Therefore, the limit as a definite integral is ∫[-2,3] f(x) dx, that is, 62.25.
To express the given limit as a definite integral, we need to use the definition of a Riemann sum and convert it into an integral.
The given limit can be expressed as
lim(n → ∞) ∑(k=1 to n) △x · (x_k)³
where △x = (b-a)/n is the width of each subinterval, with a = -2 and b = 3 being the endpoints of the interval [-2, 3]. We can rewrite (x_k)³ as f(x_k) and interpret the limit as the definite integral of f(x) over the interval [-2, 3]
lim(n → ∞) ∑(k=1 to n) △x · (x_k)³ = ∫[-2,3] f(x) dx
where f(x) = x³. Using the Fundamental Theorem of Calculus, we can evaluate the integral as
∫[-2,3] f(x) dx = F(3) - F(-2)
where F(x) is the antiderivative of f(x) = x³, which is F(x) = (1/4) x⁴ + C, where C is a constant of integration.
Thus, the definite integral is
∫[-2,3] f(x) dx = F(3) - F(-2) = (1/4) (3⁴ - (-2)⁴) = 62.25
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Sofia is making a batch of slime using a recipe that includes glue, water, and salt solution. She uses 7.5 fl. oz of glue. The amount of glue is 12.5 times the amount of salt solution. There are 4.2 less fluid oz of salt solution in the water. How many fluid ounces of water does Sofia use to make a batch of slime?
Answer:
its 4.8
Step-by-step explanation:
i divided 7.5 by 12.5 and got 0.6 and added 4.2 and got 4.8
Equation is formed of two expressions. The amount of water that is used by Sofia to make the slime is 4.8 fl oz.
What is an equation?
An equation is formed when two equal expressions are equated together with the help of an equal sign '='.
Let the amount of salt solution used by Sofia be x, and the amount of water used by Sofia be y.
As it is given that the amount of glue used is 12.5 times the amount of salt solution used, and the amount of glue used is 7.5 fl oz. Therefore, the equation can be written as,
Amount of glue = 12.5 x Amount of salt solution
[tex]7.5 = 12.5 \times x\\\\x = \dfrac{7.5}{12.5}\\\\x = 0.6[/tex]
Also, it is given that the amount of water used is 4.2 fl oz. less than the amount of salt solution used, since we know the amount of salt solution used, therefore, the equation can be written as,
Amount of salt solution = Amount of water - 4.2 oz
[tex]x = y -4.2\\\\(0.6)=y-4.2\\\\y = 4.8[/tex]
Hence, the amount of water that is used by Sofia to make the slime is 4.8 fl oz.
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What is the surface area of this square pyramid? Please help
most kitchens have a working triangle between the sink (S), the counter (C), and the oven (O). the distance between the sink in the oven is 8 feet. The distance between the sink and the counter is 6 feet, and the distance between the oven and the counter at 7 feet. in the kitchen triangle, where is the largest angle?
a. all of the angles are congruent
b. sink
c. counter
d. oven
Answer:
b my
Step-by-step explanation:
valentines
bbg
In the kitchen triangle, the largest angle will be at counter
What is Triangle?A triangle is a three-sided polygon that consists of three edges and three vertices.
Given that kitchens have a working triangle between the sink (S), the counter (C), and the oven (O).
The distance between the sink in the oven is 8 feet.
The distance between the sink and the counter is 6 feet,
The distance between the oven and the counter at 7 feet.
we need to find in the kitchen triangle, where is the largest angle
SO=8
SC=6
OC=7
Hence, in the kitchen triangle, the largest angle will be at counter
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Find the area of the triangle below.
Be sure to include the correct unit in your answer.
Please
Answer:
Step-by-step explanation:
John worked "x" hours this week and Shanique worked eight hours less than John. Write an algebraic expression for the number of
hours that Shanique worked this week.
A) 8x
B) X-8
C)8-X
D)X - 8
Answer:
B or D, I can't tell from these answer choices. X-8
Step-by-step explanation:
Shanique has 8 less questions than John (X).
An icecream factory makes 140 quarts in 10 hours . How much can be made in 12 ?
Find quarts per hour:
140 quarts / 10 hours = 14 quarts per hour.
Now multiply quarts per hour by hours:
14 quarts per hour x 12 hours = 168 quarts
hi please help i’ll give brainly but please show your work.
Answer:0.028
Step-by-step explanation: 0.07 x 0.4= 0.028
The question is 3(2x +4) = 6x + 10
Answer:
no solution
Step-by-step explanation:
when you distribute, the 6x on each side cancel out leaving there to be no solution to the problem. I attached a photo of my work as well.
