Use your calculator to evaluate the limit from x equals 0 to 2 of the sine of x squared, dx. Give your answer to the nearest integer.
Answers
[tex]\int_{0}^{2}sin(x^{2})dx \approx 1units^2[/tex]
Step-by-step explanation:First of all, the graph of the function [tex]f(x)=sin(x^2)[/tex] is shown in the first figure below. We need to calculate the area under the curve which is in fact the definite integral. From calculus, we know that [tex]f(x)=sin(x^2)[/tex] is non integrable, that is, it doesn't have a primitive, so we must use calculator to evaluate [tex]\int_{0}^{2}sin(x^{2})dx[/tex]. To do so, calculator uses the Taylor Series, so:
[tex]sin(x^{2})=\sum_{n=-\infty}^{+\infty}\frac{(-1)^{n}}{(2n+1)!}x^{4n+2}$[/tex]
You an use a calculator or any program online, and the result will be:
[tex]\int_{0}^{2}sin(x^{2})dx=0.804units^2[/tex]
Since the problem asks for rounding the result to the nearest integer, then we have:
[tex]\boxed{\int_{0}^{2}sin(x^{2})dx \approx 1units^2}[/tex]
The area is the one in yellow in the second figure.
The value of the integral is approximately 0.8380, rounded to the nearest integer, is 1.
Evaluating the given integral involves using numerical methods since the antiderivative of sin(x²) doesn't have a simple closed-form expression in terms of elementary functions. One common numerical method is to use numerical integration techniques like Simpson's rule or the trapezoidal rule.
Let's approximate the integral using Simpson's rule with n=4 subintervals.
The interval of integration is [0, 2].
The width of each sub-interval is (2 - 0)/n = (2 - 0) / 4 = 0.5
The endpoints of the sub-intervals are:
x₀ = 0,
x₁ = 0 + 0.5 = 0.5,
x₂ = 0 + 2(0.5) = 1.0
x₃ = 0 + 3(0.5) = 1.5
x₄ = 0 + 4(0.5) = 2.0
Evaluate the function at these points:
f(x₀) = sin(0²) = 0
f(x₁) = sin(0.5²) = 0.2474
f(x₁) = sin(1²) = 0.8415
f(x₃) = sin(1.5²) = 0.7781
f(x₄) = sin(2²) = -0.7568
Apply Simpson's rule:
[tex]\int_{0}^{2} \sin(x^2) \, dx \approx \frac{h}{3} \left( f(x_0) + 4 \sum_{i \text{ odd}} f(x_i) + 2 \sum_{i \text{ even, } i \neq 0, n} f(x_i) + f(x_n) \right)\\ = \frac{0.5}{3} \left( f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + f(x_4) \right)\\ = \frac{0.5}{3} \left( 0 + 4(0.2474) + 2(0.8415) + 4(0.7781) + (-0.7568) \right) \\ = \frac{0.5}{3} \left( 0 + 0.9896 + 1.6830 + 3.1124 - 0.7568 \right) \\ = \frac{0.5}{3} \left( 5.0282 \right)\\ =0.8380[/tex]
Thus, the value of the given integral is approximately 0.8380.
Round the result to the nearest integer, which is 1.
Complete question:
Use your calculator to evaluate [tex]\int_{0}^{2} \sin(x^2) dx[/tex]. Give your answer to the nearest integer.
Related Questions
Please answer this multiple choice question correctly for 30 points and brainliest!!
Answers
Answer:
B Step 2
Step-by-step explanation:
he was suppose to collect the like terms
5x +x = 5x + 1x
collect the like terms by adding their coefficients
(5 + 1)
add the numbers
(5+1)x = 6x
Answer:
A. Step 1
Step-by-step explanation:
The correct result of eliminating parentheses in Step 1 is ...
[tex]\dfrac{5x+10-8+x}{2}[/tex]
Distributing the minus sign in the second term changes the signs of both of the terms in parentheses:
-(8 -x) = -8 +x
A population of geese grow in such a way that each generation is simply 2.2 times the previous generation. There were 3 geese in the first generation, how many will there be by the 15th generation?
