dy/dx = y/x^2

dy/y = dx/x^2

ln y * ln c = -1/x

cy = e^(-1/x)

y = ce^(-1/x)

Answer:

D) y2 = -5x - 8

Step-by-step explanation:

A function is a rule between two sets A and B so that each element of set A (Domain) corresponds to a single element of set B (Range). Mathematically they can be expressed as:

[tex]f: A\rightarrow B\\\\a\rightarrow b =f(a)[/tex]

So:

[tex]y^2=-5x-8\\\\y=\pm \sqrt{-5x-8}[/tex]

Isn't a function because the elements of the domain don't corresponds to a single element of the range.

We can see this in an easier way using the vertical line test, which is a visual way to determine if a curve is a graph of a function or not. If a vertical line intersects a curve in an xy plane more than once, then for a value of x the curve has more than one value of y, and then the curve does not represent a function.    

I attached you the graphs, so you can corroborate the results visually.

A number that divides evenly into another number is known as factor.

The correct answer is

B.) factor.

A factor is a number that divides evenly into another number. To find the factors of a number, you can perform a process of division or use other mathematical methods.

Let's take an example to illustrate:

If we consider the number 12, its factors are the numbers that divide it evenly without leaving a remainder. We can start by dividing 12 by 1:

12 ÷ 1 = 12

Since there's no remainder, 1 is a factor of 12. Now, let's try dividing 12 by 2:

12 ÷ 2 = 6

Again, there's no remainder, so 2 is also a factor of 12. Continuing this process, we find that 3 is also a factor of 12:

12 ÷ 3 = 4

And finally, 12 ÷ 4 = 3

So, the factors of 12 are 1, 2, 3, 4, 6, and 12.

Each of these numbers divides evenly into 12, making them factors. Therefore, the correct answer is B.) factor.

Factors are essential in various mathematical concepts, including prime factorization, finding greatest common divisors, and simplifying fractions. Understanding factors helps in solving problems related to divisibility and multiplication. By identifying factors, we can break down complex numbers into simpler components, aiding in various mathematical computations and analyses. Hence, recognizing factors is fundamental in mathematics.

Complete question:

A number that divides evenly into another number is known as which of the following?

A.) prime number

B.)factor

C.) multiple

D.) Integer

The rate of crickets chirping is linearly dependent on the temperature and can be represented by the equation y = 4x -100, whereby y is the number of chirps per minute, and x is the temperature in °F.

This is essentially a linear equation problem where the rate of cricket chirps is related to the temperature.

Firstly, we list our two points: (59, 76) and (65, 100).

We can then calculate the slope (m). m = (y2-y1) / (x2-x1) = (100-76)/(65-59) = 24/6 = 4.

We have the rate of cricket chirps increasing by 4 chirps per increase in 1°F. Our equation is now of the form y= 4x + b.

Finally, we'll calculate the y-intercept (b) by plugging one of our points into the equation. If we choose the point (59, 76), we get 76 = 4*59 + b. Solving for b, we get b = -100.

So, the equation is y = 4x -100, where y is the number of chirps per minute, and x is the temperature in °F.

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The area that a dog can roam, attached to a 49-foot rope at the corner of a 42x51 foot garden, was initially calculated as a quarter-circle's area with mistakes acknowledged in the approach. The correct calculation involves more intricate geometrical considerations, considering the question's unique constraints.

To determine the area that a dog can wander, given it's attached to a 49-foot rope fastened to the outside corner of a fenced-in garden measuring 42 feet by 51 feet, we need to consider the shape of the area the dog can cover. The dog can cover a quarter-circle around each of two adjacent sides of the garden and a semi-circle along the side opposite the corner where the rope is fastened, provided the rope's length exceeds the garden's dimensions on those sides.

Calculating the area involves adding the areas of these three parts:

Therefore, the total area that the dog can wander is effectively the area of a quarter-circle with a radius of 49 feet. This can be calculated as:

Area = 1/4 * π * (49)^2 square feet

Calculating this gives an area of 1884.96 square feet (rounded to two decimal places).

