we prove the following step using proof by induction
What is proof by induction?
For each and every natural number n, a statement, theorem, or formula can be proved using the mathematical induction technique. By generalizing this into the "Principle of Mathematical Induction," which we would utilize to demonstrate any mathematical assertion.
We are given a statement 1+3+5+. +(2n-1)=n^2
Now first let us prove the statement for n=1 where n is the number of terms
On the left hand side = 1
And on the right hand side = 1^2 = 1
Hence the statement is proved for n =1
Now let us prove the statement for n= k
Therefore we have 1+3+5+....+k=k^2
Now let us prove the statement for n= k+1
Hence we have
1+3+5+....+k+(k+1)=
k^1+k+1
= (k+1)^2
Hence proved
Therefore we prove the following step using proof by induction
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Let point P be (2,4) and Q be (-7, -2). (a) What are the coordinates of the midpoint of segment PQ? Use slope to verify that this point is on the line through P and Q, and use the distance formula to show that this point is just as far from P as it is from Q.
Answer:
Step-by-step explanation:
The midpoint of a line segment with endpoints (x1, y1) and (x2, y2) is:((x1 + x2)/2, (y1 + y2)/2)Substituting the coordinates of points P and Q into the formula, we get:((2 + (-7))/2, (4 + (-2))/2) = (-2.5, 1)The coordinates of the midpoint of segment PQ are (-2.5, 1).To verify that this point is on the line through P and Q, we can find the slope of the line using the coordinates of points P and Q:slope = (y2 - y1)/(x2 - x1) = (-2 - 4)/(-7 - 2) = -6/-9 = 2/3We can then use the point-slope form of a linear equation to write the equation of the line through points P and Q:y - y1 = m(x - x1)
y - 4 = (2/3)(x - 2)We can substitute the coordinates of the midpoint (-2.5, 1) into this equation to see if it is a solution:1 - 4 = (2/3)(-2.5 - 2)
-3 = -10/3The equation is true, so the midpoint (-2.5, 1) is on the line through points P and Q.To show that this point is just as far from P as it is from Q, we can use the distance formula to find the distance from the midpoint to points P and Q:distance = √((x2 - x1)^2 + (y2 - y1)^2)distance from (-2.5, 1) to (2, 4) = √((2 - (-2.5))^2 + (4 - 1)^2) = √(4.5^2 + 3^2) = √(20.25 + 9) = √(29.25) = 5.4distance from (-2.5, 1) to (-7, -2) = √((-7 - (-2.5))^2 + (-2 - 1)^2) = √(4.5^2 + 3^2) = √(20.25 + 9) = √(29.25) = 5.4. Since the distance from the midpoint to points P and Q is the same, the midpoint is just as far from P as it is from Q.
Write an equation of a line in slope-intercept form for the table
The equation of the line in slope-intercept form is y = -2x+8
How to write the equation of a line in slope-intercept form?The equation of a line is an algebraic form of representing the set of points, which together form a line in a coordinate system
The slope-intercept form of the equation of a line is y = mx+b,
where m is the slope and b is the y-intercept
Given the table, pick any two points in order to find the slope:
point 1: x₁ = 0, y₁ = 8
point 1: x₂ = -2, y₂ = 12
slope (m) = (y₂-y₁) / (x₂-x₁)
slope (m) = (12-8)/(-2-0) = 4/(-2) = -2
Using y = mx+b and x₁ = 0, y₁ = 8:
8 = -2(0) + b
b = 8
y = -2x+8
Thus, the equation of the line is y = -2x+8
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How do I find the sum of the measures of the marked angles?
Answer: Total angles on the 4
Step-by-step explanation:
The sum of the marked angles = Total angles on the 4 straight lines – Sum of the interior angles of the quadrilateral. Sum of interior angles of polygon of 4 sides = (4 – 2) *1800 =3600
Hence, sum of the marked angles = 4 * 180° – 360° = 720° – 360° = 360°
A rectangular garden has a length that is 3 more than twice the width. The perimeter of the
rectangular garden is 36 ft.
+ xp
a) Write a system of equations that represents the situation.
b) What is the length and width of the garden?
