Answer:
$6
Step-by-step explanation:
50% x 2 = 100%
3 x 2 = $6
If triangle ABC has the following measurements, find the measure of side c:
a = 17
b=23
C = 76°
O a
Ob
Od
37.56
7.69
31.74
25.08
If triangle ABC has the following measurements, than the measure of side c is 25.08
What is triangle ?
A triangle in which it contains three sides and three angles and the sum of three angles be 180 degrees.
Given ,
a = 17
b = 23
C = 76 degrees.
So,
we know that,
c = [tex]\sqrt{a^{2}+b^{2}-2abcosc }[/tex]
c = [tex]\sqrt{17^{2} + 23^{2} - 2*17*23*cos76 }[/tex]
c = [tex]\sqrt{289 + 529 - 884*cos76}[/tex]
c = [tex]\sqrt{818-189}[/tex]
c = [tex]\sqrt{629}[/tex]
c = 25.08
hence , If triangle ABC has the following measurements, than the measure of side c is 25.08
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Substitution Property of Equality
y=-5 and 7x + y =11
Answer:
x=2.28
Step-by-step explanation:
7x +y=11 we substitute y by ( -5)
7x +(-5)=11 (+)×(-)= --
7x-5=11
7x=11+5 when we move -5 to the opposite it become +5
7x =16
x=16 ÷7
x=2.28
15. A savings account earns interest. The account initially had $1,500 deposited in it. The worth of the account
after t-years can be calculated using the formula:
A(t)=1500e^0.3r
So, on solving the question we can say that the linear equation A(t)=1500e^0.3r = A(t) = 1232
What is a linear equation?A linear equation is one that has the form y=mx+b in algebra. B is the slope, and m is the y-intercept. It's usual to refer to the previous clause as a "linear equation with two variables" because y and x are variables. The two-variable linear equations known as bivariate linear equations. There are several instances of linear equations: 2x - 3 = 0, 2y = 8, m + 1 = 0, x/2 = 3, x + y = 2, and 3x - y + z = 3. It is referred to as being linear when an equation has the form y=mx+b, where m stands for the slope and b for the y-intercept. When an equation has the formula y=mx+b, with m denoting the slope and b the y-intercept, it is referred to as being linear.
A(t)=1500e^0.3r
the linear equation
A(t) = 1232
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Eric traveled 78 mile. He rode his bike 45 of the distance.
Eric used this expression to find the distance he rode his bike.
78×45
How far did Eric ride his bike?
Enter your answer as a fraction in simplest form by filling in the boxes.
Answer: 7/10 of a mile
Step-by-step explanation:
As we can use the expression that Eric used to solve this question!
7/8 x 4/5 = 7/10 of a mile
If we combine like terms in 4b - 2v + 4b + 6v + 13 what would be our resulting expression?
hurry please
The resulting expression would be 8b + 4v + 13.
What is an Expression?
An expression consists of one or more numbers or variables along with one more operation.
If we combine like terms in 4b - 2v + 4b + 6v + 13 the resulting expression would be, 8b + 4v + 13.
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The resulting expression would be 8b + 4v + 13.
What does an expression exactly mean?An expression is created by combining one or more variables or numbers with another operation. A mathematical expression is made up of a statement, at least two integers or variables, and one or more arithmetic operations. This mathematical operation enables the multiplication, division, addition, or subtraction of numbers. The following is the structure of an expression: Expression: (Number/Variable, Math Operator, Math Operator)
How do you do math expressions?You must substitute a number for each variable and carry out the arithmetic operations in order to evaluate an algebraic expression. Since 6 + 6 equals 12, the variable x in the example above is equal to 6. If we are aware of the values of our variables, we can substitute those values for the original variables before evaluating the expression.
If we combine like terms in 4b - 2v + 4b + 6v + 13 the resulting expression would be, 8b + 4v + 13.
