Please help in math 75 points

The true statement is all irrational and rational numbers have decimal expansions. (second option)

A rational number is a number that can be expressed as a fraction of two integers . Rational numbers have terminating decimals and they have decimal expansions. Examples of rational numbers are 1.5, 2 and 4.25

A irrational number is a number that cannot be expressed as a fraction of two integers . Irrational numbers have repeating decimals. Examples of irrational numbers are 22/7, 1/3.

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Answer:

mean 36.2 median is 33

Step-by-step explanation:

For median you put the numbers in least to greatest and then find the middle which in this case is 33 and 34 so 33 is in between those two numbers so that would be the median. For the mean you add all the number together which is 362 and then divide by the number of numbers that there is which is 10

The value of n will be 12.

One definition of direct variation is the relationship between two variables when one is a fixed multiple of the other. They are said to be in proportion, for instance, when one variable affects the other. The equation has the form b = ka if b is directly proportional to a. (where k is a constant)

The direct variation includes the points (-6, 18) and (-4, n).

We know that, direct variation = b= ka

where, a is the x coordinate, b is the y coordinate and k is the constant.

So, For (-6, 18),

we have, 18 = -6k

So, k = -3

Now, For (-4, n),

we have, n = -4k

               n  =  -4×-3 ( putting k= -3)

                   = 12

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Answer:

3/5

Step-by-step explanation:

The greatest common factor (GCF) of the numerator (15) and the denominator (25) is 5

GCF(15,25) = 5

15/25=

15 ÷ 25 ÷ 25 ÷ 5= 3/5

The graph representing the inequality 2.5 is less than the product of −0.5 and a number is number line with open circle on point negative 5 and arrow shaded to the left.

Inequalities are defined as those expressions which are connected by inequality signs like >, <, ≤, ≥ and ≠.

Given inequality is,

2.5 is less than the product of −0.5 and a number.

2.5 < -0.5x

We have to isolate x on one side.

Dividing both sides by -0.5,

-5 > x

The inequality sign changes when a negative number is divided.

Or x < -5

So the solutions of the inequality are all the numbers less than -5.

So the graph of the inequality will be a number line where an open circle is at -5, since the point -5 is not included.

Since the solutions are less than -5, arrow is shaded to the left.

Hence the second option is the correct one.

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The first step is to recall the equation for an arithmetic sequence: an = a1 + (n - 1)d, where a1 is the first term, an is the nth term, and d is the common difference.

Given the sum of three consecutive terms of the sequence, we know that a3 + a2 + a1 = 6. We can also recall that the product of the terms is a1 x a2 x a3 = -42.

Now, using the two equations, we can solve for the three terms:
a3 = (6 - a2 - a1)/2,
a2 = 6 - a1 - 2a3,
a1 = 6 - a2 - 2a3

Substituting the second and third equation into the first, we get:
a3 = -14/2 = -7

Therefore, the three consecutive terms of the arithmetic sequence are: a1 = -5, a2 = 2, a3 = -7.

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According to Descartes' Rule of Signs, the possible number of positive real zeros for a polynomial function is determined by the number of sign changes in the coefficients of the terms in the function.

Similarly, the possible number of negative real zeros is determined by the number of sign changes in the coefficients of the terms in the function when x is replaced with -x.

For the polynomial function P(x)=6x^(4)-x^(3)+5x^(2)-x+9, there are 2 sign changes in the coefficients of the terms: from 6 to -1, and from 5 to -1. Therefore, the possible number of positive real zeros is 2 or 0.

For the polynomial function P(-x)=6(-x)^(4)-(-x)^(3)+5(-x)^(2)-(-x)+9, there are 0 sign changes in the coefficients of the terms. Therefore, the possible number of negative real zeros is 0.

In conclusion, the possible number of positive real zeros for the polynomial function P(x)=6x^(4)-x^(3)+5x^(2)-x+9 is 2 or 0, and the possible number of negative real zeros is 0.

