 55

For f(3) put f{n}=f{2}, f(2+1)=3 f(2)+7

f(3) = 3(55) + 7 = 172

For f(4) put n = 3, f(3+1) = 3 f(3)+7

f(4) = 3(172)+7 = 523

For f(5) put n = 4, f(4+1) = 3 f(4)+7

f(5) = 3(523)+7 = 1576

c) f(n+1) = f(n)²-2 f(n)-2

For f(1) put n = 0, f(0+1) = f(0)²-2 f(0)-2

f(1) = 3²-2(3)-2 = 1

For f(2) put n = 1, f(1+1) = f(1)²-2 f(1)-2

f(2) = 1²-2(1)-2 = -3

For f(3) put n = 2, f(2+1) = f(2)² - 2 f(2) - 2

f(3) = -3² - 2(-3) - 2 = 13

For f(4) put n = 3, f(3+1) = f(3)² - 2 f(3) - 2

f(4) = 13² - 2(13) - 2 = 141

For f(5) put n=4, f(4+1) = f(4)² - 2 f(4) - 2

f(5) = 141² - 2(141) - 2 = 19597

d) f(n+1)[tex]=3^{\frac{f(n)}{3}}$[/tex]

For f(1) put n = 0, f(0+1) [tex]$=3^{\frac{f(0)}{3}}$[/tex]

f(1) = [tex]3^{\frac{3}{3}}[/tex] = 3

For f(2) put n = 1, f(1+1) [tex]=3^{\frac{f(1)}{3}}$[/tex]

f(2) [tex]=3^{\frac{3}{3}}[/tex] = 3

For f(3) put n = 2, f(2+1) [tex]=3^{\frac{f(2)}{3}}$[/tex]

f(3) [tex]=3^{\frac{3}{3}}[/tex] = 3

For f(4) put n = 3, f(3+1) [tex]$=3^{\frac{f(3)}{3}}$[/tex]

[tex]$f(3)=3^{\frac{3}{3}}=3$[/tex]

For f(4) put n = 3, f(3+1) = [tex]$3^{\frac{f(3)}{3}}$[/tex]

f(4) [tex]$=3^{\frac{3}{3}}[/tex] = 3

For f(5) put n = 4, f(4+1) = [tex]3^{\frac{f(4)}{3}}$[/tex]

f(5) [tex]=3^{\frac{3}{3}}[/tex] = 3

The complete question is:

Find f( 1 ) , f( 2 ) , f( 3 ) , f( 4 ) , and f( 5 ) if f(n) is defined recursively by f( 0 ) = 3 and for n = 0 , 1 , 2 ,... a) f(n + 1 ) = − 2 f(n) . b) f(n + 1 ) = 3 f(n) + 7. c) f(n + 1 ) = f(n) 2 − 2 f(n) − 2. d) f(n + 1 ) = 3 f( slader

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f(2)=4, f(3)=7, f(4)=19, f(5)=40

What is a function?

In mathematics, a function is an expression, rule, or law that establishes the relationship between an independent variable and a dependent variable (the dependent variable). In mathematics, functions exist everywhere, and they are crucial for constructing physical links in the sciences. German mathematician Peter Dirichlet initially provided the contemporary concept of function in 1837.

Given, the function is  f(n+1) = f(n) + 3f(n-1)

Calculating f(2) by substituting n=1 in the above equation.

f(2) = f(1) + 3f(0)

⇒f(2) = 1 + 3

⇒f(2) = 4

f(3) is calculated by substituting n=2

f(3) = f(2) + 3f(1)

⇒f(3) = 4 + 3

⇒f(3) = 7

f(4) is calculated by substituting n=3

f(4) = f(3) + 3f(2)

⇒f(4) = 7 + 12

⇒f(4) = 19

f(5) is calculated by substituting n=4

f(5) = f(4) + 3f(3)

⇒f(5) = 19 + 21

⇒f(5) = 40

f(2)=4, f(3)=7, f(4)=19, f(5)=40

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The following are the answer to each statement;

1. The book is $18

2. Giselle can earn up to $18 pulling weeds for her grandma.

3. Giselle's grandma pays her $1 for every 5 weeds she pulls

4. Giselle does not have to use any of her own money towards the purchase of the book when she pulls 90 weeds.

5. Giselle only needs to pull 18 weeds to not use any of her own money

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The angular velocity of the sphere at the bottom of the incline is 88.1 rad/s.

In physics, angular velocity or angular speed, also recognized as angular frequency vector, is a pseudovector portrayal of how fast the angular position or orientation of an object changes over time.

So, angular velocity will be:

mg(h sinθ) = 1/2 Iω² + 1/2mv²

I of solid sphere = 2/5 mr²

Hence,

mg(h sinθ) = 1/2 (2/5 mr²) ω² + 1/2 mv²

Now remove the mass:

g h sin θ = 1/5 r² ω² + 1/2 v²

ω = v/r, v = ωr

Now, calculate angular velocity as follows:
g h sin θ = 1/5 r² ω² + 1/2 (ωr)²

g h sin θ = (7/10) r² ω²

ω² = 10 g h sin θ/7 r²

ω = √10 g h sin θ/7 r²

  = √10 (9.8) (2.1) sin 25° / 7 (0.04)²

  = 88.1 rad/s

Therefore, the angular velocity of the sphere at the bottom of the incline is 88.1 rad/s.

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Complete question:
An 8.0 cm diameter, 400 g sphere is released from rest ta the tip of a 2.1 m long, 25-degree incline. It rolls, without slipping, to the bottom. What is the sphere's angular velocity at the bottom of the incline?

We obtain x1=10 and x2=10 by solving the one-step matrix equation.