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simplify 7a+8a^2+7a^0-4a^2-9a
Answer:
4a^2-2a+7
Step-by-step explanation:
combine like terms
7a+-9a= -2a
8a^2 - 4a^2 =4a^2
and anything to the power of 0 is a 1
so 7x1 is 7
so the answer is 4a^2 -2a+7
Find the area of the rectangle.
The area of the rectangle is
(3x-4) yards
(Simplify your answer.)
(3x+4) yards
The area of rectangle with length 3x + 2 and width 3x - 4 is 9x² - 6x - 8.
What is the area of rectangle?Once the length and width of a rectangle are known, the area is calculated. The area of a rectangle can be calculated by multiplying its length and width.
Here given that,
The rectangle's length is (3x + 2) units.
The width of the rectangle is (3x - 4).
We must determine the area of a rectangle.
Now,
Since the area of a rectangle equals length x width .
As a result, the area of a rectangle equals,
A = (3x + 2) (3x - 4)
By simplifying the expression we get,
= (3x)(3x) + (3x)(-4) + 2(3x) + 2(-4)
= 9x² - 12x + 6x - 8
= 9x² - 6x - 8
Therefore the area of rectangle will be 9x² - 6x - 8.
The complete question:
"The length of a rectangle is 3x + 2 and the width is 3x – 4. What is the area of the rectangle in terms of x?"
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How many solutions does the system of linear equations represented in the graph have?
Coordinate plane with one line that passes through the points 0 comma negative 1 and 1 comma negative 3.
One solution at (−1, 0)
One solution at (0, −1)
No solution
Infinitely many solutions
The answer in the system of linear equations represented in the graph have is b) One solution at (0, -1)
How is this determined?The system of linear equations represented in the graph has one solution, which is the point of intersection of the line with the coordinate plane.
This line can be represented by the equation y = mx + b where m is the slope and b is the y-intercept.
To find the slope, we can use the point slope formula which is m = (y2 - y1) / (x2 - x1)
We have the points (0,-1) and (1,-3)
So m = (-3 - (-1)) / (1 - 0) = -2/1 = -2
y = -2x + b is the equation of the line where x and y are the coordinates and b is the y-intercept.
b can be found by substituting the point (0,-1) in the equation.
-1 = -2(0) + b
b = -1
So the equation of the line is y = -2x - 1
therefore the solution is the point of intersection of the line with the coordinate plane which is (0, -1)
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through (2,4) parrallel to y=3x+
The slope y is 3x - 10.When the line to be examined's slope is known, and the provided point also serves as the y intercept.
The slope intercept form of a line's equation can be found by using this method?When the line to be examined's slope is known, and the provided point also serves as the y intercept, the slope intercept formula, y = mx + b, is utilized (0, b). B stands in for the y value of the y-intercept point in the formula.
According to question:-
In the equation of a line's slope-intercept form, y = mx + b, we get y = 3x + 2, m1 = m2 gives us y = 3x +2, and m = 3 gives us the point (2 -4) where x = 2 and y = -4.
4 6 + b deducts 6 from both sides.
-10 = b, b = -10
After all, y = 3x - 10.
The complete question is,
y=3x+2 via (2,-4) parallel to
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Z is a standard normal randomvariable. The P(z > 2.11)equals
The standard normal random variable P(z > 2.11) = 1(1/2(1+erf(1.45))
A standard normal random variable Z has a mean of 0 and a standard deviation of 1.
P(Z > 2.11) can be found using the cumulative distribution function (CDF) of the standard normal distribution.
The CDF of the standard normal distribution is provided by:
F(z) = P(Z 2.11) we can use the complement rule:
P(Z > 2.11) = 1 - P(Z 2.11)
= 1 - F(2.11)
= 1 - (1/2) * (1 + erf(2.11/sqrt(2)))
= 1 - (1/2) * (1 + erf(1.45))
Where erf is the error function.
This value can be calculated using a calculator or a standard normal table by looking up the value of erf(1.45) and subtracting it from 1/2.
Therefore, the resulting value is the probability that a standard normal random variable is greater than 2.11
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-40
5
-5
40
x
Select all of the following ordered pairs that are in
the solution set of the inequality shown.