Answers
Answer:
186655 geese
Step-by-step explanation:
In this question , the next generation is calculated by multiplying the current generation by 2.2. Let's do a simple math;
1st generation = 3
2nd generation= 3×2.2=6.6
3rd generation=6.6×2.2=14.52
4th generation=14.52×2.2=31.94
5th generation=31.94×2.2=70.28
6th generation= 70.28×2.2=154.61
7th generation=154.62×2.2=340.14
8th generation=340.14×2.2=748.31
9th generation=748.31×2.2=1646.28
10th generation=1646.28×2.2=3621.81
11th generation=3621.81×2.2=7967.98
12th generation=7967.98×2.2=17529.55
13th generation=17529.55×2.2=38565.0
14th generation=38565.0×2.2=84843.02
15th generation=84843.02×2.2=186654.64
=186655 geese
OR
If current generation is calculated as the previous generation multiplied by a factor then you should see that this is an exponential function.In this case
Original number of geese=3
Factor of multiplication is =2.2
Exponential function for first generation is =0
How? see here
[tex]1st=3*2.2^0=3*1=3\\\\2nd=3*2.2^1=3*2.2=6.6\\\\3rd=3*2.2^2=3*4.84=14.52[/tex]
So from the above you can derive an expression for 15th generation
You notice for 1st generation , the exponent is 0
For the 2nd generation the exponent is is 1
For the 3rd generation the exponent is 2
Thus for the nth generation the exponent will be n-1
Your expression for nth generation can now be;
[tex]nth=3*2.2^{n-1}[/tex]
For 15th generation
[tex]15th=3*2.2^{15-1} =3*2.2^{14} =186654.63\\\\\\=186655[/tex]
A bag contains 5 blue marbles, 2 black marbles and 3 red marbles. A marble is randomly draw from the bag, what's the probability of drawing a red marble
Answers
Answer:
3/10 or 0.3
Step-by-step explanation:
You have 3 red marbles and 10 marbles in total, so you put how many red marble in the numerator and your total marbles in the denominator.
PLEASE HELPPPPP!!!!!!!!
Answers
Answer:
7x = 2x + r
Step-by-step explanation:
On the left side you have 7 boxes. So let's call those boxes "x" and we have 7x.
On the right side we have 2 of them, 2x, and 1 r.
Putting them together gives you
7x = 2x + r
You randomly draw a marble from a bag of marbles that contains 333 blue marbles, 444 green marbles, and 555 red marbles. What is \text{P(draw a blue marble})P(draw a blue marble)P, left parenthesis, d, r, a, w, space, a, space, b, l, u, e, space, m, a, r, b, l, e, right parenthesis? If necessary, round your answer to 222 decimal places.
Answers
The P(draw a blue marble)=0.25
Step-by-step explanation:The bag has:
3 blue marbles, 4 green marbles, and 5 red marbles.
This means that the total number of marbles in bag are:
3+4+5=12 marbles.
We are asked to find the probability that the marble that is randomly drawn from a bag is: Blue marble.
Let P denote the probability of an event.
We know that the probability of drawing a blue marble is equal to the ratio that the marble drawn is blue to the total number of marbles in the bag.
Hence,
[tex]\text{P(blue\ marble)}=\dfrac{3}{12}\\\\i.e.\\\\\text{P(blue\ marble)}=\dfrac{1}{4}\\\\\\\text{P(blue\ marble)}=0.25[/tex]
Hence, the answer is:
P(draw a blue marble)=0.25
Certainly! Let's solve this step by step.
**Step 1: Determine the number of blue marbles.**
We have 333 blue marbles in the bag.
**Step 2: Determine the total number of marbles in the bag.**
The bag contains:
- 333 blue marbles,
- 444 green marbles, and
- 555 red marbles.
To find the total number of marbles, we add these numbers together:
Total marbles = 333 (blue) + 444 (green) + 555 (red)
**Step 3: Calculate the total number of marbles.**
Now, we add up each type of marble to get the total:
Total marbles = 333 + 444 + 555
**Step 4: Calculate the probability of drawing a blue marble.**
The probability of an event is calculated by dividing the number of favorable outcomes by the number of total possible outcomes. In this case, the favorable outcomes are drawing a blue marble, and the total outcomes are the total number of marbles.