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Answer:

Refer to pic..........

The given indicates that the decimal of growth or decay is 0.69 for this problem. Decimal of growth or decay can be determined using the formula A = P(1+r)ⁿ.

Decimal of Growth is the change in the size of an entity over a period of time, expressed as a percentage. It is a measure of the rate at which something grows or shrinks.

Decimal of growth or decay can be determined using the formula A = P(1+r)ⁿ, where A is the final amount, P is the initial amount, r is the rate of growth or decay, and n is the number of intervals.

In this case, A=7,545, P=7,545, r=0.96, and n=5. Plugging these values into the formula, we get 7,545=(7,545)(1+0.96)⁵.

Using a calculator, we can determine that (1+0.96)⁵=0.69, resulting in A=7,545(0.69)=5,203. This indicates that the decimal of growth or decay is 0.69 for this problem.

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One possible way to achieve this task is by using the following method:
1. Toss the biased coin twice.
2. If the result is two heads or two tails, discard the result and toss the coin again.
3. If the result is one head and one tail, record the result as a 1 if the first toss was a head and a 0 if the first toss was a tail.

This algorithm works because the probability of getting heads followed by tails or tails followed by heads is equal to 2p(1-p), which is symmetric around p=1/2. Therefore, the probability of outputting 1 is equal to the probability of outputting 0, which is 1/2. This method of generating a random process with a desired probability distribution from another random process is known as rejection sampling.

This method will produce a random process with outputs 1's and 0's with probability 1/2, regardless of the bias of the coin. This is because the probability of getting one head and one tail in two tosses is the same for both biased and unbiased coins, and we are only recording the result when we get one head and one tail. Therefore, this method will produce an unbiased random process from a biased coin.

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The answers to each question are:

1. the perimeter is 10 ft.

2. the area of the chest’s base is 6,

3. This wouldn’t work.

4. The chest weighs 956 pounds when filled with sand

What is surface area?

The space occupied by a two-dimensional flat surface is called the area. It is measured in square units. The area occupied by a three-dimensional object by its outer surface is called the surface area.

1. The base of the treasure chest is the bottom edge

Because most chests are rectangles, we can assume that the base is also a rectangle

This means that the perimeter would be 3+3+2+2, which is 10 ft

2.

To find the area of the chest’s base, you must multiply 3 by 2

3*2 is 6, so the answer is 6 feet squared

3.

It looks like it is asking if you can find the surface area by multiplying the base by one of the side lengths (again, not sure which one because it’s not labeled)

This wouldn’t work regardless of the side length, however, because base times height is equal to the volume of the chest and not the surface area

4. To do this, it would be easier to find how much sand can fit in the chest, and then do weight after you have that figure

There are 12 inches in a foot, so the measurements of how much sand can fit in the chest could be found by multiplying 12 by the side length

This would leave you with measurements of 36, 24 and 24

Volume is base * height, so first multiply the two base measurements (36 and 24)

This leaves you with 624

Multiply that by height (24) and you get 14,976 cubic inches of sand

Because each cubic inch of sand weighs about 1 ounce, we know that the weight of the sand is 14,976 ounces

To find out how much that is in pounds, divide 14,976 by 16

This leaves you with 936 lbs

Add the additional 20 that comes from the chest and you get 956 lbs total

The chest weighs 956 pounds when filled with sand

Hence, the answers to each question are:

1. the perimeter is 10 ft.

2. the area of the chest’s base is 6,

3. This wouldn’t work.

4. The chest weighs 956 pounds when filled with sand

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Answer:

See below for proof.