(-10,-10)
(0, 0)
(0, 10)
(10,0)
(20, 0)
(100, 100)
The only solution that satisfies the inequality is the point (0,0)
What is inequality and give step by step explanation for above answer?An inequality is a statement that compares two expressions using mathematical symbols such as <, >, <=, >=, !=. It is used to represent a range of values that a variable can take.x^2 + y^2 <=40 is an equation of a circle with center (0,0) and radius of 40.The inequality means that for all points on and inside the circle, x^2 + y^2 is less than or equal to 40.Now we will check each ordered pair provided to see if it satisfies the inequality:(-10,-10): x^2 + y^2 = (-10)^2 + (-10)^2 = 100 + 100 = 200, which is greater than 40, so this ordered pair does not satisfy the inequality and is not in the solution set.(0, 0): x^2 + y^2 = 0 + 0 = 0, which is less than or equal to 40, so this ordered pair satisfies the inequality and is in the solution set.(0, 10): x^2 + y^2 = 0 + 10^2 = 100, which is greater than 40, so this ordered pair does not satisfy the inequality and is not in the solution set.(10,0): x^2 + y^2 = 10^2 + 0 = 100, which is greater than 40, so this ordered pair does not satisfy the inequality and is not in the solution set.(20, 0): x^2 + y^2 = 20^2 + 0 = 400, which is greater than 40, so this ordered pair does not satisfy the inequality and is not in the solution set.(100, 100): x^2 + y^2 = 100^2 + 100^2 = 10000+ 10000 = 20000, which is greater than 40, so this ordered pair does not satisfy the inequality and is not in the solution set.To learn more about inequality refer:
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A band wants to tend a rehearsal space. They will be charged a cleaning fee and hourly rate for use of the room.
Room A: Cleaning fee: $50
Rate: $24 per hour
Room B: Cleaning fee: $80
Rate: $18 per hour
The band has $250 to spend. Let t be the time (in hours) the band rents the room.
a. Write and solve an inequality to represent the time (in hours) the band could rent room A.
b. Write and solve an inequality to represent the time (in hours) the band could rent room B
c. Which room should the band rent? Explain your answer.
Answer:
band could rent room B for a maximum of 9.44 hours.
Step-by-step explanation:
To represent the time (in hours) the band could rent room A, we need to consider both the cleaning fee and the hourly rate. The total cost of renting room A for t hours is given by the expression "50 + 24t". Since the band has $250 to spend, we can write an inequality to represent the maximum time they could rent the room:
50 + 24t <= 250
We can solve this inequality for t by subtracting 50 from both sides:
24t <= 200
Then, by dividing both sides by 24, we get:
t <= (200/24) hours
t <= 8.33 hours
So, the band could rent room A for a maximum of 8.33 hours.
b. To represent the time (in hours) the band could rent room B, we can use a similar approach as above. The total cost of renting room B for t hours is given by the expression "80 + 18t". Using the same budget of $250, we can write the following inequality:
80 + 18t <= 250
We can solve this inequality for t by subtracting 80 from both sides:
18t <= 170
Then, by dividing both sides by 18, we get:
t <= (170/18) hours
t <= 9.44 hours
So, the band could rent room B for a maximum of 9.44 hours.
c. From the above results, we can see that the band could rent room B for a slightly longer period of time compared to room A. However, the hourly rate for room B is cheaper than that of room A. So even though the band will get to use room B for a longer period of time, they will pay less overall to rent it. Additionally, the cleaning fee is cheaper in room B. Based on this information, the band should rent room B. They can stay in the room longer, pay less per hour, and pay less in cleaning fees.
2 divided by 3xy raised to power 2 when x = negative 1 divided by 3, and y = one half
The value for expression 2/3xy² when x = -1/3 and y = 1/2 is 8.
What is an expression?
Mathematical expressions consist of at least two numbers or variables, at least one arithmetic operation, and a statement. It's possible to multiply, divide, add, or subtract with this mathematical operation.
The given expression is 2/3xy².
