Answers
Substitute x = √7 sin(t) and dx = √7 cos(t) dt. Then
∫ √(7 - x²) dx = ∫ √(7 - (√7 sin(t))²) • √7 cos(t) dt
… = √7 ∫ √(7 - 7 sin²(t)) cos(t) dt
… = 7 ∫ √(1 - sin²(t)) cos(t) dt
… = 7 ∫ √(cos²(t)) cos(t) dt
We require that -π/2 ≤ t ≤ π/2 in order for the substitution we made to be reversible. Over this domain, cos(t) ≥ 0, so
√(cos²(t)) = |cos(t)| = cos(t)
and the integral reduces to
… = 7 ∫ cos²(t) dt
Recall the half-angle identity for cosine:
cos²(t) = (1 + cos(2t))/2
Then the integral is
… = 7/2 ∫ (1 + cos(2t)) dt
… = 7/2 (t + 1/2 sin(2t)) + C
… = 7t/2 + 7/4 sin(2t) + C
Get the antiderivative back in terms of x. Recall the double angle identity for sine:
sin(2t) = 2 sin(t) cos(t)
We have t = arcsin(x/√7), which gives
sin(t) = sin(arcsin(x/√7)) = x/√7
cos(t) = cos(arcsin(x/√7)) = √(7 - x²)/√7
Then
∫ √(7 - x²) dx = 7/2 arcsin(x/√7) + 7/4 • 2 sin(arcsin(x/√7)) cos(arcsin(x/√7)) + C
… = 7/2 arcsin(x/√7) + x/2 √(7 - x²) + C
Related Questions
if z1= 3+3i and z2=7(cos(5pi/9) + i sin (5pi/9)), then z1/z2= blank
Answers
[tex]z1=\stackrel{a}{3}+\stackrel{b}{3}i~~ \begin{cases} r = \sqrt{a^2+b^2}\\ r = \sqrt{18}\\[-0.5em] \hrulefill\\ \theta =\tan^{-1}\left( \frac{b}{a} \right)\\ \theta =\frac{\pi }{4} \end{cases}~\hfill z1=\sqrt{18}\left[\cos\left( \frac{\pi }{4} \right) i\sin\left( \frac{\pi }{4} \right) \right] \\\\[-0.35em] ~\dotfill[/tex]
[tex]\cfrac{z1}{z2}\implies \cfrac{\sqrt{18}\left[\cos\left( \frac{\pi }{4} \right) i\sin\left( \frac{\pi }{4} \right) \right]} {7\left[\cos\left( \frac{5\pi }{9} \right) i\sin\left( \frac{5\pi }{9} \right) \right]} \\\\[-0.35em] ~\dotfill\\\\ \qquad \textit{division of two complex numbers} \\\\ \cfrac{r_1[\cos(\alpha)+i\sin(\alpha)]}{r_2[\cos(\beta)+i\sin(\beta)]}\implies \cfrac{r_1}{r_2}[\cos(\alpha - \beta)+i\sin(\alpha - \beta)] \\\\[-0.35em] ~\dotfill[/tex]
[tex]\cfrac{z1}{z2}\implies \cfrac{\sqrt{18}}{7}\left[\cos\left( \frac{\pi }{4}-\frac{5\pi }{9} \right)+i\sin\left( \frac{\pi }{4}-\frac{5\pi }{9} \right) \right] \\\\\\ \cfrac{\sqrt{18}}{7}\left[\cos\left( \frac{-11\pi }{36} \right) +i\sin\left( \frac{-11\pi }{36} \right) \right]\implies \cfrac{\sqrt{18}}{7}\left[\cos\left( \frac{83\pi }{36} \right) +i\sin\left( \frac{83\pi }{36} \right) \right] \\\\[-0.35em] ~\dotfill\\\\ ~\hfill \cfrac{z1}{z2}\approx 0.348~~ + ~~0.496i~\hfill[/tex]
The value of z1/z2 is √18/ 7 (cos ( 11π/36 ) - isin ( 11π/36 )).
What is complex number?"A complex number is the sum of a real number and an imaginary number and it is of the form x + iy and is usually represented by z".
For the given situation,
z1= 3+3i and
z2= 7(cos(5π/9) + i sin (5π/9))
To divide the complex numbers, both should be in same form.