Answer:
The answer is False meaning it's no solution
Step-by-step explanation:
1. Distribute 3 through the parentheses which will be 6x+12=6x+10
2.Cancel equal terms on both sides of the equation and you will get 12=10 which isn't a true statement.
2. A cup of coffee costs $2.50. If the total cost
is $17.50, then how many cups of coffee were
purchased?
a. Write an equation:
b. What is the constant of proportionality?
Help please
Answer:
The constant is 2.50
I believe the equation would look like 2.5x=17.5
and x would equal 7
What is the least common multiple of 8, 7, and 2?
Answer:
1 i know its one hope it help
Step-by-step explanation:
List all prime factors for each number.
Prime Factorization of 2 shows:
2 is prime => 21
Prime Factorization of 7 shows:
7 is prime => 71
Prime Factorization of 8 is:
2 x 2 x 2 => 23
For each prime factor, find where it occurs most often as a factor and write it that many times in a new list.
The new superset list is
2, 2, 2, 7
Multiply these factors together to find the LCM.
LCM = 2 x 2 x 2 x 7 = 56
In exponential form:
LCM = 2^3 x 7^1 = 56
LCM = 56
Therefore,
LCM(2, 7, 8) = 56
rs10,000 for 1 year at 8% per annum compounded half yearly.
Step-by-step explanation:
Answered 1 year, 9 months ago
Given,
Principal, P =Rs.10000, Rate, R = 8% per annum compounded half yearly for 1 year.
Now, There are two half years in a year. Therefore compounding has to be 2 times.
And rate = half of 10% = 5% half yearly.
Therefore, the required amount = = Rs. 10816
And Compound Interest, C.I. = Amount - Principal = Rs. (10816 - 10000) = Rs. 816.
hope it helps you
What is the area, in square inches, of the trapezoid below? 15.7 in 8 in 6.6 in
Answer:
What is the area, in square inches, of the trapezoid below? 15.7 in 8 in 6.6 in-by-step explanation:
The area, in square inches, of the trapezoid, is 89.2 in².
We have given that,
A = 89.2 in²
The area (A) of a trapezoid is calculated as,
What is the area trapezoid?A = 1/2 h (b₁ + b₂ )
where h is the perpendicular height and b₁, b₂ the parallel bases
Here h = 8, b₁ = 6.6, b₂ = 15.7 , then
A = 1/2 × 8 × (6.6 + 15.7)
A= 4 × 22.3
A= 89.2 in²
The area, in square inches, of the trapezoid, is 89.2 in².
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in a standard deck of cards, what is the probability that you would draw an ace
Answer:
Step-by-step explanation:
Since there are four aces and 52 total cards.
P(A)=4/52=1/13
Answer:
The probability that you would draw an ace in a deck of cards would be 8%
please answer the 2 question
l will mark as brainiest
Answer:
1. A=πr2=π·4.62≈66.4761
A≈66.48
2. A=πr2=π·22≈12.56637
A≈12.57
please mark as brainiest
Answer:
1. 66.48
2. 12.57
Step-by-step explanation:
Formula is A = ㅠr^2
The radius is that half line :))
Algebra 1 Find the GCF
Answer:
[tex]c. \\ x(x+3)[/tex]
Step-by-step explanation:
Answer:
C)
Step-by-step explanation:
If you were solving 17 x 23 using the area model, which numbers would be partial products you would use to find the product
Answer:
The number to use are 10, 7, 20 and 3.
Step-by-step explanation:
Given
[tex]17 * 23[/tex]
Required
Which numbers would be used using the area model or partial products
The number to use are 10, 7, 20 and 3.
This is shown below:
[tex]\begin{array}{ccc}{ } & {10} & {7} & {20} & { } & { } \ \\ {3} & { } & { } \ \end{array}[/tex]
When the numbers are multiplied, we get:
[tex]\begin{array}{cccc}{ } & {10} & {7} & {} & {20} & {200} &{140} &{=340}\ \\ {3} & {30 } & {21 } &{=51} \ \end{array}[/tex]
[tex]Total = 340 + 51[/tex]
[tex]Total = 391[/tex]
Please help will give brainliest
Answer:
another one ayy ok let me think
Step-by-step explanation:
b
Answer:
I believe is A
Step-by-step explanation:
20/4=5
4/4=1
because the ratio is friends to fluid ounces the answer would be 1:5
What percent of 56 is 42?