The value of x is given as x = -1/3.
The value of y is given as y = 1/2.
To find the value of the expression, substitute the value of x and y in the expression -
= 2/3xy²
Expand using the power rule -
= [2/(3 × (-1/3) × (1/2)²)]
Use the arithmetic operation of multiplication -
= [2/(3 × (-1/3) × (1/4)]
= [-2/(-3/12)]
= -2/(-1/4)
The denominator will reciprocate -
= -2 × -4
= 8
Therefore, the final value is obtained as 8.
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4. If we want to find the volume of the ice cream shown below, we will need to find the volumes of which 3D solids? (Mark all that apply)
A Sphere/Hemisphere
B Prism
C Cylinder
D Pyramid
E Cone
F Cube
Answer:
A, E
Step-by-step explanation:
The lower part, the cone, is shaped like a cone.
The upper part, the ice cream, is shaped like a hemisphere.
Answer: A, E
The number line represents the solution to which inequality?
Answer: Choice G [tex]3\text{x} \le 21[/tex]
======================================================
Reason:
The inequality for choice G solves to the following shown below.
[tex]3\text{x} \le 21\\\\3\text{x}/3 \le 21/3\\\\\text{x} \le 7\\\\[/tex]
In the second step, I divided both sides by 3 to undo the multiplication going on when saying "3x".
Once we arrive at [tex]\text{x} \le 7[/tex], the graph will have a closed endpoint at 7 and shading to the left. This is to visually describe all values of x that are 7 or smaller. The endpoint is included as part of the solution set.
This shows why choice G is the answer.
---------------------
Something like choice F has the inequality [tex]\text{x}-2 \le 9[/tex] solve to [tex]\text{x} \le 11[/tex] after adding 2 to both sides. The graph of [tex]\text{x} \le 11[/tex] involves a closed endpoint at 11 and shading to the left, which doesn't match the given number line graph. This allows us to rule out choice F.
Choices H and J are ruled out for similar reasoning.
Indicate in standard form the equation of the line given the following information: The line that contains the point Q(1, -2) and is parallel to the line whose equation is y - 4 = 2/3 (x - 3) Enter your answer into the blank equation box.
The equation of the parallel line will be y = (2/3)x - 8/3.
What is the equation of a parallel line?Let the equation of the line be ax + by + c = 0. Then the equation of the parallel line that is parallel to the line ax + by + c = 0 is given as ax + by + d = 0.
The equation of the line is given below.
y - 4 = 2/3 (x - 3)
The equation of the line that is parallel to the given line will be written as,
y = (2/3)x + c
The equation of the line that passes through (1, -2), then we have
- 2= (2/3)(1) + c
- 2 = 2 /3 + c
c = - 8 / 3
Then the equation of the parallel line will be y = (2/3)x - 8/3.
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I need to show my work help me out
The distance of the flag pole from the end of the shadow is 23.32 feet.
How to find the side of a right triangle?A 20 foot flag pole is casting a shadow that is 12 feet long. Therefore, the distance of the top of the flag pole to the tip of the shadow can be calculated as follows:
The situation forms a right angle triangle.
Hence, using Pythagoras's theorem, we can find the distance.
Therefore,
c² = a² + b²
where
c = hypotenusea and b are the legsTherefore,
20² + 12² = c²
400 + 144 = c²
c = √544
c = 23.3238075794
Therefore,
distance of the flag pole to the end of the shadow = 23.32 feet
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Write a system of linear inequalities represented by the graph.
Linear inequality represented by the graph is y > x/3
along with linear equations: y = x/3 and y = (x/3) - 2
What is Inequality?A relationship between two expressions or values that are not equal to each other is called 'inequality.
The first line passes through
(0, 0) and (3, 1)
The slope of line is m = 1-0/3-0 = 1/3
y intercept = 0
So the equation of line is y = x/3
As the shading is outside so y > x/3
Now let us find for second line
(-3, -3) and (3, -1)
slope m' = -1 - -3/3 - - 3 = 2/6 = 1/3
Now y intercept
-2 = m(0) + c
c = - 2
So equation is y = x/3 - 2
Hence, linear inequality y > x/3 and linear equations: y = x/3, y = x/3 - 2 represent the graph.