However, upon reevaluation, it's clear that a mistake was made in the initial explanation; the correct approach is to consider not just a quarter-circle but also the additional areas the rope covers beyond the garden's dimensions.

The exact area that the dog can wander is [tex]{2401\pi} \)[/tex] square feet.

Identify the radius: The radius of the circle that the dog can wander is the length of the rope, which is 49 feet.

Calculate the area of the circle: Use the formula for the area of a circle, [tex]\( A = \pi r^2 \)[/tex], where [tex]\( r = 49 \)[/tex] feet.

[tex]\[ A = \pi \times 49^2 \][/tex]

Compute [tex]\( 49^2 \)[/tex]:

[tex]\[ 49^2 = 2401 \][/tex]

Multiply to find the area:

[tex]\[ A = \pi \times 2401 \][/tex]

The exact area that the dog can wander is [tex]{2401\pi} \)[/tex] square feet.

This calculation confirms that the dog can wander an area of [tex]\( 2401\pi \)[/tex] square feet within the fenced-in garden while attached to the 49-foot rope.

The volume of the prism is 22.5 cubic inches.

To find the volume of the prism, we need to first find the area of the base and then multiply it by the height. In this case, the base is a 45-45-90 triangle with a leg of 5 inches. The area of a triangle is given by the formula 1/2 * base * height, so the area of the base is 1/2 * 5 * 5 = 12.5 square inches. The height of the prism is given as 1.8, so we can now calculate the volume by multiplying the area of the base by the height: 12.5 * 1.8 = 22.5 cubic inches.

The number of cakes she can make is 9.

The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.

Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.

Given that Stella has 6 pounds of chocolate. She will use 2/3 pounds of the chocolate to make one cake.

The number of cakes will be calculated as,

Number = 6 / ( 2 / 3 )

Number = ( 6 x 3 ) / 2

Number = 9

Therefore, the number of cakes will be 9.

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Answer:

mos def 15

Step-by-step explanation:

To solve this problem, we are going to use a proportion, letting the variable x represent the unknown number of cups of oatmeal that we need.

1.5 cups/36 cookies = x cups/60 cookies

To simplify a proportion like this, we can use cross-multiplication. This means multiplying the numerator of one fraction by the denominator of the other fraction and setting it equal to the product of the other numerator and denominator. This is modeled below:

(36)(x) = (1.5)(60)

To simplify, we begin by multiplying out each side.

36x = 90

Finally, we divide both sides of the equation by 36.

x = 2.5

This means that we need 2.5 cups of oatmeal to make the cookies.

Hope this helps!

The range of f(x) is a single value which is the y-coordinate of any point on the horizontal line representing f(x).

The range of a function is the set of possible values its output can have. In this case, the function f(x) is a horizontal line. Since f(x) is restricted to the portion between x = 0 and x = 20, inclusive, the range of f(x) is the set of all y-values within that range. Since the graph of f(x) is a horizontal line, the range of f(x) is a single value, which is the y-coordinate of any point on the line.

Based on the graph shown below, the range of f(x) include the following: B. { f(x)| –∞ < f(x) ≤ 4}

In Mathematics and Geometry, a range is the set of all real numbers that connects with the elements of a domain.

Additionally, the vertical extent of any graph of a function represents all range values and they are always read and written from smaller to larger numerical values, and from the bottom of the graph to the top.

By critically observing the graph shown in the image attached below, we can reasonably and logically deduce the following range:

Range = (-∞, 4] or {f(x)| –∞ < f(x) ≤ 4}

Complete Question;

What is the range of f(x)?

{ f(x)| –∞ < f(x) < ∞}

{ f(x)| –∞ < f(x) ≤ 4}

{ f(x)| 4 < f(x) < ∞}

{ f(x)| 0 ≤ f(x) < ∞}