Step-by-step explanation:

Given equations for x and y:

  [tex]x=v_0 \cdot t \cdot \cos \theta[/tex]

  [tex]y=-16 \cdot t^2 + v_0 \cdot t \cdot \sin \theta[/tex]

Rearrange the equation for x to isolate t by dividing both sides of the equation by v₀ · cos θ:

[tex]\implies t=\dfrac{x}{v_0 \cdot \cos \theta}[/tex]

Square the equation for x:

[tex]\implies x^2=(v_0 \cdot t \cdot \cos \theta)^2[/tex]

[tex]\implies x^2={v_0}^2 \cdot t^2 \cdot \cos^2 \theta[/tex]

Rearrange to isolate t² by dividing both sides of the equation by v₀² · cos²θ:

[tex]\implies t^2= \dfrac{x^2}{{v_0}^2 \cdot \cos^2 \theta}[/tex]

Now we have created expressions for t and t² in terms of x.

Substitute these into the equation for y:

[tex]\implies y=-16 \cdot t^2 + v_0 \cdot t \cdot \sin \theta[/tex]

[tex]\implies y =-16 \cdot \left(\dfrac{x^2}{{v_0}^2 \cdot \cos^2 \theta}\right)+ v_0 \cdot \left(\dfrac{x}{v_0 \cdot \cos \theta}\right) \cdot \sin \theta[/tex]

Simplify:

[tex]\implies y =-\dfrac{16}{{v_0}^2} \cdot \left(\dfrac{x^2}{\cos^2 \theta}\right)+\left(\dfrac{x}{\cos \theta}\right) \cdot \sin \theta[/tex]

[tex]\implies y =-\dfrac{16}{{v_0}^2} \cdot \left(\dfrac{1}{\cos^2 \theta}\right) \cdot x^2+x \cdot \left(\dfrac{\sin \theta}{\cos \theta}\right)[/tex]

[tex]\textsf{Use\;the\;identities\;\;$\sec^2\theta=\dfrac{1}{\cos^2\theta}$\;\;and\;\;$\tan\theta=\dfrac{\sin\theta}{\cos\theta}$}:[/tex]

[tex]\implies y=-\dfrac{16}{{v_0}^2} \cdot \sec^2 \theta \cdot x^2+x \cdot \tan \theta[/tex]

[tex]\textsf{Hence\;proving\;that\;\;$y=-\dfrac{16}{{v_0}^2} \cdot \sec^2 \theta \cdot x^2+x \cdot \tan \theta$}.[/tex]

The equation can be written as,

[tex]y = \dfrac{-15x^2} { (v_o^2) + (\dfrac{1} { v_o^2})tan^{-1}(\dfrac{y} { x})}[/tex].

An object or particle that is projected in a gravitational field, such as from the surface of the Earth, and moves along a curved path while only being affected by gravity is said to be in projectile motion.

To begin with, we know that the horizontal displacement of the projectile is given by [tex]x = v_ot cos(\theta)[/tex], where vo is the initial velocity of the projectile and θ is the angle it makes with the positive x-axis. Therefore, we can rearrange this equation to solve for time t:

[tex]t = \dfrac{x} { (v_o cos(\theta))}[/tex]

Next, we can use the equation for vertical displacement under constant acceleration due to gravity: [tex]y = -16t^2+ v_ot sin(\theta)[/tex]. Substituting the expression for t derived above, we obtain:

[tex]y = -16[\dfrac{x} { (vo cos(\theta)}]^2 + v_o([\dfrac{x} { (v_o cos(\theta)}] sin(\theta)[/tex]

Simplifying this expression, we get:

[tex]y = \dfrac{-16x^2} { (v_o^2 cos^2(\theta)) + x tan(\theta)}[/tex]

Now, we need to eliminate the angle θ in terms of x and y. We can do this by using the fact that tan(θ) = y / x, which gives:

[tex]\theta= tan^{-1}(\dfrac{y} { x})[/tex]

Substituting this expression for θ into our previous equation, we obtain:

[tex]y =\dfrac{ -16x^2 }{(v_o^2 cos^2(tan^{-1}(\dfrac{y} { x}))} + x\ tan(tan^{-1}(\dfrac{y} { x})[/tex]