Convert z1 in polar form.
z is of the form x+iy, so, r=[tex]\sqrt{x^{2}+y^{2} }[/tex]
⇒[tex]r=\sqrt{3^{2}+3^{2} }[/tex]
⇒[tex]r=\sqrt{18}[/tex]
θ = [tex]tan^{-1}(\frac{b}{a} )[/tex]
⇒[tex]tan^{-1}(\frac{3}{3} )[/tex]
⇒[tex]tan^{-1}(1)[/tex]
⇒[tex]45[/tex]°
The polar form is of the form, z= r (cosθ + i sinθ),
⇒ z1 = [tex]\sqrt{18}[/tex] (cosπ/4 + isinπ/4)
The formula for dividing complex number is
z1/z2 = r1(cos θ1 + isin θ1) / r2(cos θ2 + isin θ2)
⇒ z1/z2 = r(cosθ + isinθ)
where, r = r1/r2 and θ = (θ1 - θ2)
z1/z2 = [tex]\sqrt{18}[/tex] (cos π/4 + isin π/4) / 7 (cos 5π/9 + isin 5π/9)
⇒ r = [tex]\sqrt{18}[/tex] / 7 and
θ = (π/4 - 5π/9 )
⇒ θ = (-11π/36)
cos(-θ) = cos θ
⇒cos( -11π/36 ) = cos ( 11π/36 )
sin(-θ) = -sin θ
⇒ sin ( -11π/36 ) = -sin ( 11π/36 )
Thus, z1/z2 = [tex]\sqrt{18}[/tex] / 7 (cos ( 11π/36 ) - isin ( 11π/36 ))
Hence we can conclude that the value of z1/z2 is
√18/ 7 (cos ( 11π/36 ) - isin ( 11π/36 )).
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Marc drove 312.8 miles on 12.5 gallons of gas. What was the car’s average mileage miles per gallon for the distance.
Answers
Answer:
25.024 mi/gal
Step-by-step explanation:
To find miles per gallon, divide miles by gallons.
(312.8 mi)/(12.5 gal) = 25.024 mi/gal
The owner of a retail store is tracking his inventory for an annual report. The graph shows the remaining inventory for a particular item and the number of days that have passed since the stock was replenished. Which type of function best models the relationship between the number of days and the inventory remaining? A. a linear function with negative slopes
Answers
Let y represent units of inventory, and x represent days since the last replenishment. We are given points (x, y) = (3, 61) and (13, 33). The line through these points can be described using the 2-point form of the equation of a line:
... y -y1 = (y2-y1)/(x2 -x1)(x -x1)
Filling in the given point values, we have ...
... y -61 = (33 -61)/(13 -3)(x -3)
Simplifying and adding 61, we get ...
... y = -2.8x +69.4
Single choice question
The expression 24(1−x) gives the discounted price of a pair of shorts, where x is the percent of the discount written in decimal form.
What does 1−x represent in the expression?
A. Discount price of the shorts.
B. Original price of shorts.
C. Percent of discount.
D. Percent of original price being paid.
Answers
Answer:
D
Step-by-step explanation:
Because the discount price is the whole equation 24(1-x),
the original price is 24 technically, and the percent of discount has been given as x...
Can anyone please help me answer this Math question ? As soon as possible !
The We-Make-It factory in the United States pays its 250 workers an average of $21 per hour, including benefits. The average number of work hours per employee in a year is 1,924. The owners of the factory are considering relocating overseas to a country where they can pay workers an average of $6 per hour, including benefits, to achieve the same production levels. Assuming that the company would hire the same number of workers for the same number of hours in a new location, how much could We-Make-It save in wages alone in the first year by moving to the proposed overseas location?
Answers
Assuming that this would not be destroying the lives of many in a community by locating overseas. The factory would save $2,886,000 in payroll wages during the first year of relocation.
Step-by-step explanation:
$15 saved in wages/working hour × 250 workers × 1924 hours per year/worker = $2,886,000
Evaluate 3x2 - 2xy + 4y2 for x = -1 and y = -2. .
Answers
Answer:
The answer is 12
Step-by-step explanation:
3x² - 2xy + 4y²
Let x = -2
y = -1
3(-2)² - 2(-2)(-1) + 4(-1)²
= 3(4) - 2(2) + 4(1)
= 12 - 4 + 4
= 12
A car is traveling at 25 miles per hour during rush hour. How far does the car travel in
2 minute and 45 seconds?
