The minimum sample size required for the estimate (to the next integer) is given as 404.
What is a sample size?Note that a sample size is the number of observations or individuals in a sample selected from a population.
The formula for Sample Size (n) is given as:
Sample size = n = [(Z[tex]_{\alpha/2 }[/tex] * σ ) / E]²
Where:
α = 1-90%
Margin of Error (E) = 0.18
Standard Deviation = σ = 2.2
At 90% confidence level the z is,
α = 1 - 90% = 1 - 0.90 = 0.10
α /2 = 0.10/2 = 0.05
Z[tex]_{\alpha/2 }[/tex] = Z₀.₀₅ = 1.645
Thus, Samples Size =
= [(1.645 * 2.2) /0.18]²
= (3.619/0.18)₂
= (20.1055555556)²
= 404.2333641993; thus
n [tex]\approx[/tex] 404
Therefore, the minimum sample size required for the estimate to the next integer is 404.
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The length of a rectangle is 97 meters and the width is 14 meters. Find the area. Give your answer without units.
Answer: The zone of a rectangle is the item of length and width in this way the zone will be 1358 square meters.
Step-by-step explanation:
Rectangle ?
A rectangle may be a geometrical figure in which inverse sides are equal. The point between any two successive sides will be 90 degrees. The edge of the rectangle = 2( length + width). It has 4 sides, 4 points, and 4 corners (vertices)
It is known that,
Area of rectangle = length × Breadth
Area = 97 x 14 = 1358 square meters
Hence
The range of a rectangle is the item of length and Breadth in this way the zone will be 1358 square meters.
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The area of the rectangle is 1358
GIVEN-
Length=97
breadth=14
TO FIND-
Area of rectangle
STEPS-
Area of rectangle= length×breadth
∴area of rectangle=97 x 14
=1358
The area of rectangle is 1358m
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5) There are 112 grams of a radioactive element and
it decays at a rate of 1.5% per year. Write an
model of this scenario.
exponential
The mass of the radioactive material for n years can be modeled as 112(1 - 1.5/100)ⁿ
How to find the decay rate of an exponential function?An exponentially decaying function has a general form f(x) = (1 -r)ˣ, where r is called the decay rate. it can be found by having any two values of x for which f(x) is known.
The initial mass of radioactive element is given as 112 grams.
And, the decay rate is 1.5%.
Then, the expression for the mass after n years can be written as,
P(1 - r/100)ⁿ
Where P is the initial value, r is the decay rate in percent.
Plug P = 112 and r = 1.5 to obtain the required model as,
⇒ 112(1 - 1.5/100)ⁿ
Hence, the exponential model for the given case is 112(1 - 1.5/100)ⁿ.
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If P(A)=0.4,P(B)=p,P(A∪B)=0.6 and A and B are given to be independent events , find the value of p
The value of p is 1/3.
P(A∪B) = P(A) + P(B) − P(A∩B)
0.6 = 0.4 + p − P(A∩B)
(A∩B) = 0.4 + p − 0.6 = p − 0.2
Since , A and B are independent events.
P(A∩B) = P(A) × P(B)
p − 0.2 = 0.4 × p
p − 0.4p = 0.2
p = 0.2/0.6 = 1/3
The chance of an event happening given that another event has already occurred (via assumption, supposition, statement, or evidence) is known as conditional probability in the field of probability theory. [1] This particular approach depends on event B happening in some way connected to another event A. In this case, a conditional probability analysis with respect to event B is possible.
The probability of A under the condition B is typically expressed as P(A|B)[2] or PB if the event of interest is A and the event B is known to have occurred or is assumed to have done so (A). This can alternatively be thought of as the percentage of possibility that B and A will collide.
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The rate of change of the volume V with respect to time t of water leaking from a tank is proportional to the cube of the volume. Which of the following is a differential equation that could describe this relationship? a. dV/dt = -1.4t^3 b. dV/dt = -0.2V^3 c. dV/dt =0.75t^3 d. dV/dt = 0.15V^3
dV/dt = -1.4t³ is differential equation that describe the relationship.
The differential equation dV/dt = kV³ where k is the constant will forecast the rate of change of volume.
We know that a tank's water leak causes its volume to fluctuate at a pace that is proportional to its cube as time passes.
The tank is leaking the volume. Therefore, the rate might be expressed negatively.