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Please help will mark brainliest!!
Answer:51
Step-by-step explanation:
7*7=49，10*10=100 100-49=51
Submarine A is 1200 feet underwater and has descended at a constant rate of 12 feet per minute. Submarine B is 1100 feet underwater but descends at a constant rate of 16 feet per minute. When will the two submarines be at the same depth?
The time when the two submarines will be at the same depth is 25 minutes.
How to calculate the value?An equation simply has to do with the statement that illustrates the variables given. In this case, it is vital to note that two or more components are considered in order to be able to describe the scenario.
In this case, Submarine A is 1200 feet underwater and has descended at a constant rate of 12 feet per minute. Submarine B is 1100 feet underwater but descends at a constant rate of 16 feet per minute
The equation will be:
1200 - 12m = 1100 - 16m
Collect the like terms
16m - 12m = 1200 - 1100
4m = 100
Divide
m = 100 / 4
= 25 minutes
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simplify
[2/11]^3 ÷ [4/22]^6
Answer:
(2/11)^3/(4/22)^6
= (2/11)^3/(2/11) ^6
=1/(2/11)^3
=(11/2)^3
Please guys i need help 80 points for this
Answer:
mario
Step-by-step explanation:
For f(x), the transformation of the function is stated in the first column. Determine the type of transformation
The first column of the transformation of the function contains information on the transformation of the functions f(x) - f(x) -2.
What is functions ?The study of mathematics comprises the study of quantities and their variations, equations and associated structures, shapes and their positions, and locations where they can be found. A combination of inputs and corresponding outputs are referred to as a "function," which describes the relationship between them. The term "function" refers to an association between inputs and outputs where each input produces a single, unique result. There are two domains, or scopes, assigned to each function. Usually, the symbol f is used to signify functions (x). input is an x. There are four basic categories of functions that are available: on functions, one-to-one functions, many-to-one functions, within functions, and on functions.
given
transformation of functions
f(x) - f(x) -2
translate 2 units down
f(x) - f(x - 2)
translate 2 units right
f(x) - f(x) + 2
translate 2 units up
The first column of the transformation of the function contains information on the transformation of the functions f(x) - f(x) -2.
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i need help asap!!! 30 points
Answer:
8 × sin(35)
Step-by-step explanation:
remember the trigonometric norm circle and the triangle inside describing with its sides the trigonometric function values of the angle at the center of the circle.
sine is the vertical side, cosine is the horizontal side.
and the longest side really creating that angle at the center is the radius (going all the way from the center to the circle arc).
do not forget, the trigonometric functions sine, cosine, ... deliver values for the norm circle only (radius = 1).
for any circle larger or smaller these function values need to be multiplied by the actual radius to get the actual side lengths of the triangle.
now, we are allowed to rotate or twist/turn a given right-angled triangle to be oriented like the basic trigonometric triangle.
in our case we can simply flip things upside-down.
and we see, BC is simply sine of 35° (multiplied by the radius 8).
it does not matter that it is originally pointing down. it is for the angle 35°, and sine of angles 0° <= angle <= 180° must be positive.
What is the interest earned in a savings account after 12 months on a balance of $5000 if the interest rate is 1% APY compounded yearly?
The interest earned in savings is given by the equation I = $ 50
What is Compound Interest?Compound interest is interest based on the initial principle plus all prior periods' accumulated interest. The power of compound interest is the ability to generate "interest on interest." Interest can be added at any time, from continuously to daily to annually.