Since tan(tan⁻¹(z)) = z, we can simplify this expression further:

[tex]y = \dfrac{-16x^2} { (v_o^2 (1 + (\dfrac{y} { x})^2))} + y[/tex]

Finally, by rearranging terms, we get the desired equation:

[tex]y = \dfrac{-15x^2} { (v_o^2) + (\dfrac{1} { v_o^2})tan^{-1}(\dfrac{y} { x})}[/tex]

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Answer: the circles diameter is 14 meters

Step-by-step explanation:                                {Formula for circumference

2TUR = 43.96                                                      of a circle : L 27UR}

2x3 . 14r = 43.96

r=7

d=2r=2x7=14                                                         {d=2r}  

Answer:

The circumference of a circle can be calculated using the formula:

C = 2πr

where C is the circumference, π is the mathematical constant pi (approximately 3.14), and r is the radius of the circle.

To find the diameter of the circle, we can use the formula:

d = 2r

where d is the diameter and r is the radius.

Given that the circumference of the circle is 43.96 meters, we can use the formula for circumference to solve for the radius:

C = 2πr

43.96 = 2 × 3.14 × r

r = 43.96 / (2 × 3.14)

r = 6.998 meters

Now we can use the formula for diameter to find the diameter:

d = 2r

d = 2 × 6.998

d = 13.996 meters

Therefore, the diameter of the circle is approximately 13.996 meters.

Step-by-step explanation:

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a) Null hypothesis: Sex and risk of a lower-extremity injury in interscholastic soccer are not related variables. Alternative hypothesis: Sex and risk of a lower-extremity injury in interscholastic soccer are related variables. So the option E is correct.

b) There is insufficient information to draw the conclusion that the variables are connected even though the value is greater than the 0.05 threshold for significance.

c) The difference between these percentage are fairly similar which suggests that sex and risk of lower-extremity Injury in  interscholastic soccer are not related.

a) The null and alternative hypotheses about sex and the risk of a lower-extremity Injury while playing interscholastic soccer is:

Null hypothesis: Sex and lower-extremity injury risk in interscholastic soccer are unrelated factors. Alternative hypothesis: sex and the likelihood of sustaining a lower-extremity injury in collegiate soccer are associated variables.

So the option E is correct.

b) The conclusion is that there is no significant relationship between the gender of the athlete and the risk of lower-extremity injuries. This conclusion is based on the fact that the chi-square statistic (1.63) and the corresponding p-value (0.202) are both greater than the 1.63 X standard for significance. This indicates that there is not enough evidence to show that the variables are related.

c) Of 997 participants in girls' soccer, 79 experienced lower-extremity injuries. Of 1,664 participants in boys' soccer, 156 experienced lower-extremity injuries.

Girls = 79/997 × 100 = 7.92%

Boys = 156/1664 × 100 = 9.38%

The difference between the two percentages is consistent with the conclusion that there is no significant relationship between the gender of the athlete and the risk of lower-extremity injuries because the difference between the two percentages is small.

This indicates that the risk of lower-extremity injuries is relatively similar for both boys and girls, which is consistent with the conclusion that there is no significant relationship between gender and the risk of lower-extremity injuries.

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The complete question is:

Researchers studied a random sample of high school students who participated in interscholastic athletics to learn about the risk of lower-extremity injuries (anywhere between hip and toe) for interscholastic athletes. Of 997 participants in girls' soccer, 79 experienced lower-extremity injuries. Of 1,664 participants in boys' soccer, 156 experienced lower-extremity injuries.

(a) Write null and alternative hypotheses about sex and the risk of a lower-extremity Injury while playing interscholastic soccer.

A. Null hypothesis: Sex and risk of a lower-extremity injury in interscholastic soccer are related variables. Alternative hypothesis: Sex and risk of a lower-extremity Injury in interscholastic soccer are not related variables.