The rate of change of volume can be expressed as follows: dV/dt is proportional to V³ that is dV/dt = kV³ for the specified conditions.
The proportionality constant in this case is k. By employing the boundary conditions, its value can be determined.
Therefore, dV/dt=kV³ where k is the constant will be the differential equation predicting the rate of change of volume.
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WILL GIVE BRAINLIEST
In the summertime the local school district tries to conserve electricity by adjusting the air conditioning temperatures. During the day the lowest the air conditioning can reach is 75° F, at night it is turned up to 92° F so the building does not reach temperatures over 92° F. Staff cannot work in the building when the temperature is above 92° F. Write an inequality to represent the acceptable temperatures the building can reach.
Answer:
Below
Step-by-step explanation:
75 °F ≤ T° ≤ 92 °F <=======given
A pie chart shows: O The frequency for each class O The relative frequency for each class O Trend in the data O The class midpoints
A pie chart shows The relative frequency for each class.
What Is a Pie ChartA pie chart is a diagram that shows a data or a number result in the form of a circle.
A pie chart is a type of chart that displays data statistics in a circular graph resembling a circle shape.
Inside the pie chart there are several slices containing numeric variables and categorical variables that represent certain data.
Pie charts are more suitable for making comparisons if the data consists of various categories or groups.
The pie chart is divided into sections according to the data presented. The area of the sections is proportional to the central angle of the circle and the amount of data presented.
This diagram is quite often used as a tool to show data.
By using this, everyone will easily visualize data so that data will be understood much more easily.
Types of Pie ChartsThere are three types of pie charts. The difference between the three types lies in the use of units to present data.
Given the wide use of pie charts, the presentation of the data can be done in various ways.
the following are types of pie charts:
Regular pie charts; is a pie chart that uses data in the form of numbers. Circle diagram degrees; is a pie chart that uses data in degrees. Usually the calculation will focus on the size of the angle of a section and the total angle, which is 360°. Percent pie charts; is a pie chart that uses data in degrees. The calculation will focus on the percentage of a part and the whole part, which is 100%.Learn more about pie chart at https://brainly.com/question/9979761.
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A restaurant offers delivery for their pizzas. The total cost includes a delivery fee added to the price of the pizzas. One customer pays $25 to have 2 pizzas delivered. Another customer pays $49 for 5 pizzas. How many pizzas are delivered to a customer who pays $81? Show your work.
A customer who pays $81 has 8 pizzas delivered.
Solving for the number of pizzas delivered to a customer who pays $81 we have:Let x be the number of pizzas delivered to a customer who pays $81.
The cost of the pizzas is $81 - delivery fee.
The delivery fee for 2 pizzas is $25 - cost of 2 pizzas = $25 - 2x.
The delivery fee for 5 pizzas is $49 - cost of 5 pizzas = $49 - 5x.
Equating the two delivery fees, we get:
$25 - 2x = $49 - 5x
Solving for x:
- 2x + 5x = $49 - $25
3x = 24
Divide both sides by the coefficient x
x = 8
Hence, a customer who pays $81 has 8 pizzas delivered.
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Select the equivalent expression
1/6 to the fifth power
If the derivative of f is given by f'(x)=e^x-3x^2 at which of the following values of x does f have a relative maximum value?
a) -0.46
b) 0.20
c) 0.91
d) 0.95
e) 3.73
The relative maximum value of f occurs at x = b) 0.20.
We can find the relative maximum value of f by finding the critical point of f, which is the value of x where the derivative of f is equal to 0.
f'(x) = e^x - 3x^2 = 0
e^x = 3x^2
ln(e^x) = ln(3x^2)
x = sqrt(ln(3x^2))/sqrt(3)
That is not a real value, however, we can find the relative max by finding the second derivative test, which is the sign of the second derivative of f at x. If the second derivative is positive, the x is a relative max.
f''(x) = e^x - 6x
To find the relative max we have to evaluate the second derivative at each option point a,b,c,d,e
a) -0.46 f''(-0.46) is negative
b) 0.20 f''(0.20) is positive
c) 0.91 f''(0.91) is negative
d) 0.95 f''(0.95) is negative
e) 3.73 f''(3.73) is negative
Therefore, the relative maximum value of f occurs at x = b) 0.20
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What type of fraction is 10 3?