Answer:
W = width
L = length = 2W + 3
P = 36 = 2W + 2L = 2W + 2(2W + 3) =
6W + 6 = 36
6W = 30
W = 5
L = 13
Exponential Function: y = 6(1)²
The value of y is 54 when x = -2
The value of y is 18 when x = -1.
The value of y is 6 when x = 0.
The value of y is 2 when x = 1.
The value of y is 2/3 when x = 2.
What is a function?
A mathematical phrase, rule, or law that establishes the link 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.
Given function is
y = 6(⅓)^x
To find the value at x = -2, -1, 0, 1, 2, put the value of x in the given equation.
Putting x = -2 in the equation y = 6(⅓)^x
y = 6(⅓)⁻²
Applying index formula (1/a)^(-m) = a^m
y = 6(3)²
y = 6 × 9
y = 54
Putting x = -1 in the equation y = 6(⅓)^x
y = 6(⅓)⁻¹
Applying index formula (1/a)^(-m) = a^m
y = 6(3)¹
y = 6 × 3
y = 18
Putting x = -1 in the equation y = 6(⅓)^x
y = 6(⅓)⁰
Applying a⁰ = 1
y = 6 × 1
y = 6
Putting x = 1 in the equation y = 6(⅓)^x
y = 6(⅓)¹
y = 6 × 1 /3
y = 2
Putting x = 2 in the equation y = 6(⅓)^x
y = 6(⅓)²
y = 6 × (1/9)
y = 6/9
Divide the denominator and numerator by 3:
y = 2/3
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HELP NOW A dugout needs to reach at least 54 feet below ground level. This means it needs to be more than 2 1/4 times its current depth. This depth is represented by the inequality 2 1/4d<−54, where d is the current depth of the dugout.
Select from the drop-down to correctly interpret the solution of this inequality.
The current depth of the dugout is *Choose..* feet below ground level.
a. at most 24
b. deeper than 24
c. less than 24
The depth of the dugout is less than 24.
What is inequality?
In mathematics, inequalities describe the relationship between two values that are not equal. Equal means to be equal, not. The "not equal symbol (≠)" is typically used to indicate that two values are not equal. But different inequalities are used to compare the values to determine whether they are less than or greater than.
Consider, the given inequality
[tex]2\frac{1}{4}d < -54[/tex]
Where d is the depth of the dugout.
We have to find the depth of the dugout.
Now to simplify the inequality.
[tex]2\frac{1}{4}d < -54[/tex]
convert improper fraction into a fraction.
[tex]2\frac{1}{4} = \frac{9}{4}[/tex]
⇒ 9/4d < -54
Multiply both sides by 4/9
⇒ d < -54 x 4/9
d < -6 x 4
d < -24
Hence, the depth of the dugout is less than 24.
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4. Express the following in exponential form: 4x 4x4 x5x5x5
Answer: 4^3 * 5^3
Step-by-step explanation: 4 repeats itself 3 times and 5 repeats itself 3 times
4^3 * 5^3
Put the numbers in numerical order
4 comes before 5
Answer:
[tex] { 4}^{3} \times {5}^{3} [/tex]
Step-by-step explanation:
I'd assume your x's stand for multiplication, so using that logic there are three fours and three fives. Each being multiplied by itself, so the simplified/exponential version of the equation is:
[tex] {4}^{3} \times {5}^{3} [/tex]
Make sure you put in numerical order (4 goes before 5).
use a single digit times a power of 10 to estimate how many times greater the amount of water is at the Georgia Aquarium
Answer:
The amount of water at the Georgia Aquarium is 20 times greater than the aquarium of the Americas hold.
Step-by-step explanation:
What is an expression?Expression in maths is defined as the collection of numbers variables and functions by using signs like addition, subtraction, multiplication, and division.
Numbers (constants), variables, operations, functions, brackets, punctuation, and grouping can all be represented by mathematical symbols, which can also be used to indicate the logical syntax's order of operations and other features.