Thus, the probability P of drawing a blue marble is:
P(draw a blue marble) = Number of blue marbles / Total number of marbles
Using the numbers provided:
P(draw a blue marble) = 333 / (333 + 444 + 555)
**Step 5: Calculate the exact probability.**
To get the exact probability, we perform the addition in the denominator and then divide:
P(draw a blue marble) = 333 / (1332)
Now, let's execute this calculation.
**Step 6: Simplify the fraction (if necessary).**
In this case, the fraction 333/1332 is already in its simplest form, so we don't need to simplify it further.
**Step 7: Calculate the approximation to 2 decimal places.**
To round the probability to 2 decimal places, we will compute the exact division and then round the result:
P(draw a blue marble) ≈ 0.2500 (rounded to 4 decimal places for illustrative purposes)
However, since we need to round our answer to 2 decimal places, the result is:
P(draw a blue marble) ≈ 0.25
Therefore, the probability of drawing a blue marble, rounded to two decimal places, is 0.25.
Jenny wants to buy a new dress for her first school dance. She borrows $25.63 from her sister to help pay for the dress. Then Jenny earns $16.33 helping her dad organize his home office. If she gives all the earnings to her sister, Jenny will still owe her sister $
Answers
Answer:
If she gives all the earnings to her sister she would still own her $9.30
Step-by-step explanation:
Subtract the $25.63 (sisters' money) fromv $16.33(earnngs from her dad) then you get 9.30
Answer:
$9.30
Step-by-step explanation:
Katie wants to hang a painting in a gallery. The painting and frame must have an area of 45 square feet. The painting is 6 feet wide by 7 feet long. Which quadratic equation can be used to determine the thickness of the frame, x?a. 4x^2 + 26x + 45 = 0b. 4x^2 + 26x − 3 = 0c. x^2 + 13x − 3 = 0d. x^2 + 13x + 45 = 0
Answers
Answer:
[tex]4x^2+26x-3 = 0[/tex]
Step-by-step explanation:
Given that the total area of Frame and Painting is 45 Sq Fts.
Dimensions of painting are 6 feet and 7 feet. Hence.
Let us assume the uniform thickness of frame is x
Hence the Length of frame with painting = 7+2x
and breadth = 6+2x
Area of the frame = length * breadth
[tex]45=(7+2x)(6+2x)\\45=42+14x+12x+4x^2\\45=42+26x+4x^2\\4x^2+26x+42-45\\4x^2+26x-3=0\\[/tex]
Hence option b is correct.
Answer: B) - 4x^2 + 26x − 3 = 0
Step-by-step explanation:
I got it right :)
given f(x)= 9x+1 and g(x)=x^3, choose the expression (f*g)(x)
Answers
[tex](f\cdot g)(x)=(9x+1)\cdot x^3=9x^4+x^3[/tex]
Identify m∠ABC. HELP ASAP!!
Answers
Answer:
m∠ABC = 120°
Step-by-step explanation:
Consider the case where point C is on the circle and long arc AC has measure 240°. Then ∠ABC is an inscribed angle, and its measure is half the measure of the intercepted arc, so is 120°.
This is still true even when point B and point C are on top of each other and BC is a tangent to the circle. That is the case shown here, so m∠ABC = 120°.
Seth worked a summer job at a camp. His salary accumulated all summer and at the end of the
summer he deposited $1890 in a savings account at 3.5 percent interest for 2.5 years. What was the total amount of money in the account at the end of this time?
$2106.25
$2450.00
$2762.50
$2055.38
Answers
Answer is D using A = Pe^(rt).
Final answer:
To calculate the total amount of money in Seth's account after 2.5 years with compound interest, use the formula [tex]A = P(1 + r/n)^{(nt)[/tex], applying the given values to find a total of $2106.25 in the account.
Explanation:
To calculate the total amount of money in the account at the end of the 2.5 years with compound interest, we use the formula
[tex]A = P(1 + r/n)^{(nt)[/tex]
where A is the total amount, P is the principal amount, r is the annual interest rate, n is the number of times that interest is compounded per year, and t is the time the money is invested for.
For this case, P = $1890, r = 3.5% = 0.035, n = 1, and t = 2.5 years.
Plugging in these values gives [tex]A = $1890(1 + 0.035/1)^{(1\times 2.5)[/tex] = $2106.25.