Answer:
It is an improper proper fraction because the denominator (bottom number) is smaller than the numerator (top number).
how to factor 14s + 7?
a question in ixl for algebra 1 : distributive property
The factor of 14s + 7 is 7 (2s +1). A factor is a number that divides another number by itself with no residual.
What is factor?The term "divisor of an integer n," also known as a "factor of n," refers to an integer m that can be multiplied by another integer to yield the value "n." That n is a multiple of m in this situation is also true. A number that divides another number by itself with no residual is known as a factor. In other words, if multiplying two whole numbers results in the creation of a product, the numbers we are multiplying are factors of the product since the product is divisible by them. Division and multiplication are the two ways to find factors. The factorization approach involves first considering the numbers 1 and 10 as factors of 10, then moving on to discover the other pair of multiples of 10, which results in the original number as the outcome.Factor 14s + 7 : 7(2s + 1)
Steps
14s + 7
Rewrite 14 as 7 × 2
Rewrite 7 as 7 × 1
=7 × 2 s + 7 × 1
Factor out common term 7
=7(2s + 1)
Hence, The factor of 14s + 7 is 7 (2s +1). A factor is a number that divides another number by itself with no residual.
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PLEASE HELP ME ASAP
The measure of angle x is x = 51°
The area of the shaded portion of the circle A = 47.17 cm²
What is a Circle?A circle is a closed two-dimensional figure in which the set of all the points in the plane is equidistant from a given point called “center”. Every line that passes through the circle forms the line of reflection symmetry. Also, the circle has rotational symmetry around the center for every angle
The perimeter of circle = 2πr
The area of the circle = πr²
where r is the radius of the circle
The standard form of a circle is
( x - h )² + ( y - k )² = r²,
where r is the radius of the circle and (h,k) is the center of the circle
Given data ,
Let the area of the shaded region of the circle be represented as A
Now , the equation will be
From the figure , the line AB is a straight line
And , 129° + x° = 180°
Subtracting 129° on both sides of the equation , we get
x° = 180° - 129°
On simplifying the equation , we get
x° = 51°
So , the measure of angle x = 51°
And ,
Area of the shaded region A = Area of the sector of circle - Area of triangle
Now , sin 51° = h / 8 ( from Pythagoras Theorem )
Multiply by 8 on both sides of the equation , we get
h = 8 x ( sin 51° )
h = 8 x 0.7771
h = 6.2168
h ≈ 6.22 cm
Now , Area of the sector of circle = ( 129π ( 8 )² ) / 360
On simplifying the equation , we get
Area of the sector = 72.05 cm²
And , Area of triangle = ( 1/2 ) x 8 x 6.22
Area of triangle = 24.88 cm²
Now , Area of the shaded region A = Area of the sector of circle - Area of triangle
Substituting the values in the equation , we get
Area of the shaded region A = 72.05 cm² - 24.88 cm²
Area of the shaded region A =47.17 cm²
Hence , the area of the shaded region of the circle A is 47.17 cm²
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PLEASE I NEED HELP ASAP I WILL MARK BRAINLEIST AND I WILL GIVE YOU 100 COINS IF YOU GIVE ME THE RIGHT ANSWER PLEASE HELP!!!
What is four over five times one over three?
Answer: There u go
Step-by-step explanation:
Rose invests $80 in an account that pays 1% annual interest, compounded monthly.
Emily invests $50 in an account that pays 2% annual interest, compounded weekly.
Whose balance is greater after 10 years?
Type Rose or Emily
using π as 3 1/7, what is the area of the circle with a diameter of 8.4 cm
if the diameter is 8.4, that means its radius must be half that, or 4.2
[tex]\textit{area of a circle}\\\\ A=\pi r^2 ~~ \begin{cases} r=radius\\[-0.5em] \hrulefill\\ r=4.2\\ \pi =3\frac{1}{7}\\[1em] \qquad \frac{22}{7} \end{cases} \implies A=\cfrac{22}{7}(4.2)^2\implies A=55.44[/tex]
Use the equation $18.62 + c = $26.00, where c is the
item cost, to find the most expensive item Trish can buy.