B. Null hypothesis: Sex and risk of a lower extremity injury in interscholastic soccer are explanatory variables. Alternative hypothesis: Sex and risk of a lower-extremity injury in interscholastic soccer are not explanatory variables.

C. Null hypothesis: Sex and risk of a lower extremity injury in interscholastic soccer are not explanatory variables. Alternative hypothesis: Sex and risk of a lower-extremity injury in interscholastic soccer are explanatory variables.

D. Null hypothesis: There is a weak relationship between sex and risk of a lower-extremity injury in interscholastic soccer. Alternative hypothesis: There is a strong relationship between sex and risk of a lower-extremity injury in interscholastic soccer

E. Null hypothesis: Sex and risk of a lower-extremity injury in interscholastic soccer are not related variables. Alternative hypothesis: Sex and risk of a lower-extremity injury in interscholastic soccer are related variables.

(b) For these data, the value of the chi-square statistic is 1. 63, and the p-value for the chi-square test is 0. 202. Based on these results, state a conclusion about the two variables in this situation and explain how you came to this conclusion. The p-value is greater than the 1. 63 X standard for significance, so there is not sufficient evidence to be able to conclude that the variables are related.

(c) For each sex separately, calculate the percent of participants who had a lower-extremity injury. (Round your answers to one decimal place. ) girls 7.9% boys 9.4%. Explain how the difference between these percentages is consistent with the conclusion you stated in part (b). These percentages are fairly similar , which suggests that sex and risk of a lower-extremity injury in interscholastic soccer are not related

Using basic arithmetic operation, the solution to the equation 7/6 + 2/3y = 4/3 is y = 3/4.

What is the basic arithmetic operation?

The four basic mathematical operations are Addition, subtraction, multiplication, and division.

To solve this equation for y, we need to isolate the variable y on one side of the equation. We can do this by performing algebraic operations to simplify the equation. Here are the steps:

Subtract 7/6 from both sides:

7/6 + 2/3y - 7/6 = 4/3 - 7/6

2/3y = 1/2

Multiply both sides by the reciprocal of 2/3:

(3/2)(2/3y) = (3/2)(1/2)

y = 3/4

Hence, the solution to the equation 7/6 + 2/3y = 4/3 is y = 3/4.

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Option D, The next step in constructing an angle bisector of angle ABC is to use a straightedge to connect point C to the point where the angle bisector intersects side AB.

To construct an angle bisector of angle ABC, the next step is to use a straightedge to draw a line from vertex C to the point where the angle bisector intersects side AB.

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Same-side interior angles are a pair of angles formed when two parallel lines are intersected by a third line, known as a transversal.

In this case, the same-side interior angles are the angles that are on the same side of the transversal and between the two parallel lines. They are called "interior" because they are inside the two parallel lines.

If you take any two same-side interior angles, they will add up to 180 degrees, meaning that they are supplementary.

The same side interior angles for this problem are given as follows:

As the same side interior angles are supplementary, the value of x is obtained as follows:

4x - 23 + 2x + 5 = 180.

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The function is transformed through a series of transformations, including compression, translation, and reflection.

The  is done vertically by a factor of 4, meaning that the function is made smaller in the y-direction. The translation is done horizontally by moving the function 2 units to the left. The reflection is done with respect to the x-axis, meaning that the function is flipped across the x-axis.

The resulting function can be represented by the equation y = -4f(x + 2), where f(x) is the original function. The negative sign in front of the 4 indicates the reflection with respect to the x-axis, the 4 indicates the vertical compression, and the (x + 2) indicates the horizontal translation to the left.

A new function that is compressed, translated, and reflected is produced after the function has undergone a number of modifications overall.

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We have the following response after answering the given question: Hence, the comparable fractions with 12 as the denominator are: 5/12= 60/144 and  1/2 and 5/6 = 10/12

What is the equation?