Therefore, the total amount of money in the account at the end of 2.5 years is $2106.25.
What is the factored form of the equation 2(x-1)(8x-3)=0? Which ordered pairs are solutions to the system? Check all of the boxes that apply. (-1, 10) (-0.375, -5.25) (0.375, 12.75) (1, 0.375) (1, 14)
Answers
Answer:
The solutions to the equation are 1, 0.375.
Step-by-step explanation:
The equation is already in factored form.
The solutions to the system are the values of x that make one or the other of the factors be zero.
x -1 = 0 . . . when x = 1
8x -3 = 0 . . . when x = 3/8 = 0.375
There is only one variable, so no "ordered pairs" are involved.
The solutions are {1, 0.375}.
Answer:
1,14
0,375,12.75
Step-by-step explanation:
Which shows the correct substitution of the values a, b, and c from the equation –2 = –x + x2 – 4 into the quadratic formula? Quadratic formula: x =
Answers
Answer:
The answer in the procedure
Step-by-step explanation:
we have
[tex]2=-x+x^{2} -4[/tex]
[tex]x^{2}-x-4-2=0[/tex]
[tex]x^{2}-x-6=0[/tex]
we know that
The formula to solve a quadratic equation of the form
[tex]ax^{2} +bx+c=0[/tex]
is equal to
[tex]x=\frac{-b(+/-)\sqrt{b^{2}-4ac}} {2a}[/tex]
in this problem we have
[tex]x^{2}-x-6=0[/tex]
so
[tex]a=1\\b=-1\\c=-6[/tex]
substitute in the formula
[tex]x=\frac{-(-1)(+/-)\sqrt{(-1)^{2}-4(1)(-6)}} {2(1)}[/tex]
[tex]x=\frac{1(+/-)\sqrt{25}} {2}[/tex]
[tex]x=\frac{1(+/-)5} {2}[/tex]
[tex]x=\frac{1(+)5} {2}=3[/tex]
[tex]x=\frac{1(-)5} {2}=-2[/tex]
3|2x-5|>21
BRAINLIEST solve for x
Answers
Answer:
(x < -1) ∪ (6 < x)
Step-by-step explanation:
Divide by 3
|2x -5| > 7
"Unfold." That is, put the negative of the constant value on the other side of the absolute value expression, using the same inequality symbol.
-7 > 2x -5 > 7 . . . . . . . . . . this notation is convenient for expressing the two disjoint inequalities, but is not strictly correct because it is not true that -7 > 7.
Add 5, then divide by 2
-2 > 2x > 12
-1 > x > 6 . . . . . . . . . . this should be interpreted as -1 > x OR x > 6. Usually a compound inequality is interpreted as having an AND connector.
The solution is ...
(x < -1) ∪ (6 < x)
_____
The more correct (and less convenient) way to "unfold" the absolute value is to write it as two distinct inequalities:
-7 > 2x -52x -5 > 7Each of these is then separately solved (by adding 5 and dividing by 2) to get ...
-1 > xx > 6A conveyor belt carries supplies from the first floor to the second floor, which is 26 feet higher. The belt makes a 60° angle with the ground. How far do the supplies travel from one end of the conveyor belt to the other? Round your answer to the nearest foot. If the belt moves at 75 ft/min, how long, to the nearest tenth of a minute, does it take the supplies to move to the second floor? Question 4 options: 37 ft; 22.5 min 30 ft; 0.4 min 45 ft; 37 min 15 ft; 1.1 min
Answers
Answer:
1. 30ft
2. 0.4 min
Step-by-step explanation:
In this question, apply sine law which states that sine Ф= opposite/hypotenuse
The general formulae; is SOHCAHTOA where ;
(SOH) sineФ=opposite/hypotenuse ; (COH) cosine Ф= adjacent/hypotenuse and (TOA) tan Ф=opposite/adjacent
In the question the height of the two floors is the " c" and the angel at the base is 60° and distance the conveyor travels is the hypotenuse
Hence;
[tex]Sin 60= \frac{26}{H} \\\\\\H=\frac{26}{Sin60} \\\\\\H=30 ft[/tex]
If 75ft=1min
30ft=?........................simple math
30/75 = 0.4 minutes
Answer:
i) 30 ft
ii) 0.4 min
Step-by-step explanation:
i)
The question will be modeled by a right-angled triangle where-by;
The length of the conveyor belt will represent the Hypotenuse
The vertical distance which is 26 feet high will be the Opposite side to the angle the belt makes the ground which is given as 60 degrees.