I need an answer ASAP
a
23%
b
42%
c
75%
d
90%
Answer:
nicki monaj is the queen of rap
Step-by-step explanation:
periodt purr
Finance the formula in exercise 17 is the formula for computing simple interest, where I is interest P is principle of amount invested, r is the interest rate and t is the same tm in years. Find the amount of interest earned if you were to invest $5000 at 6% interest for 3 years.
Answer:
$900
Step-by-step explanation:
Given that:
Principal, p = 5000
Interest, r = 6%
Period, t = 3 years
Using the relation :
Simple interest = Principal * Rate * Time
Simple interest = $5000 * 0.06 * 3
Simple interest = $900
Hence, amount of interest earned = $900
eddie who earns $5 per hour worked two hours longer zach who earns $12 per hour. together they earn $30
a cyclist travels for 5 days and covers 375km in total.
He covers 67km 78km 46km and 89km on the fist 4 days .How far does he travel on day 5
Answer:
95km
Step-by-step explanation:
5 days = 375km
4 days = 67km + 78km + 46km + 89km= 280km
5 days - 4days = the 5th day
375km - 280km = 95km
Miss Smith took her bird to the vet 20 weighed 1 and 3/10 pounds the vet said that tweety weighed 4/10 pound more last year how much did tweety way last year
Answer:
1 and 7/10 pounds?
Step-by-step explanation:
because if its 4/10 more you add 3/10 and 4/10 im pretty sure
Answer:
PLEASE MARK BRAInliest
Step-by-step explanation:
tweety weighed 17/10 pounds last year.
Find the coordinates of the turning point when y=5x^3 - 6x^2 - 4X + 8
Given:
The equation is
[tex]y=5x^3-6x^2-4x+8[/tex]
To find:
The coordinates of the turning point.
Solution:
We have,
[tex]y=5x^3-6x^2-4x+8[/tex]
Differentiate with respect to x.
[tex]\dfrac{dy}{dx}=5(3x^2)-6(2x)-4(1)+(0)[/tex]
[tex]\dfrac{dy}{dx}=15x^2-12x-4[/tex]
For the turning points [tex]\dfrac{dy}{dx}=0[/tex].
[tex]15x^2-12x-4=0[/tex]
Using quadratic formula:
[tex]x=\dfrac{-b\pm \sqrt{b^2-4ac}}{2a}[/tex]
[tex]x=\dfrac{-(-12)\pm \sqrt{(-12)^2-4(15)(-4)}}{2(15)}[/tex]
[tex]x=\dfrac{12\pm \sqrt{144+240}}{30}[/tex]
[tex]x=\dfrac{12\pm \sqrt{384}}{30}[/tex]
Now,
[tex]x=\dfrac{12+\sqrt{384}}{30}\text{ and }x=\dfrac{12-\sqrt{384}}{30}[/tex]
[tex]x\approx 1.053\text{ and }x\approx-0.253[/tex]
Putting x=1.053 in the given equation, we get
[tex]y=5(1.053)^3-6(1.053)^2-4(1.053)+8[/tex]
[tex]y\approx 2.973[/tex]
Putting x=−0.253 in the given equation, we get
[tex]y=5(-0.253)^3-6(-0.253)^2-4(-0.253)+8[/tex]
[tex]y\approx 8.547[/tex]
Therefore, the turning points are (1.053,2.973) and (-0.253, 8.547).
12.) Joe sold 56 tickets to the school play and collected $215. Tickets
for adults cost $4.50 and student tickets cost $3.50 each. How many
of each did he sell?
Answer:
I DONT KNOW!!
Step-by-step explanation:
(3a - 1/2) (3a - 1/2)
Answer: The answer is 9a^2 - 3a + 1/4
Step-by-step explanation:
Hope this helpfull.
But Im so sorry if this not helpfully
someone plz help!!!!