In a mathematical equation, the equals sign (=), which connects two claims and denotes equality, is utilized. In algebra, an equation is a mathematical statement that proves the equality of two mathematical expressions. For instance, in equation 3x + 5 = 14, the equal sign separates the numbers by a space. Mathematical expressions can be used to describe the relationship between the two sentences on either side of a letter. The logo and the particular piece of software frequently correspond. like, for instance, 2x - 4 = 2.

2 x 2 x 3 is the prime factorization of 12.

2 x 1 is the prime factorization of the number 2.

Two by three is the prime factorization of six.

The LCD is the sum of all the prime factors with the largest powers, giving us 2 × 2 × 3

5/12 = 5/(12 * 1) = 5/12 * 12/12 = 60/144

1/2 = 1/2 * 6/6 = 6/12

5/6 = 5/6 * 2/2 = 10/12

Hence, the comparable fractions with 12 as the denominator are:

5/12 = 60/144

1/2  = 6/12

5/6 = 10/12

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The amount of Kingston's birth weight that was much bigger than Karmichael would be = 1 pound 4 ounces

The weight of Kingston at birth = 9 pounds 2 ounces

The weight of Karmichael at birth = 7 pounds 11 ounces.

Next, the weights are converted to ounces such as:

1 pound = 16 ounces

For 9 pounds = 16×9 = 144 ounces+ 2 ounces = 146 ounces

For 7 pounds = 16×7 = 112 ounce + 11 ounces = 123 ounce

The difference between the birth weight of Kingston and Karmichael = 146-123 = 23 ounce

Convert 23 ounce to pounds = 23/16 = 1 pound 4 ounces

Therefore, amount of Kingston's birth weight that was much bigger than Karmichael would be = 1 pound 4 ounces.

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Ann average speed for the entire walk was 1 mile per hour.

Time (hours) Distance from Home (miles)

0                                0

1                                 2

2                                4

3                                6

4                                5

5                                4

6                                3

7                                 2

8                                 1

9                                0

Ann walked for three hours, covering a distance of 6 miles, the furthest she had walked from her house.

2. When did Ann make her lunch break?

At 4 hours, Ann took a break for lunch.

3. What time did Ann arrive at her house?

After the nine-hour mark, Ann went back home.

4. What was Ann's average walking speed for the first three hours?

The total distance Ann walked during the first three hours of her walk (6 miles) divided by the time required to complete that distance can be used to determine her average pace (3 hours). She moved at an average speed of 2 miles per hour for the first three hours.

During the entire trek, Ann travelled a total of 9 miles (since she returned to her starting point). It took 9 hours to travel the entire trip. She walked at a mile an hour on average for the entire distance.

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Answer:

1440

Step-by-step explanation:

hope this helps.......

The slope of this situation is equal to 5.48.

In Mathematics, the slope-intercept form of the equation of a straight line is given by this mathematical expression;

y = mx + c

Where:

Based on the information, an equation that models the situation is given by this mathematical expression;

y = mx + c

y = 5.48x + 3.50

In conclusion, we can reasonably infer and logically deduce that the slope is 5.48 and the y-intercept is equal to 3.50.

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Answer:

1. 468

2. 529

3. 513

4. 176

Step-by-step explanation:

The expanded form of (10a - b)(10a + b) is 100a^2 - b^2.

To expand the given expression, we need to distribute the first term inside the parentheses to the second term inside the parentheses. This can be done by multiplying each term inside the first parentheses by each term inside the second parentheses. Here are the steps:

1. 10a * 10a = 100a^2
2. 10a * b = 10ab
3. -b * 10a = -10ab
4. -b * b = -b^2

Now, we can combine like terms and simplify the expression:

100a^2 + 10ab - 10ab - b^2 = 100a^2 - b^2

So, the final answer is:

100a^2 - b^2

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To complete the sentence, we need to choose the expression that correctly represents the number of books owned by the university library. The expression given in parentheses is "nine million three hundred thirty-nine thousand five hundred sixty" or "9,339,560".