Given that the tanks at the Georgia Aquarium hold approximately 8.4 × 10⁶ gallons of water. The tanks at the Audubon Aquarium of the Americas hold about 400,000 gallons of water.The number of the time will be calculated as below:-= 8.4 × 10⁶ / 400,000= 20.1
Therefore, the amount of water at the Georgia Aquarium is 20 times greater than the aquarium of the Americas hold.To know more about an expression followbrainly.com/question/15772941#SPJ2
Kayla is working two summer jobs, making $10 per hour lifeguarding and $21 per
hour tutoring. Last week Kayla worked 4 times as many hours tutoring as she worked
lifeguarding and earned a total of $188. Write a system of equations that could be
used to determine the number of hours Kayla worked lifeguarding last week and the
number of hours she worked tutoring last week. Define the variables that you use to
write the system.
Answer:
10L + 21t = 188
t = 4L
Step-by-step explanation:
Let L be the number of hours spent lifeguarding and t be the number of hours spent tutoring. Kayla earned $188 total, so 10L+ 21t = 188(the number of hours spent lifeguarding/tutoring times how much she gets paid every hour). We also know that she spent 4 times longer tutoring than lifeguarding, so t = 4L. These are our systems of equations.
A gardener drew this diagram to find the area of land inside a right-triangular-shaped fence. The length of fence b is 130 feet. The measure of angle A is 42°. What is the area of land inside the fence rounded to the nearest square foot?
The area of land inside the fence is 7600 square feet.
What is the area of the triangle?
If the height of the triangle is 'h' and the base is 'b', then the area of the triangle can be computed by using the formula,
Area = 1/2 * base * height
To find the area of the land inside the fence, we can use the formula for the area of a right triangle:
Area = (1/2) * a * b
where a is the length of one of the legs of the triangle and b is the length of the hypotenuse.
We are given that the length of fence b is 130 feet and that the measure of angle A is 42°. Since the triangle is right, angle A is complementary to angle 90°, so the measure of angle A is 90° - 42° = 48°. Since the measure of angle A is 48°, the other leg of the triangle (which we will call "a") is the side opposite angle A.
We can use the sine function to find the length of side a:
a = b * sin A
= 130 * sin 48°
= approximately 117.3 feet
Substituting these values into the formula for the area of a right triangle gives:
Area = (1/2) * 117.3 feet * 130 feet
= approximately 7600.5 square feet
Rounded to the nearest square foot, the area of land inside the fence is 7600 square feet.
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A bag contains 10 red marbles, 7 blue marbles, and 3 green marbles. How many blue marbles need to be added to the bag so that the probability of randomly drawing a blue marble is 3/4?.
In order to increase the likelihood that a blue marble would be drawn at random to 3/4, we must thus add 32 more blue marbles.
What is probability?The area of mathematics known as probability deals with numerical representations of the likelihood that an event will occur or that a statement is true. An event's probability is a number between 0 and 1, where, roughly speaking, 0 denotes the event's impossibility and 1 denotes certainty. A probability is a numerical representation of the likelihood or chance that a specific event will take place. Both proportions ranging from 0 to 1 and percentages ranging from 0% to 100% can be used to express probabilities.
Here,
Let the number of blue marbles added be x,
Probability=3/4
Total marbles=20
Number of blue marbles=7
(7+x)/(20+x)=3/4
4(7+x)=3(20+x)
28+4x=60+3x
x=60-28
x=32
So, we need to add 32 more blue marbles to make the probability of randomly drawing a blue marble be 3/4.
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The triangle congruency theorem can be used to prove △ ≅ △onp.
The triangle congruency theorem which can be used to prove
△ ≅ △onp is ASA (angle side angle)
Let the triangles be ONP and GHL
The congruence theorem, which will demonstrate that "GHL" and "ONP ," are congruent is to be discovered.
As can be seen, side lengths GH of GHL and ON of ONP are equal.
Furthermore, we are informed that mONP = mHKJ.
Since line segments GK and JL connect to produce them, we can see that GHL is the vertical opposite angle of ONP . Therefore, using the vertical angles theorem, mONP congruence mKHJ.
We can see that both triangles contain two congruent angles and one congruent included side; as a result,
GHL ONP by ASA congruence.