A matrix is a set of numbers arranged in rows and columns. Learn about the dimensions and components of matrices by starting your first introduction. A rectangular grid of numbers organized into rows and columns is known as a matrix. Matrix A, as an illustration, has two rows and three columns.

An arrangement of numbers, occasionally expressions and symbols, in rows and columns is called a matrix. Quantum mechanics, optics, mathematics, and other mathematical functions are all solved using matrix formulas.

Matrices are very expensive items that are used in a variety of applications. Matrix applications are employed in a variety of scientific disciplines in addition to mathematics.

x1 + (-2) = 8

x2 +( 5) = 15

solve 1

x1 - 2 = 8

On both sides of the equation, add 2.

⇒ x1 -2 + 2 = 8 +2

⇒ x1 =8 +2

⇒ x1 = 10

Equation 2 resolution

x2 + 5 = 15

On both sides of the equation, deduct 5

⇒x2 + 5 -5 = 15 - 5

⇒ x2 = 15 - 5

⇒ x2 = 10

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The distance on the map where the scale is 1 inch to 40 miles is 1.75 inches.

First we know that the cities are at 2.8 inches apart on a map whose scale is 1 inch to 25 miles, then the real distance between the two cities is:

D = (2.8)*25mi = 70 mi

Now we move to a map where each inch is 40 miles, to find the distance in inches, we can take the quotient between the real distance and the scale, it gives:

d = (70mi/40mi) inches

d = 1.75 inches

The distance on the other map is 1.75 inches.

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The distance on the map where the scale is 1 inch to 40 miles is 1.75 inches.

First we know that the cities are at 2.8 inches apart on a map whose scale is 1 inch to 25 miles, then the real distance between the two cities is:

D = (2.8)*25mi = 70 mi

Now we move to a map where each inch is 40 miles, to find the distance in inches, we can take the quotient between the real distance and the scale, it gives:

D = (70mi/40mi) inches

D = 1.75 inches

Therefore, distance on the other map is 1.75 inches. Learn more about map scales:

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Answer:

Measures are SV=9 units., SY=14 units, YW= [tex]\sqrt{75units}[/tex], YW=[tex]\sqrt{27units}[/tex]

Step-by-step explanation:

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

Explanation:

The circumcenter is found by intersecting the perpendicular bisectors of the triangle.

Segments VY, WY, and XY are perpendicular bisectors of triangle STU.

A perpendicular bisector has two properties:

Perpendicular bisector WY divides side TU into equal pieces TW and WU

In short:

TW = WU

So

TW = 11

since WU is equal to 11 as well

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Focus on triangle TWY.

This is a right triangle with the sides of...

We'll use the pythagorean theorem to determine the unknown leg.

Plug in a = 11 and c = 14. Then let's solve for b.

[tex]a^2+b^2 = c^2\\\\11^2+b^2 = 14^2\\\\121+b^2 = 196\\\\b^2 = 196-121\\\\b^2 = 75\\\\b = \sqrt{75}\\\\b \approx 8.66025\\\\b \approx 8.7\\\\[/tex]

Therefore, segment YW is roughly 8.7 units.

Two or more figures are said to be similar when they have common properties. The value of MP is 18 units.

More than one figure are said to be similar if they have some relatable common properties, either considering their length of sides or measure of their internal angles. But similarity is not the same as congruence.

In the given question, it can be deduced that ΔMNP is similar to ΔQRP. Thus, relating the corresponding sides of the two triangles we have;

[tex]\frac{12}{30}[/tex] = [tex]\frac{5x-2}{11x+1}[/tex]

cross multiply to have

30(5x-2) = 12(11x+1)

150x - 60 = 132x + 12

collect like terms

150x - 132x = 12 + 60

18x = 72

x = 4

So that,

MP = 5x - 2

      = 5(4) - 2

      = 20 -2

MP = 18 units

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The average rate of change of the function at the interval -2 < x < 8 is 1.5

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

g(x) = 3/2x + 1

The interval is given as

-2 < x < 8

This interval can be represented as

x = -2 and x = 8

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

g(-2) = 3/2(-2) + 1 = -2

g(8) = 3/2(8) + 1 = 13

This means that

g(-2) = -2 and g(8) = 13

The average rate of change is then calculated as

Average rate of change = [g(8) - g(-2)]/[8 + 2]

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

Average rate of change = [13 + 2]/[8 + 2]

Evaluate the expression

Average rate of change = 1.5

Hence, the rate is 1.5

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At x = 0 and x = 2, the curve defined by y² = x³ - 3x² has a horizontal tangent.

We have,

First, let's find the derivative of y with respect to x.

We can rewrite the equation as:

y² = x³ - 3x²

y = (x³ - 3x²)^(1/2)

Now, differentiating y with respect to x using the chain rule:

dy/dx = (1/2) * (x³ - 3x²)^(-1/2) * (3x² - 6x)

Next, we set the derivative equal to 0 to find the values of x where the curve has a horizontal tangent:

(1/2) * (x³ - 3x²)^(-1/2) * (3x² - 6x) = 0

We can disregard the constant factor of (1/2) since it doesn't affect the solution.

Thus, we have:

(x³ - 3x²)^(-1/2) * (3x² - 6x) = 0

To have a horizontal tangent, the slope of the curve should be 0, which means the derivative is equal to 0.

Now, we have two factors to consider:

(x³ - 3x²)^(-1/2) = 0, which means the expression inside the parentheses is equal to 0. However, this does not yield a real solution.

(3x² - 6x) = 0, which means the quadratic term equals 0.

Solving this quadratic equation:

3x² - 6x = 0

3x(x - 2) = 0

This equation has two solutions: x = 0 and x = 2.

Therefore,

At x = 0 and x = 2, the curve defined by y² = x³ - 3x² has a horizontal tangent.

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