Answer:
C=$7.38
Step-by-step explanation:
$18.62 + c = $26.00
$26-$18.62=$7.38
c=$7.38
if g(x)=x-2 and h(x)=x^2-1 , what is ( h+g)(x)
Answer:
[tex]x^{2} + x -3[/tex]
Step-by-step explanation:
1. Write out the equation
[tex](x-2)+( x^{2} -1)\\[/tex]
2. Add like-terms
[tex]x^{2} +x-3[/tex]
Step-by-step explanation:
(X^2-1+x-2)
(X^2+x-2x-2)
X(x+1)-2(x+1)
(X-2)(x+1)
lisa needs to sell at least 140 cards to make a profit. monday-thursday she sells 15,7,14,45 cards. how many more will she need to sell on friday to make a profit?
To make a profit, Lisa must sell 59 cards on Friday
How to determine how many more she will need to sell on Friday?From the question, we have the following parameters that can be used in our computation:
Sales that gives profit = at least 140 cards
Sales on Monday to Thursday = 15,7,14,45 cards
This means that
Friday = Sales that gives profit - Sales on Monday to Thursday
Substitute the known values in the above equation, so, we have the following representation
Friday = 140 - (15 + 7 + 14 + 45)
Evaluate
Friday = 59
Hence, the sales is 59
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Suppose a cube and a sphere
have the same volume.how do you think their surface areas would compare?
Note:S.A sphere = [tex]4\pi r[/tex]² and V sphere = [tex]4\pi r[/tex]³ /3
When a cube and a sphere have the same volume, their surface areas would compare as sphere will have a larger surface area than the cube.
How would the surface areas of the cube and sphere compare?The area or region that an object's surface occupies is known as its surface area. Volume, on the other hand, refers to how much room an object has. There are numerous shapes and dimensions in geometry, including sphere, cube, cuboid, cone, cylinder, etc. Each shape has its own volume and surface area.
If a cube and a sphere have the same volume, the sphere will have a larger surface area than the cube. This is because the surface area of a sphere is determined by the radius of the sphere, while the surface area of a cube is determined by the length of its sides. Since the radius of the sphere is greater than the length of the sides of the cube, the surface area of the sphere will be larger.
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the cook at the elmView School cafeteria made grilled chicken and fried chicken yesterday for lunch he made five pieces of grilled chicken for every two pieces of fried chicken if the cook made 80 pieces of fried chicken how many pieces of chicken did he make all together
Using simple unitary method and the provided conditions and values, The answer is he made 280 chicken all together.
What is unitary method?The unitary method is a method used in mathematics to solve problems involving units of measurement. It involves expressing the units of measurement as a fraction or multiple of a single unit and then using this fraction or multiple to solve the problem. The method is particularly useful in solving problems involving conversions between different units of measurement. The unitary method is particularly useful in solving word problems where the conversion of units is required. It involves expressing the given information in terms of a single unit and then using that information to find the unknown quantity.
Why is unitary method important?The unitary method is a useful tool for solving problems involving units of measurement, it simplifies the problem and improves understanding of conversions. It also helps to develop critical thinking skills and is a useful tool for problem-solving in a variety of subjects.
5 grilled chicken for 2 fried chicken.
5/2 grilled chicken for 1 fried chicken.
5/2 * 80 grilled chicken for 80 fried chicken.
200 grilled chicken for 80 fried chicken.
total = 200 +80 =280
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Find the missing values in the ratio table. Then write the equivalent ratios in the order they appear in the table.
Meters 8, 4, ,
Minutes 1/3, ,1/4,5/12
The equivalent ratios are 8:1/3, 4:1/6, 6:1/4, 10:5/12.
What is Ratio?A ratio is an ordered pair of numbers a and b, written a / b where b does not equal 0.
Let us consider the table as below
Meters 8 4 y z
Minutes 1/3 x 1/4 5/12
Since all the columns are pertaining same ratio; thus we have:
8/(1/3)=4/x=y/(1/4)=z/(5/12)
8/(1/3)=4/x
24=4/x
x=4/24=1/6
So the value of x is 1/6
24=y/(1/4)
24=4y
6=y
So the value of y is 6
24=z/(5/12)
24=12z/5
10=z
Hence, the equivalent ratios are 8:1/3, 4:1/6, 6:1/4, 10:5/12.
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Which invesment is better $1200 at 9% 2 years or $ 1200 at 8% for 2 years
Question 7 a data analyst is working with a data frame named salary_data. They want to create a new column named wages that includes data from the rate column multiplied by 40. What code chunk lets the analyst create the wages column?.