Answer:
7/8 > 3/4
Step-by-step explanation:
Find a common denominator first and here it is 8. To get 3/4 to 8ths, you multiply the numerator and denominator by 2 to get 6/8 and 7/8 is greater
Plz help due tomorrow
Answer:
It's C) -6y
Step-by-step explanation:
I know someone already answered but I figured I could explain since it looks like you are getting a little confused. The reason it's not A is because its
-5 + 5 which is zero. So it gets rid of x so your just left with -6y. If it was
5x+5x then it would 10x but the negative makes it 0
Can u answer these for me with the work shown
Answer:
[tex]\frac{x^3 + 2x^2 -9x-18}{x^3-x^2-6x}= \frac{(x+3)}{x}[/tex]
[tex]\frac{3x^2 - 5x - 2}{x^3 - 2x^2} = \frac{3x + 1}{x^2}[/tex]
[tex]\frac{6 - 2x}{x^2 - 9} * \frac{15 + 5x}{4x}=-\frac{5}{2x}[/tex]
[tex]\frac{x^2 -6x + 9}{5x - 15} / \frac{5}{3-x} = \frac{-(x-3)^2}{25}[/tex]
[tex]\frac{x^3 - x^2 -x + 1}{x^2 - 2x+1}= x +1[/tex]
[tex]\frac{9x^2 + 3x}{6x^2} = \frac{3x + 1}{2x}[/tex]
[tex]\frac{x^2-3x+2}{4x} * \frac{12x^2}{x^2 - 2x} / \frac{x - 1}{x} = 3x[/tex]
Step-by-step explanation:
Required
Simplify
Solving (1):
[tex]\frac{x^3 + 2x^2 -9x-18}{x^3-x^2-6x}[/tex]
Factorize the numerator and the denominator
[tex]\frac{x^2(x + 2) -9(x+2)}{x(x^2-x-6)}[/tex]
Factor out x+2 at the numerator
[tex]\frac{(x^2 -9)(x+2)}{x(x^2-x-6)}[/tex]
Express x^2 - 9 as difference of two squares
[tex]\frac{(x^2 -3^2)(x+2)}{x(x^2-x-6)}[/tex]
[tex]\frac{(x -3)(x+3)(x+2)}{x(x^2-x-6)}[/tex]
Expand the denominator
[tex]\frac{(x -3)(x+3)(x+2)}{x(x^2-3x+2x-6)}[/tex]
Factorize
[tex]\frac{(x -3)(x+3)(x+2)}{x(x(x-3)+2(x-3))}[/tex]
[tex]\frac{(x -3)(x+3)(x+2)}{x(x+2)(x-3)}[/tex]
Cancel out same factors
[tex]\frac{(x+3)}{x}[/tex]
Hence:
[tex]\frac{x^3 + 2x^2 -9x-18}{x^3-x^2-6x}= \frac{(x+3)}{x}[/tex]
Solving (2):
[tex]\frac{3x^2 - 5x - 2}{x^3 - 2x^2}[/tex]
Expand the numerator and factorize the denominator
[tex]\frac{3x^2 - 6x + x - 2}{x^2(x- 2)}[/tex]
Factorize the numerator
[tex]\frac{3x(x - 2) + 1(x - 2)}{x^2(x- 2)}[/tex]
Factor out x - 2
[tex]\frac{(3x + 1)(x - 2)}{x^2(x- 2)}[/tex]
Cancel out x - 2
[tex]\frac{3x + 1}{x^2}[/tex]
Hence:
[tex]\frac{3x^2 - 5x - 2}{x^3 - 2x^2} = \frac{3x + 1}{x^2}[/tex]
Solving (3):
[tex]\frac{6 - 2x}{x^2 - 9} * \frac{15 + 5x}{4x}[/tex]
Express x^2 - 9 as difference of two squares
[tex]\frac{6 - 2x}{x^2 - 3^2} * \frac{15 + 5x}{4x}[/tex]
Factorize all:
[tex]\frac{2(3 - x)}{(x- 3)(x+3)} * \frac{5(3 + x)}{2(2x)}[/tex]
Cancel out x + 3 and 3 + x
[tex]\frac{2(3 - x)}{(x- 3)} * \frac{5}{2(2x)}[/tex]
[tex]\frac{3 - x}{x- 3} * \frac{5}{2x}[/tex]
Express [tex]3 - x[/tex] as [tex]-(x - 3)[/tex]
[tex]\frac{-(x-3)}{x- 3} * \frac{5}{2x}\\[/tex]
[tex]-1 * \frac{5}{2x}[/tex]
[tex]-\frac{5}{2x}[/tex]
Hence:
[tex]\frac{6 - 2x}{x^2 - 9} * \frac{15 + 5x}{4x}=-\frac{5}{2x}[/tex]
Solving (4):
[tex]\frac{x^2 -6x + 9}{5x - 15} / \frac{5}{3-x}[/tex]
Expand x^2 - 6x + 9 and factorize 5x - 15
[tex]\frac{x^2 -3x -3x+ 9}{5(x - 3)} / \frac{5}{3-x}[/tex]