The formula for calculating Compound Interest is
A = P ( 1 + r/n )ⁿᵇ
where A = Final Amount
P = Principal
r = rate of interest
n = number of times interest is applied
b = number of time periods elapsed
Given data ,
Let the interest be represented as I
Now , the amount invested A = $ 5000
The number of months = 12 months = 1 year
The rate of interest = 1 %
The interest I = A - P = P ( 1 + r/n )ⁿᵇ - P
So , the compound interest I = 5000 ( 1 + 1/100 )¹ - 5000
On simplifying the equation , we get
The compound interest I = 5000 ( 1 + 0.01 ) - 5000
The compound interest I = 5000 ( 1.01 ) - 5000
The compound interest I = 5050 - 5000
The compound interest I = $ 50
Hence, the interest is $ 50
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4) Match the polynomial with its factored form.
a. (a - b)(a - b)
b. (a + b)(a - b)
c. (a+b) (a²-ab+6²)
d. (a-b) (a² + ab +6²)
e. (a + b)(a + b)
a³ - 6³
a³ + b³
a²-6²
a² + 2ab + b²
a² - 2ab + b²
Calculations may be made for the polynomials (a - b)(a - b) = a2 - 2ab + b2, (a + b)(a - b) = a2-62, and (a+b)(a2-ab+62) = a3 - 63.
what is polynomial ?The only operations used in a polynomial are addition, subtraction, multiplication, and non-negative integer exponentiation of the variables. Polynomials are mathematical expressions made up of variables (also known as indeterminates) and coefficients. A mathematical expression known as a polynomial is made up of two or more algebraic terms that are added, subtracted, or multiplied (never divided!). In polynomial expressions, which often also have at least one variable, constants and positive exponents are frequently utilized. The equation x2 4x + 7 denotes a polynomial.
given
a) (a - b)(a - b) = a² - 2ab + b²
b. (a + b)(a - b) = a²-6²
c. (a+b) (a²-ab+6²) = a³ - 6³
d. (a-b) (a² + ab +6²) = a³ - 6³
e. (a + b)(a + b) = a² + 2ab + b²
Calculations may be made for the polynomials (a - b)(a - b) = a2 - 2ab + b2, (a + b)(a - b) = a2-62, and (a+b)(a2-ab+62) = a3 - 63.
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The complete question is :- Match the following polynomial by its factored form .
a. (a - b)(a - b)
b. (a + b)(a - b)
c. (a+b) (a²-ab+6²)
d. (a-b) (a² + ab +6²)
e. (a + b)(a + b)
The inequality 2c−3<9 represents the amount of money a student can spend on c candy bars. Select the values that best complete the sentence. The solution to the inequality is , and it represents that the student can buy a maximum of whole candy bars.
The number of candy bars that the student can buy is given by the inequality c < 6 which means less than 6.
What is inequality?It shows a relationship between two numbers or two expressions.
There are commonly used four inequalities:
Less than = <
Greater than = >
Less than and equal = ≤
Greater than and equal = ≥
We have,
The inequality represents the amount of money a student can spend on c candy bars.
2c - 3 < 9
2c < 9 + 3
2c < 12
c < 6
This means,
c is less than 6.
Thus,
The number of candy bars that the student can buy is less than 6.
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Which of the following correctly uses exponents to write "6factors of 2
The mathematical expression which correctly uses exponents to write "6 factors of 2" include the following: B. 2⁶
What is an exponent?In Mathematics, an exponent can be defined as a mathematical operation that is used in conjunction with an algebraic expression to raise a quantity to the power of another and it is generally written as bⁿ.
What are factors?In Mathematics, factors can be defined as the fundamental building blocks of a number. This ultimately implies that, factors simply refers to numbers which can be multiplied together to get another number.
Next, we would translate the word statement into an algebraic expression as follows;
6 factors of 2 = 2 × 2 × 2 × 2 × 2 × 2
6 factors of 2 = 2⁽¹ ⁺ ¹ ⁺ ¹ ⁺ ¹ ⁺ ¹ ⁺ ¹⁾
6 factors of 2 = 2⁶
In this context, we can reasonably infer and logically deduce that "6 factors of 2" can be correctly written by using an exponent as 2⁶.