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Consider what you know about the sampling distribution of the sample proportion. This sampling distribution:
a. will become more variable as the sample size increases.
b. will be Normal in shape only if the sample size is at least 100.
c. will have a center equal to the population proportion, or p.
d. has a shape that is skewed to the right, regardless of sample size.
e. is a collection of the parameters of all possible samples of a particular size taken from a particular populatio
From the sampling distribution we know:
a. will become more variable as the sample size increases.
What is sampling distribution?A sampling distribution is the probability distribution of a statistic that was acquired from more samples taken from a certain population. The distribution of frequencies for a variety of potential outcomes that could happen for a population statistic makes up the sampling distribution of a specific population.
When it comes to the spread of all sample proportions, theory predicts behaviour far more accurately than just stating that larger samples have a smaller spread. In actuality, the sample size, n, and the standard deviation of each sample proportion.
Thus, option (a) correctly describe about sampling distribution of the sample proportion.
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how many license plates can be made using either 3 digits followed by 3 letters, or 3 letters followed by 3 digits?
Answer:
infinite
Step-by-step explanation:
the law in most states in the usa states there must be 3 letters and 3 numbers so there could be infinite
-hope this helps-
Find the value of X and 2x
Answer:
x=125,2x=250
Step-by-step explanation:
A hexagon sum of angles is 720 degrees.
The sum of the hexagon's angles is 345+3x
345+3x=720
3x=375
x=125
2x=250
7 Determine whether the equation below does or does
not represent a proportional relationship. Support your
answer with a table or a graph.
Equation: y = 3x + 2
pls help
Answer:
Step-by-step explanation:
3
h is a trigonometric function of the form h(x)=acos(bx+c)+d.
Below is the graph h(x). The function has a maximum point at (-pi/2,5) and has a minimum point at (pi/4,-4)
Find the Formula for h(x). Give an exact expression
The Formula for h(x) is 7.5cos(51.4x - 154.3)-1.5.
What is trigonometric function ?
Trigonometric functions are referred to as Circular Functions will be merely outlined because the functions of associate degree angle of a triangle.
The difference in y values from max to min = 15. 15/2 = a
max cosine = 1, min cosine =-1
7.5 x 1 - 1.5 = 6
7.5x -1 -1.5 = -7.5-1.5= -9
h(x) = 7.5cos(bx+c) -1.5
now just get a period or frequency that matches the x coordinates 3 and 6.5 to get b and c
at (3,6) cosine = 1 to get a maximum value. cosine is at a max when cos(bx+c) = 1 and bx+c=0, 360, 720...
if x=3, then 3b +c=0 and c=-3b
for (6.5,-9) to be a minimum h value, then cos(6.5b+c) =-1 that happens when the angle = 180. +n360 for any integer n. 6.5b+c = 180
c= 180-6.5b
c=-3b
180-6.5b =-3b
add 6.5b to both sides to get
180= 3.5b
b = 180/3.5 = 51 1/7
c=-3b = -153 3/7
h(x) = 7.5cos(51.4x - 154.3)-1.5
hence , Formula for h(x) is 7.5cos(51.4x - 154.3)-1.5
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Mast 1 broadcast radio signals that can be heard within a radius of 15km.
Mast 2 broadcast radio signals that can be hear within a radius of 20km.
The difference in the areas of the signals based on the information given is 549.5km².
How to calculate the area?The formula to calculate the area of a circle will be:
= πr²
where r = radius
Mast 1 broadcast radio signals that can be heard within a radius of 15km. The area will be:
= πr²
= 3.14 × 15²
= 706.5km²
Mast 2 broadcast radio signals that can be hear within a radius of 20km. The area will be:
= πr²
= 3.14 × 20²
= 1256km²
The difference in the areas will be:
= 1256km² - 706.5km²
= 549.5km²
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Complete question
Mast 1 broadcast radio signals that can be heard within a radius of 15km.
Mast 2 broadcast radio signals that can be hear within a radius of 20km.
Calculate the difference in the areas of the signals.
3. Students are taking three busses to go on a field trip. On the first bus there were 14 students, on the second, 18 students, and on the third bus there were 12 students. The total number of students would then be: 14+18+12 To calculate the total students, their chaperone, Ms. Flores decides to add 18 and 12 to get 30 and then adds 14 to get 44. What property is Ms. Flores using to add these numbers in this order?