To write this number in words, we can break it down into its different place values. The digit in the millions place is 9, so we can say "nine million."The digit in the hundred thousands place is 3, so we can say "three hundred thousand." The digit in the ten thousands place is 3, so we can say "thirty thousand."The digit in the thousands place is 9, so we can say "nine thousand." The digit in the hundreds place is 5, so we can say "five hundred.

"The digit in the tens place is 6, so we can say "sixty." The digit in the ones place is 0, so we can say "zero."Putting it all together, we can say that the university library owns more than "nine million, three hundred thirty-nine thousand, five hundred sixty" books.

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The present value is $7,307.31.

To calculate the present value, we can use the formula: PV = A / (1 + r/n)^(n*t) where PV is the present value, A is the future value, r is the annual interest rate, n is the number of compounding periods per year, and t is the number of years.

In this case, we have A = $35,000, r = 15% = 0.15, n = 4 (since it is compounded quarterly), and t = 10.

Plugging these values into the formula, we get:

PV = $35,000 / (1 + 0.15/4)^(4*10)

PV = $35,000 / (1.0375)^40

PV = $35,000 / 4.79

PV = $7,307.31

Therefore, the present value is $7,307.31.

I ignored the "Ineedarim?" part of the question as it is irrelevant to the calculation.

$7,307.31.

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There are 432 possible outcomes when tossing a coin and rolling three number cubes.

Yes, it is possible to calculate the total number of outcomes when tossing a coin and rolling three number cubes using the basic counting method.

According to the basic counting principle, if there are m ways to do one task and n ways to complete another, then there are m * n methods to complete both tasks sequentially.

There are two possible outcomes from this coin toss (heads or tails), and each of the three number cubes can be rolled six different ways (1 through 6). As a result, according to the basic counting principle, there are:

2 * 6 * 6 * 6 = 432

Hence, when rolling three number cubes and tossing a coin, there are 432 possible results.

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By answering the above question, we may state that The rectangle's equation measurements are 9/4 miles by 9/20 miles.

A mathematical equation links two statements and utilises the equals sign (=) to indicate equality. In algebra, an equation is a mathematical assertion that proves the equality of two mathematical expressions. For instance, in the equation 3x + 5 = 14, the equal sign separates the numbers by a gap. A mathematical formula may be used to determine how the two sentences on either side of a letter relate to one another. The logo and the particular piece of software are usually identical. like, for instance, 2x - 4 = 2.

A is equal to (9/20) sq miles wide at x miles.

[tex]A = (9/20) (width x length) = x(x/5)\\9x/20 = x^2/5\\45x = 20x^2\sx^2 - 45x/20 = 0[/tex]

[tex]x=(-b \sqrt(b2 - 4ac)) / 2a[/tex]

Here, an equals 1, b equals -45/20, and c equals 0. By entering these values, we obtain:

[tex]x = (-(-45/20) +\sqrt((-45/20)^2 - 4(1)(0))) / 2(1) x = (45/20 ± sqrt(2025/400)) / 2\sx = (45/20 ± 45/40) / 2\sx = (9/4) or (1/4)[/tex]

We may disregard the negative answer since a rectangle's length cannot be negative. As a result, the rectangle has a length of 9/4 miles and a width of (1/5) x (9/4) = 9/20 miles.

The rectangle's measurements are 9/4 miles by 9/20 miles.

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The equation of the line with gradient 2 and that passes through the point (1,1) is y= 2x -1.

A line's slope is determined by how its y coordinate changes in relation to how its x coordinate changes. y and x are the net changes in the y and x coordinates, respectively. Therefore, it is possible to write the change in y coordinate with respect to the change in x coordinate as,

m = Δy/Δx where, m is the slope

Given:

Gradient = 2

and, line passes through the point (1,1).

We know using Slope intercept form y= mx+ b

and, y= 2x+ b

Now, line passes through the point (1,1).

1 = 2(1) + b

b= -1

So, the equation of line is y= 2x -1.

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