Code chunk is created the wages column using analyst technique. mutate(salary_data, hourly_salary = wages/40) .
By using data analyst technique in which the mutate() function is used to create a new column for wages which is divided by 40 known as hourly_salary. The mutate() function in which it adds a new column and leaves existing columns .
A data analyst who collects, cleans, and interprets large amounts and contents of data in order to simplify ,answer questions to a particular problem in a standard manner . The team people work in a wide range of industries, including business, finance, criminal justice, science, medicine, and government of utilization of process.
Role of a Data Analyst makes more crucial in modern technician businesses, assisting in the modern reflection of the company's work and customer related base, determining in such way that how these factors have been affected by profits, and advising modern leadership on to move forward to grow the business in the modern world .
hence , code chunk is discussed based on the analyst at the column of wages and also detailed about how does the data analyst is working with the data frame wages which includes data at the rate of column and multiplied by 40.
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A code chunk which lets the analyst create the wages column include the following: A. mutate(salary_data, wages = rate * 40).
What is a mutate function?In Computer programming, a mutate function can be defined as a type of function in R programming that is primarily used for the creation of a new variable from a list of data set while preserving existing variables or existing columns.
What is an executable code in R Studio?In this scenario, the code chunk which would let the data analyst create the wages column from the rate column should be written in R Studio (R programming) as follows:
mutate(salary_data, wages = rate * 40).
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Complete Question:
A data analyst is working with a data frame named salary_data. They want to create a new column named wages that includes data from the rate column multiplied by 40. What code chunk lets the analyst create the wages column?
A. mutate(salary_data, wages = rate * 40)
B. mutate(salary_data, wages = rate + 40)
C. mutate(wages = rate * 40)
D. mutate(salary_data, rate = wages * 40)
a 5 ft long ladder is leaning against a wall, and the top of the ladder is originally at 4 ft above the ground. the bottom of the ladder is being pulled away from the wall at a constant rate. how fast is the bottom of this ladder being pulled away from this wall if the top of the ladder is sliding down at a rate of 1 ft/second?
The speed at which the bottom of the ladder is being pulled away from the wall is equal to the rate at which the top is sliding down, which is 1 ft/second.
The bottom of the ladder is being pulled away from the wall at the same rate at which the top of the ladder is sliding down the wall. As the top of the ladder slides down at a rate of 1 ft/second, the bottom of the ladder is also being pulled away at a rate of 1 ft/second.
The bottom of the ladder is being pulled away from the wall at a rate of 100 ft/second if the top of the ladder is sliding down at a rate of 100 ft/second.
The speed at which the bottom of the ladder is being pulled away from the wall is equal to the rate at which the top is sliding down, which is 1 ft/second.
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In each of the following equations…
The values of variables are
a. x = -3/2; b. x = 0; c. y = 5; and d. a = 0 and b = 0.
A mathematical definition of an equation is a statement that two expressions are equal and joined by the equals sign "=". In algebra, an equation only satisfies a definite value of the variable.
To solve an equation we need to do operations like adding and subtracting the same integers or dividing or multiplying with the same integers.
Given equations can be solved as follows
a. 2x + 3 = 0
Add -3 on both sides
=> 2x + 3 - 3 = 0 - 3
=> 2x = - 3
=> x = -3/2
b. 7x = 0
dividing by 0 into both sides
=> 7x/7 = 0/7
=> x = 0 [ ∵ 0/x = 0 ]
c. 7(y - 5) = 0
divide by 7 into both sides
=> 7(y - 5)/7 = 0/7
=> (y - 5) = 0
Add 5 on both sides
=> y - 5 + 5 = 0 + 5
=> y = 5
d. ab = 0
If we divided by 'a' into both sides
=> ab/a = 0/a
=> b = 0
If we divided by 'b' into both sides
=> ab/b = 0/b
=> a = 0
Therefore,
The values of variables are
a. x = -3/2; b. x = 0; c. y = 5; and d. a = 0 and b = 0.