Ms. Flores using to add these numbers in this order is follow Commutative Property of Addition.
What is Commutative Property of Addition?
No matter what sequence the addends are in when we add two or more integers, the result is always the same. This characteristic may be expressed as A + B Equals B + A. In this case, 2+4=4 + 2 = 6. Due to the fact that the total of both phrases is 6, 2+4 is equivalent to 4+2. The Commutative Property of Addition refers to this.
On the first bus there were 14 students, on the second, 18 students, and on the third bus there were 12 students. The total number of students would then be: 14+18+12=44.
Ms. Flores decides to add 18 and 12 to get 30
18+12= 30
And then adds 14
30+14=44
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Explain how you would use a number line to find the absolute value of –12.
Answer:
12 notches from both left and right of 0
you want the absolute value of (-12). all that means is how far is the number away from 0. in this case it's just 12
Evaluate
-3/4 - (-1/5)
SIMPLEST FORM
The sum of the fractions (- 3/5 + 1/5) in simplest form is - 11/20.
What is a fraction?A fraction is written in the form of p/q, where q ≠ 0.
Fractions are of two types they are proper fractions in which the numerator is smaller than the denominator and improper fractions where the numerator is greater than the denominator.
Given, Are two proper fractions - 3/4 and - 1/5, and we are required to take the difference which is,
= - 3/4 - (- 1/5).
Now the negative signs will distribute to positive which is,
= - 3/4 + 1/5.
Now, the LCM of 4 and 5 is 20.
= {(-3×5) + (1×4)}/20.
= (- 15 + 4)/20.
= - 11/20.
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Adam records the increase in the height, in centimeters, of a plant each week for 10 weeks. His data are shown.
4.5, 3.2, 8.0, 9.1, 10.2, 6.8, 7.2, 95, 10.0, 8.8
Adam wants to display the data on a box plot.
What are the values of the lower quartile, median, and upper quartile that Adam should use for his box plot?
Lower quartile
Median =
Upper quartile =
The values needed for the box plot that Adams wants to display are:
Lower quartile = 6.8
Median = 8.4
Upper quartile = 9.5
How to Find the Lower Quartile, Median, and Upper Quartile?Given the following data that represents the records of the increase in the height of a plant for 10 weeks, 4.5, 3.2, 8.0, 9.1, 10.2, 6.8, 7.2, 9.5, 10.0, 8.8, in order to draft a box plot, the following steps are what Adams should take to find each of the values needed.
Step 1: Order the data set from least to the greatest.
3.2, 4.5, 6.8, 7.2, 8.0, 8.8, 9.1, 9.5, 10.0, 10.2
Step 2: Find the median.
The median is the middle of the data set = (8.0 + 8.8)/2 = 16.8/2 = 8.4
Step 3: Find the lower quartile which is the center of the first part of the data that is divided by the median.
Lower quartile = 6.8
Step 4: Find the upper quartile which is the center of the second part of the data that is divided by the median.
Upper quartile = 9.5
Therefore, the values that Adam needs for his box plot are:
Lower quartile = 6.8
Median = 8.4
Upper quartile = 9.5
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Salma got offered two different job with telecommunication companie with different citie. She ha the option to elect which city he would like to work. A job in Milan offer €70,000 per year. A job in New York offer $102,900 per year. The exchange rate are:
£1 = €1. 18
£1 = $1. 72
Which job ha the higher alary?
According to the exchange rates £1 = €1. 18, £1 = $1. 72 the job in New York will pay Salma the higher salary.
What is greater value mean?In mathematics, the term "more than" refers to an inequality between two values, quantities, or numbers, where one is greater than the other. In other terms, we say that A is more than B when A is greater in contrast to B.
A job in Milan offer €70,000 per year;
And exchange rate are:
£1 = €1. 18
so she will get
£=70,000x1. 18
£=82600
on other hand job in New York offer $102,900 per year. The exchange rate are:
£1 = $1. 72
so she will get
£= 102,900x1. 72
£=176988
So the job in New York will pay her higher salary.
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flips a fair coin five times, what is the probability that there are 3 heads in a row somewhere in the sequence of five flips.
flips a fair coin five times, 5/16 is the probability that there are 3 heads in a row somewhere in the sequence of five flips.