The horizontal distance which is the ground will be the Adjacent side to the angle 60 degrees.
Therefore, we have a right angled triangle in which one angle and the length of the opposite side are given. To determine the length of the conveyor belt, the hypotenuse, we use the sine definition of an acute angle;
Remember the mnemonic. SOHCAHTOA
sine 60 = (opposite side)/(hypotenuse)
Hypotenuse = (opposite side)/sin 60
Hypotenuse = 26/sin 60
Hypotenuse = 30.02
The supplies thus travel 30 ft from one end of the conveyor belt to the other.
ii)
The speed of the belt is given as 75 ft/min
The length of the belt has been found to be approximately 30 ft
We are required to determine the time it takes for the supplies to move to the second floor.
Distance, Time and speed are related according to the following equation;
Distance = Time * Speed
Time = Distance/Speed
Time = 30/75
Time = 0.4 min
Need help asap
Solve using the properties of logarithms:
5^㏒(base x) 7 = 7
(See picture for a written version of the equation.)
Answers
Answer:
x = 5
Step-by-step explanation:
Taking logarithms to the base 5, we get ...
[tex]\log_x{7}=\log_5{7}[/tex]
Comparing forms gives x = 5.
Answer:
x=5
Step-by-step explanation:
18. The supplement of an angle of 72° is an angle measuring
A. 144°.
B. 18°.
C. 108°.
D. 288°.
Answers
Your answer is C
Answer: OPTION C.
Step-by-step explanation:
First, it is important to remember the definition of "Suplementary angles".
By definition, two angles are supplementary if they add up 180 degrees.
Therefore, the measure of the supplement angle of 72 degrees can be found by setting up the following expression:
[tex]x+72\°=180\°[/tex]
Being "x" the measure of the supplement angle of 72 degrees.
Finally, you need to solve for "x":
[tex]x=180\°-72\°\\\\x=108\°[/tex]
This matches with the option C.
A triangle has two sides of length 4 and 7 what could the length of the third side be
Answers
We can solve this problem using a Pythagorean theorem.
[tex]c^2=a^2+b^2[/tex]
Where a is 7 and b is 4 and c is the hypothenuse we are looking for.
[tex]
c^2=7^2+4^2\Longrightarrow c=\sqrt{49+16}=\boxed{8.06}
[/tex]
The length of the third side is approximately 8.
Hope this helps.
r3t40
In some? country's Congress, the number of Representatives is 20 less than five times the number of Senators. There are a total of 520 members. Find the number of Senators and the number of Representatives.
Answers
Answer:
Senators: 90Representatives: 430Step-by-step explanation:
Let s represent the number of Senators in the Congress. Then the number of Representatives is (5s-20) and the total number of members is ...
s + (5s -20) = 520
6s = 540 . . . . . . . . . add 20, simplify
s = 90 . . . . . . . . . . . divide by 6
The number of Senators is 90; the number of Representatives is 430.
_____
You can find the number of Representatives either as 520 -90 = 430 or as 5·90 -20 = 430.
Rebecca has 45 coins, all nickels and dimes. The total value of the coins is $3.60. How many of each type of coin does Rebecca have?
Answers
Answer:
Rebecca have 18 nickels coins and 27 dime coins
Step-by-step explanation:
Remember that
1 nickel =$0.05
1 dime=$0.10
so
Let
x -----> the number of nickel coins
y -----> the number of dime coins
we know that
x+y=45
x=45-y ------> equation A
0.05x+0.10y=3.60 -----> equation B
Solve the system by substitution
Substitute equation A in equation B and solve for y
0.05(45-y)+0.10y=3.60
2.25-0.05y+0.10y=3.60
0.05y=3.60-2.25
0.05y=1.35
y=27
Find the value of x
x=45-y ---->x=45-27=18
therefore
Rebecca have 18 nickels coins and 27 dime coins
Rebecca has 18 nickels and 27 dimes.