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Complete Question:
Which of the following correctly uses exponents to write "6 factors of 2"? A. 6² B.2⁶ C. 6 ∙ 2 D. 12
what is the value of 5 to the six power
Answer:
15,625
Step-by-step explanation:
5 to the 6 power = 5 × 5 × 5 × 5 × 5 × 5 = 15,625
Find a formula for R n for the function f ( x ) = ( 3 x ) 2 on [ − 1 , 5 ] in terms of n .
The formula for Rn for the given function is[tex]\int\limits^5_{-1} {9x^2} \, dx =\sum_{i=1}^{n}\frac{54}{n}\left(1-\frac{12i}{n}+\frac{36i^2}{n^2}\right)[/tex]. And the area is 378 units².
The approximate area under the graph of a function f(x) throughout the range [a, b] can be calculated using a Riemann sum. Then, the formula to determine the length of each subinterval is given by [tex]\Delta x=\frac{b-a}{n}[/tex].
Substituting the given interval in the above formula, we get,
[tex]\begin{aligned}\Delta x &=\frac{5+1}{n}\\&=\frac{6}{n}\end{aligned}[/tex]
Now, the sample point [tex]x_i[/tex] is given as
[tex]\begin{aligned}x_i&=a+i\Delta x\\&=-1+i\left(\frac{6}{n}\right)\\&=\frac{6i}{n}-1\end{aligned}[/tex]
The formula for Rₙ is calculated as follows,
[tex]\begin{aligned}R_n&=\sum_{i=1}^{n}f(x_i)\Delta x\\&=\sum_{i=1}^{n}9\left(-1+\frac{6i}{n}\right)^2\frac{6}{n}\\&=\sum_{i=1}^{n}\frac{54}{n}\left(1-\frac{12i}{n}+\frac{36i^2}{n^2}\right)\end{aligned}[/tex]
Then, the area under the curve is computed as follows,
[tex]\begin{aligned}\int\limits^{5}_{-1} {9x^2} \, dx&= \lim_{n \to \infty} \sum_{i=1}^{n}\frac{54}{n}\left(1-\frac{12i}{n}+\frac{36i^2}{n^2}\right)\\&=\lim_{n \to \infty}\left( \sum_{i=1}^{n}\frac{54}{n}-\sum_{i=1}^{n}\frac{648i}{n^2}+\sum_{i=1}^{n}\frac{1944i^2}{n^3}\right)\\&=\lim_{n \to \infty}\left(\frac{54}{n}\sum_{i=1}^{n}(1)-\frac{648}{n^2}\sum_{i=1}^{n}(i)+\frac{1944}{n^3}\sum_{i=1}^{n}(i^2)\right)\end{aligned}[/tex]
Solving this we get,
[tex]\begin{aligned}\int\limits^5_{-1} {9x^2} \, dx &=\lim_{n \to \infty}\left(\frac{54}{n}(n)-\frac{648}{n^2}\left(\frac{n(n+1)}{2}\right)+\frac{1944}{n^3}\left(\frac{n(n+1)(2n+1)}{6}\right)\right)\\&=\lim_{n \to \infty}\left(54-\frac{324(n+1)}{n}+\frac{324(n+1)(2n+1)}{n^2}\right)\\&=\lim_{n \to \infty}\left(54-324\left(1+\frac{1}{n}\right)+324\left(2+\frac{1}{n}+\frac{2}{n}+\frac{1}{n^2}\right)\right)\end{aligned}[/tex]
As the value of n tends to ∞, then 1/n is zero. Substituting these n values, we get,
[tex]\begin{aligned}\int\limits^5_{-1} {9x^2} \, dx&=54 -324+324(2+0)\\&=\mathrm{378\;units^2} \end{aligned}[/tex]
The required answers are [tex]\int\limits^5_{-1} {9x^2} \, dx =\sum_{i=1}^{n}\frac{54}{n}\left(1-\frac{12i}{n}+\frac{36i^2}{n^2}\right)[/tex] and 378 units².
The complete question is -
Find a formula for Rₙ for the function f (x) = (3x)² on [− 1, 5] in terms of n. Compute the area under the graph as a limit.
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