Factorize
[tex]\frac{x(x -3) -3(x-3)}{5(x - 3)} / \frac{5}{3-x}[/tex]
[tex]\frac{(x -3)(x-3)}{5(x - 3)} / \frac{5}{3-x}[/tex]
Cancel out x - 3
[tex]\frac{(x -3)}{5} / \frac{5}{3-x}[/tex]
Change / to *
[tex]\frac{(x -3)}{5} * \frac{3-x}{5}[/tex]
Express [tex]3 - x[/tex] as [tex]-(x - 3)[/tex]
[tex]\frac{(x -3)}{5} * \frac{-(x-3)}{5}[/tex]
[tex]\frac{-(x-3)(x -3)}{5*5}[/tex]
[tex]\frac{-(x-3)^2}{25}[/tex]
Hence:
[tex]\frac{x^2 -6x + 9}{5x - 15} / \frac{5}{3-x} = \frac{-(x-3)^2}{25}[/tex]
Solving (5):
[tex]\frac{x^3 - x^2 -x + 1}{x^2 - 2x+1}[/tex]
Factorize the numerator and expand the denominator
[tex]\frac{x^2(x - 1) -1(x - 1)}{x^2 - x-x+1}[/tex]
Factor out x - 1 at the numerator and factorize the denominator
[tex]\frac{(x^2 - 1)(x - 1)}{x(x -1)- 1(x-1)}[/tex]
Express x^2 - 1 as difference of two squares and factor out x - 1 at the denominator
[tex]\frac{(x +1)(x-1)(x - 1)}{(x -1)(x-1)}[/tex]
[tex]x +1[/tex]
Hence:
[tex]\frac{x^3 - x^2 -x + 1}{x^2 - 2x+1}= x +1[/tex]
Solving (6):
[tex]\frac{9x^2 + 3x}{6x^2}[/tex]
Factorize:
[tex]\frac{3x(3x + 1)}{3x(2x)}[/tex]
Divide by 3x
[tex]\frac{3x + 1}{2x}[/tex]
Hence:
[tex]\frac{9x^2 + 3x}{6x^2} = \frac{3x + 1}{2x}[/tex]
Solving (7):
[tex]\frac{x^2-3x+2}{4x} * \frac{12x^2}{x^2 - 2x} / \frac{x - 1}{x}[/tex]
Change / to *
[tex]\frac{x^2-3x+2}{4x} * \frac{12x^2}{x^2 - 2x} * \frac{x}{x-1}[/tex]
Expand
[tex]\frac{x^2-2x-x+2}{4x} * \frac{12x^2}{x^2 - 2x} * \frac{x}{x-1}[/tex]
Factorize
[tex]\frac{x(x-2)-1(x-2)}{4x} * \frac{12x^2}{x(x - 2)} * \frac{x}{x-1}[/tex]
[tex]\frac{(x-1)(x-2)}{4x} * \frac{12x^2}{x(x - 2)} * \frac{x}{x-1}[/tex]
Cancel out x - 2 and x - 1
[tex]\frac{1}{4x} * \frac{12x^2}{x} * \frac{x}{1}[/tex]
Cancel out x
[tex]\frac{1}{4x} * \frac{12x^2}{1} * \frac{1}{1}[/tex]
[tex]\frac{12x^2}{4x}[/tex]
[tex]3x[/tex]
Hence:
[tex]\frac{x^2-3x+2}{4x} * \frac{12x^2}{x^2 - 2x} / \frac{x - 1}{x} = 3x[/tex]
Mr. Velasquez built a playground. It has one section with a slide and another section with a swing set. A pathway connects the two areas. The slide is 2 meters high. The ares and the pathway are covered in asphalt. Mr velasquez want to know how much of the playground is covered in asphalt
Answer:
[tex]A_1 = 24m^2[/tex] -- Swing set
[tex]A_2 = 16m^2[/tex] -- Slide
[tex]A_3 = 5m^2[/tex] --- Pathway
Step-by-step explanation:
Missing part of the question:
Fill in the table above to show the area of each section (see attachment for table)
The area of each section is calculated as:
[tex]Area = Length * Width[/tex]
For the Swing set
[tex]A_1 = 8m * 3m[/tex]
[tex]A_1 = 24m^2[/tex]
For the Slide
[tex]A_2 = 8m * 2m[/tex]
[tex]A_2 = 16m^2[/tex]
For the Pathway
[tex]A_3 = 5m * 1m[/tex]
[tex]A_3 = 5m^2[/tex]
The total area is:
[tex]Area=A_1 + A_2 + A_3[/tex]
[tex]Area = 24m^2 + 16m^2 + 5m^2[/tex]
[tex]Area = 45m^2[/tex]
HELLLPPPPP WITHHHH MATHHHH
Answer:
54 inches
Step-by-step explanation:
6/18=7/x
6x=7(18)
6x=126
x=21
21+15=18=54