Solving the Problem
Rebecca has 45 coins, composed of nickels and dimes, with a total value of $3.60. We can set up a system of linear equations to solve this problem.
Step-by-Step Explanation
Let x represent the number of nickels and y represent the number of dimes.The total number of coins equation is: x + y = 45The total value equation is: 0.05x + 0.10y = 3.60First, solve the total number of coins equation for y: y = 45 - xSubstitute y in the value equation: 0.05x + 0.10(45 - x) = 3.60Simplify and solve for x: 0.05x + 4.50 - 0.10x = 3.60-0.05x + 4.50 = 3.60
-0.05x = -0.90
x = 18Substitute x back into y = 45 - x: y = 45 - 18
y = 27
Therefore, Rebecca has 18 nickels and 27 dimes.
HELP!!
Polygon ABCDE has the vertices A(2, 8), B(4, 12), C(10, 12), D(8, 8), and E(6, 6). Polygon MNOPQ has the vertices M(-2, 8), N(-4, 12), O(-10, 12), P(-8, 8), and Q(-6, 6).
A transformation or sequence of transformations that can be performed on polygon ABCDE to show that it is congruent to polygon MNOPQ is a
If polygon MNOPQ is translated 3 units right and 5 units down, it will coincide with a congruent polygon, VWXYZ, with its vertices at
Answers
Answer:
Part 1: The polygon ABCDE reflected across y-axis to get the polygon MNOPQ. So, the polygon ABCDE congruent to polygon MNOPQ.
Part 2: The vertices of polygon VWXYZ are V(1, 3), W(-1, 7), X(-7, 7), Y(-5, 3), and Z(-3, 1).
Step-by-step explanation:
Part 1:
The vertices of the polygon ABCDE are A(2, 8), B(4, 12), C(10, 12), D(8, 8), and E(6, 6).
The vertices of the polygon MNOPQ are M(-2, 8), N(-4, 12), O(-10, 12), P(-8, 8), and Q(-6, 6).
We need to find the transformation or sequence of transformations that can be performed on polygon ABCDE to show that it is congruent to polygon MNOPQ.
The relation between the vertices of ABCDE and MNOPQ are defined as
[tex](x,y)\rightarrow (-x,y)[/tex]
It means the polygon ABCDE reflected across y-axis to get the polygon MNOPQ. So, the polygon ABCDE congruent to polygon MNOPQ.
Part 2:
If polygon MNOPQ is translated 3 units right and 5 units down, then
[tex](x,y)\rightarrow (x+3,y-5)[/tex]
[tex]M(-2,8)\rightarrow V(-2+3,8-5)=V(1,3)[/tex]
[tex]N(-4,12)\rightarrow W(-4+3,12-5)=W(-1,7)[/tex]
[tex]O(-10,12)\rightarrow X(-10+3,12-5)=X(-7,7)[/tex]
[tex]P(-8,8)\rightarrow Y(-8+3,8-5)=Y(-5,3)[/tex]
[tex]Q(-6,6)\rightarrow Z(-6+3,6-5)=Z(-3,1)[/tex]
Therefore the vertices of polygon VWXYZ are V(1, 3), W(-1, 7), X(-7, 7), Y(-5, 3), and Z(-3, 1).
The following frequency table summarizes last week's bed sales at Cloud Nine Furniture.
Based on this data, what is the reasonable estimate of the probability that the next bed sold is a twin bed?
CHOOSE 1 ANSWER:
A.) 0.20
B.) 0.25
C.) 0.27
D.) 0.40
Answers
Answer:
A.) 0.20
Step-by-step explanation:
There were 3+6+4+2 beds sold, or 15 beds sold
There were 3 twin beds sold
P(twin) = number of twin/ total
= 3/15 = 1/5 = .2
Answer:
Option A. 0.20
Step-by-step explanation:
Observations from the given frequency table are :
Total number of beds sold at Cloud Nine Furniture = 3 + 6 + 4 + 2
= 15
Number of twin beds sold = 3
Then, probability that the next bed sold will be a twin bed,
[tex]P=\frac{\text{favorable outcome}}{\text{total number of possible outcomes}}[/tex]
[tex]=\frac{3}{15}[/tex]
= 0.20
Therefore, the probability that the next bed sold is a twin bed will be 0.20 or 20%.