What is probability?The number of positive outcomes to all possible outcomes of an event is the ratio, which is known as probability.
The number of positive outcomes may be expressed by x for an experiment with n number of outcomes.
The following equation can be used to determine an event's probability:
Probability(Event) = Favorable outcomes / Total Outcomes = x/n
Considering a fair coin, after 5 flips, there are [tex]2^5[/tex] = 32 different arrangements of heads and tails.
To get exactly 3 heads,
[tex]^5C_3[/tex] = 5! / (5 - 3)! 3!
= 5!/ 3! 2!
= 10 ways
P(exactly 3 heads) = 10/32 = 5/16
Therefore, the probability of exactly 3 heads is 5/16.
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Determine the slope of the line.
A. 2/1
B. -2/1
C. 1/2
D -1/2
Someone please help me!!!!
Answer: A 2/1
Step-by-step explanation:
Determine the slope of the line by finding 2 points
(2,-1),(4,3)
use slope formula
y2-y1/x2-x1
plug in and solve
3+1/4-2
4/2
2
the slope is 2 or 2/1
Solve for x ---> -2x - 48 ≥ 10x – 24
Answer:
x = 2
Step-by-step explanation:
-2x - 48 ≥ 10x - 24
-48 + 24 ≥ 10x + 2x
24 = 12x
x = 2
Joseph and Raquel are building a scale model of the Alamo Mission for their Texas history project using a scale in which 2 inches represents 30 feet. The enclosed area of the Alamo Mission was 480 feet long and 160 wide.
What should the length of the enclosed area of the scale model be in inches?
Responses
Representative fraction is a form of scale that can be used to change the size of an object to form an image. The required length is 32 inches.
A scale is a fraction of numbers that can be used to either increase or decrease the size of a given object. It is majorly referred to as representative fraction in scale drawing.
Representative fraction = [tex]\frac{length on drawing}{original length}[/tex]
In the given question, we have;
12 inches = 1 feet
x inches = 30 feet
So that:
x = 30 x 12 inches
= 360 inches
representative fraction = [tex]\frac{2 inches}{360 inches}[/tex]
= 5.56 x [tex]10^{-3}[/tex]
Thus, given that the length of the Alamo Mission is 480 feet long. Then:
480 feet = 480 x 12 inches
= 5760 inches
So that,
the required length in inches = 5760 x 5.56 x [tex]10^{-3}[/tex]
= 32 inches
Therefore the expected length of the Alamo Mission is 32 inches.
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A model satellite has a scale of 1 in: 6 ft.
If the model satellite is 3 in wide, then
how wide is the real satellite?
A) 9 ft
C) 1 ft
B) 18 ft
D) 2 ft
Answer:
B) 18 ft
Step-by-step explanation:
The scale is 1 in. model : 6 ft real.
6/1 = 6
To go from model size to real size, multiply the model size by 6 and change from inches to feet.
3 in. model = 3 × 6 ft = 18 ft
(b) The perimeter of a rectangular pool is 302 m. If the width of the pool is 54 m, what is its length?
Length of the pool:
L= 97
Step-by-step explanation:
The perimeter is all of the sides of the rectangular pool added up. We know the both widths are gonna be 54m. So to find both widths together you need to multiply 54 by 2 to give us 108. The we need to find both lengths together, so subtracting 302-108 will get us 194. Both lengths together, 194, divided by 2 is 97, which is our length.
The graph shows the rate at which Natasha paints a wall.
Enter values to complete each statement below based on the graph.
Answer:
Step-by-step explanation:
The unit rate is 400 square feet per gallon.
What is unitary method ?
Unitary method is a mathematical way of deriving a single unit and then finding the required unit by multiplying.
A graph is shown in the diagram and two points are given x co-ordinate represents paint used per gallon and y co-ordinate represents wall painted in square feet.
Given (1/4,100) implies 1/4th of a gallon paint is needed to pain 100 square feet of wall.
∴ In 1 gallon (100)÷1/4 square feet of wall can be painted which is 400 square feet.
And the unit rate is 400 square feet per gallon.