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Can u pls help me with me math
Answer:
Step-by-step explanation:
answer the prenthese first then multiply the numbers that are not in the prenthese. if u dont understand it let me know
What’s the answer to (-4, 7), (-6, -4
The slope of the points (-4, 7) and (-6, -4) when a line passes through them is 5.5
How to calculate the slope of the line?From the question, we have the following parameters that can be used in our computation:
(-4, 7), (-6, -4
When the above points are expressed properly, we have the following ordered pairs
(x, y) = (-4, 7) and (-6, -4)
The formula for the slope of the line is then represented using the following slope equation
Slope = (y₂ - y₁)/(x₂ - x₁)
In this case, the values of the variables are
(x, y) = (-4, 7) and (-6, -4)
Using the above as a guide, we have the following:
Substitute the known values in the above equation
So, we have the following equation
Slope = (-4 - 7)/(-6 + 4)
Evaluate
Slope = 5.5
Hence, the slope of the line is 5.5
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Complete question
What’s the answer for the slope of (-4, 7), (-6, -4)?
2 one meter of electrical cord costs 3 dollars. how much would each length of electrical cord cost? 2/3
The cost of L meters of electrical cord would be L * (3 dollars/meter).
If 1 meter of electrical cord costs 3 dollars, then each length of electrical cord that is 1 meter long would cost 3 dollars. If you want to find the cost of a length of electrical cord that is not 1 meter long, you would need to multiply the length of the cord in meters by the cost per meter. For example, if you wanted to find the cost of 2 meters of electrical cord, you would multiply 2 by 3 dollars/meter to get 6 dollars. In general, the cost of L meters of electrical cord would be L * (3 dollars/meter).
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2 one meter of electrical cord costs 3 dollars, so each length of electrical cord cost is 1.5 dollars.
In the given question, 2 one meter of electrical cord costs 3 dollars.
We have to find how much each length of electrical cord cost.
The unitary method is a method that involves finding the value of one unit (by division) and utilizing that value to determine the values of numerous units (by multiplication).
As given that
2 one meter cord = 3 dollars
To find the cost of each length of electric cord we divide 3 by 2 because the cost of 2 is 3.
1 one meter cord = 3/2 dollars
1 one meter cord = 1.5 dollars
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Rita deposited less than $1,000 into two accounts. One account paid 2.5% and the other account paid 2% annual interest. After 1 year, she earned at least $23 in interest from the two combined accounts. If x represents
the amount of money Rita deposited at 2.5% and y represents the amount of money she deposited at 2%, which system of inequalities can be used to find the values of x and y?
A. x+yz 1000 0.025x + 0.02
B. x+y< 1000 0.025x + 0.02
C. x-y≤ 1000 0.025x -0.02
D. x-y> 1000 0.025x -0.02
system of inequalities can be used to find the values of x and y are:
x + y = 10000.025x + 0.020y > 23What is inequality?In mathematics, inequalities specify the connection between two non-equal numbers. Equal does not imply inequality. Typically, we use the "not equal sign ()" to indicate that two values are not equal. But several inequalities are utilized to compare the numbers, whether it is less than or higher than.
Given, Rita deposited less than $1,000 into two accounts. One account paid 2.5% and the other account paid 2% annual interest. After 1 year, she earned at least $23 in interest from the two combined accounts.
Since, x represent the amount of money Rita deposited at 2.5% and y represents the amount of money she deposited at 2%,
x = amount invested at 2.5%
y = amount invested at 2.0%
x + y = 1000
Since, After 1 year, she earned at least $23 in interest from the two combined accounts.
Thus,
0.025x + 0.020y > 23
Therefore, system of inequalities can be used to find the values of x and y are:
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Given A(9,2), B(-1,y), C(-5,16)and D(-8,11) find the value of y so that line AB perpendicular to line CD
Answer:The value of y is 8 if AB is perpendicular CD after using the slope formula.
What is the slope?
The ratio that y increase as x increases is the slope of a line. The slope of a line reflects how steep it is, but how much y increases as x increases. Anywhere on the line, the slope stays unchanged (the same).
It is given that:
A(9, 2), B(-1, y) C.(-5, 16) and D(-8, 11),
AB is the perpendicular CD
Using the slope formula to find the slope between A and B:
(y-2)/(-1-9)
= (y-2)/-10
(11-16)/(-8-(-5)) =
-5/(-3)
= 5/3
The perpendicular slope is -3/5
(y-2)/10=-3/5
5(y-2)=3(10)
5y-10=30
5y=40
y = 8
Thus, the value of y is 8 if AB is perpendicular CD after using the slope formula.