Option A. is the correct answer.
Solve The Complex Number System
[tex]x^{3}-125=0[/tex]
Show Work Please
Answers
[tex]\bf \textit{difference and sum of cubes} \\\\ a^3+b^3 = (a+b)(a^2-ab+b^2)~\hfill a^3-b^3 = (a-b)(a^2+ab+b^2) \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ x^3-125=0\implies x^3-5^3=0\implies (x-5)(x^2+5x+5^2)=0 \\\\[-0.35em] ~\dotfill\\\\ x-5=0\implies \boxed{x=5} \\\\[-0.35em] ~\dotfill\\\\ ~~~~~~~~~~~~\textit{quadratic formula}[/tex]
[tex]\bf \stackrel{\stackrel{a}{\downarrow }}{1}x^2\stackrel{\stackrel{b}{\downarrow }}{+5}x\stackrel{\stackrel{c}{\downarrow }}{+25} \qquad \qquad x= \cfrac{ - b \pm \sqrt { b^2 -4 a c}}{2 a} \\\\\\ x=\cfrac{-5\pm\sqrt{5^2-4(1)(25)}}{2(1)}\implies x=\cfrac{-5\pm\sqrt{25-100}}{2} \\\\\\ x=\cfrac{-5\pm\sqrt{-75}}{2}~~ \begin{cases} 75=&3\cdot 5\cdot 5\\ &3\cdot 5^2 \end{cases}\implies x=\cfrac{-5\pm\sqrt{-3\cdot 5^2}}{2}[/tex]
[tex]\bf x=\cfrac{-5\pm 5\sqrt{-3}}{2}\implies x=\cfrac{-5\pm 5\sqrt{-1\cdot 3}}{2}\implies x=\cfrac{-5\pm 5\sqrt{-1}\cdot \sqrt{3}}{2} \\\\\\ x=\cfrac{-5\pm 5i\sqrt{3}}{2}\implies \boxed{x= \begin{cases} \frac{-5+ 5i\sqrt{3}}{2}\\\\ \frac{-5-5i\sqrt{3}}{2} \end{cases}}[/tex]
To solve the complex number equation x^3-125=0, we rearrange to x^3=125 and apply the rule for cube roots in the complex system. This yields three solutions: x = 5, x = 5*omega, and x = 5*omega^2.
Explanation:The given equation is x^{3}-125=0. It belongs to the category of complex number equations within the complex system. We can solve this equation step-by-step by using the cube root of unity in the complex number system.
Firstly, move '125' to the right side of the equation: x^{3}=125. We find that x is the cube root of 125. The solutions to this kind of equation are given by a rule in the complex number system that the cube roots of a number a are given by [a^(1/3), a^(1/3)*omega, a^(1/3)*omega^2], where omega is a cube root of unity and equals -1/2+sqrt(3)i/2 and omega^2=-1/2-sqrt(3)i/2 .
Therefore for x^{3}=125, we get x = 5, 5*omega, and 5*omega^2. These are the solutions in the complex number system.
Learn more about Complex Number System here:https://brainly.com/question/35258703
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A 1-liter bottle of orange juice used to cost $2.85 at food market. Its price has increased; the new price is 110 percent of the original price. What is the new price of the juice? Round your answer to the nearest cent.
Answers
Answer:
$3.14
Step-by-step explanation:
Original cost = $2.85
new cost is 110% of original cost
or
new cost is [tex]\frac{110}{100}[/tex] x original cost
= [tex]\frac{110}{100}[/tex] x $2.85
= $3.135
= $3.14 (rounded to nearest cent)
Answer:
D/ 3.14
Step-by-step explanation:
Susu is solving the quadratic equation 4x2 – 8x – 13 = 0 by completing the square. Her first four steps are shown in the table. In which step did Susu first make an error? Step 1 Step 2 Step 3 Step 4
Answers
Answer:
Step 3
Step-by-step explanation:
when solving by completing the square you are supposed to do (b/2)^2 and then add that to both sides.
In your example (-8/2)^2=16
You are supposed to add sixteen to both sides not 64
Susu forgot to divide 8 by 2 before she raised it to the to the second powerm that is why she got 64, which is incorrect.
Answer:
Step 2
Step-by-step explanation:
Susu is solving the quadratic equation [tex]4x^2 - 8x - 13 = 0[/tex] by completing the square
To apply completing the square method we need to get x^2 alone
Susu added 13 on both sides . So step 1 is correct
When we factor out 4, we divide both terms by 4
[tex]4(\frac{4x^2}{4} -\frac{8x}{2} )=13[/tex]
[tex]4(x^2-2x)=13[/tex]
Susu forgot to divide 8 by 4
So , error in step 2
A cell phone company charges $10/month plus $0.75 per text message and $1 per minute of talk. Data is unlimited. Cameron send 450 texts messages, has 60 minutes of talk, and uses 2.1 gigs of data. How much is his bill? (4.1)
a. $397.50
b. $407.50
c. More than $500
d. Not enough information
Answers
Answer: b. $407.50
Step-by-step explanation:
Multiply 450 by 0.75= 337.5
add 60: 337.5+60=397.5
add 10: 397.5+10= 407.5
Therefore B) is the correct answer
Which is the angle of depression from the lighthouse on the boat ?
Answers
Answer:
∠4
Step-by-step explanation:
An angle of depression is measured from the horizontal downwards.
This is ∠4 in the diagram
Answer:
The angle of depression from the light house to the boat is <4
Step-by-step explanation:
From the figure we can see a light house and a boat.
The angle of depression means that angle from the horizontal downward to an object.
From the figure we can see that, <4 is the angle of depression from the light house to the boat.
Therefore the answer is <a
The length of a flower garden is 9 meters. The width of the garden, w, is unknown. If the area of the garden is greater than 45 square meters, what are the possible values of w, its width, in meters?
Answers
Answer:
6,7,8,9,....
Step-by-step explanation:
the minimum width can be 5 because 9 times 5 is 45 the area using the formula length times width but the question is asking greater than 45 and 9 times any number greater than 5 is greater than 45
Answer:
W > 36
Step-by-step explanation:
i got it right on edgeinuity
PLEASE HELP ME WITH THIS MATH QUESTION
Answers
Answer:
Step-by-step explanation:
1) add the total number of male student
=14
2) divide the number of juniors by the total
2/14 = 1/7 = 0.14285
3)Multiply by 100 to get percentage
0.14285*100=14.285
4)Round
this rounds down to 14%
What’s half of 1/3?????
Answers
Answer: Half of 1/3 would be 1/6 :)
The decimal form of it would be 0.166666
ANSWER
[tex] \frac{1}{6} [/tex]
EXPLANATION
The given fraction is
[tex] \frac{1}{3} [/tex]
We want to find half of this fraction.
We can write this expression mathematically as:
[tex] \frac{1}{2} \: of \: \: \frac{1}{3} [/tex]
The 'of' in this expression means multiplication.
This implies that
[tex] \implies \: \frac{1}{2} \: of \: \: \frac{1}{3} = \frac{1}{2} \: \times \: \: \frac{1}{3} [/tex]
[tex] \implies \: \frac{1}{2 } \: of \: \: \frac{1}{3} = \frac{1 \times 1}{2 \times 3} [/tex]
We multiply out to get
[tex] \implies \: \frac{1 }{2} \: of \: \: \frac{1}{3} = \frac{1}{6} [/tex]
A student wrote the matrix below to represent the solution to a system of equations.
Which of the following describes the solution?
A. No solution
B. An infinite number of solutions
C. (-2, 0, -1)
D. (5, -2, -1)
Answers
Answer:
Option A is correct.
Step-by-step explanation:
The following matrix represent the augmented matrix
In augmented matrix the left side represent the co-efficient of the variables and right side represent the value of variables.
Since the matrix has been solved and reduced so, the solution of variables x, y and z is written on right side of augmented matrix
The given matrix is:
[tex]\left[\begin{array}{cccc}1&0&5&-2\\0&0&3&0\\0&0&0&-1\end{array}\right][/tex]
Since the last row is all zero and we have a non zero entry for the solution, the matrix is inconsistent.
As the variables have zero values, there cannot be any solution
So, there is no solution.
So, Option A is correct.
Answer:
A) No solution!
Step-by-step explanation:
Edge 2021 :)