Answers
Answer: 1.14583333333 miles
Step-by-step explanation: divide 25/21.81818181818182
The car will travel 1.15 miles in 2 minutes 45 secs
if a car is traveling at 25 miles per hour, the amount of time the car travels in secs is 3600secs
25 miles = 3600secs
To get the distance traveled in 2 minutes 45 secs, we can write;
x = 165 secs
Divide both expressions
25/x = 3600/165
3600x = 25 * 165
3600x = 4125
x = 1.15 miles
Hence the car will travel 1.15 miles in 2 minutes 45 secs
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QUESTION 4 PLEASE HELP ME ASAP
Answers
Answer:
Part 1: The entry fee is $5
Part 2: Each minute costs 25 cents
Step-by-step explanation:
In the expression, the '+5' represents the fee added to '.25m' the cost per each minute.
A TON OF POINTS AND BRAINLIEST IF ANSWERED CORRECTLY! I RLY NEED HELP PLZZZ ANSWER
Use the function f(x) = 2x^2 − 3x − 5 to answer the questions.
Part A: Completely factor f(x). (2 points)
Part B: What are the x-intercepts of the graph of f(x)? Show your work. (2 points)
Part C: Describe the end behavior of the graph of f(x). Explain. (2 points)
Part D: What are the steps you would use to graph f(x)? Justify that you can use the answers obtained in Part B and Part C to draw the graph. (4 points)
Answers
Answer:
Step-by-step explanation:
For
f(x) = 2x2-3x -5
For Part A
Set f(x) = 0
Factor the Quadratic
Set the factors = 0
Solve each factor for x to find your x intercepts
For Part B
You need to use the coefficients in your Quadratic
List them
a = 2 , b = -3 and c = -5
Since a is positive, your parabola opens up
Find the x coordinate of the vertex it is -b/2a
One you get the x coordinate, plug it into to your Quadratic to get f(x) your y coordinate of the vertex
For Part C
You can plot the vertex, both x intercepts on a coordinate grid
Which of the following sets of ordered pairs represents a function?
A (510), (0.15). (
510)
(3.5) (3,9). (4.2)
22302 79717263393-
D
(0.6) (12.10). (O.-6)
Answers
Answer: your answer is A) & D)
Step-by-step explanation: Hope this helps :)
Answer:
A) (-5,10), (0,15), (5,10)
Step-by-step explanation:
In order for an ordered pair to be a function, no pair of coordinates can have the same x variable.
I WILL GIVE BRAINLEST AND 60 POINTS I KNOW ITS NOT ALOT
Melanie was training for a race. The line plot below shows the number of miles Melanie ran last week.
2. How many times did Melanie run ½ a mile?
Answers
Answer:
Melanie and 2 times ½ a mile
Hopes This helps
let x represent the larger number4x+(x-2)^2=16
Answers
I have no clue I just need the points have a good day tho :)
Solve the following inequality 17 less than or equal to negative 3X +2 less than or equal to 26.. what is the answer
Answers
Answer:
-5 less than or equal to x less than or equal to -8
Step-by-step explanation:
You have to isolate x.
Start with removing +2.
Take 2 away from both sides
15 less than or equal to -3x less than or equal to 24
Then, the only thing you have to isolate after that is -3, you have to remove it from x.
Since -3x is multiplying, you have to divide by -3 on the other sides.
15 divided by -3 is -5
24 divided by -3 is -8
Therefore,
The answer is...
-5 less than or equal to x less than or equal to -8
Consider the equation of a circle x^2-2x+y^2-4y-4=0. If the line 2x-y+a=0 is its diameter. Then find the value of a.
Answers
Answer:
a=0
Step-by-step explanation:
x²-2x+y²-4y-4=0
(x²-2x+1)+(y²-4y+4)-9=0
(x-1)² + (y-2)² = 3²
Center: (1,2) radius: 3
(1,2) on line 2x-y+a=0
2*1 - 2 + a = 0
a = 0
If Alex is 14 years old, and Ben is 17 years old, in how many years will the sum of their ages be 169 years?
Answers
Answer:
69 years
Step-by-step explanation:
Sum of Alex's and Ben's current age
= 14 +17
= 31 years
For every year, Alex and Ben will each be one year older. Thus, 2 years would be added to the sum of their ages with each passing year.
Difference in the sum of their ages
= 169 -31
= 138 years
Number of sets of 2 years
= 138 ÷2
= 69
Thus, the sum of their ages will be 169 in 69 years.
Divide (x^3+2x-6x-9) by (x+3)
Answers
Answer:
x² - 3x + 5 - [24÷(x+3)]
Step-by-step explanation:
1. Expand (x³ + 2x - 6x - 9)
= x³ - 4x - 9
2. Divide [x³ - 4x - 9 by x+3]
= x² + [(-3x^2-4x-9) ÷ (x+3)]
3. Divide [(-3x^2-4x-9) by (x+3)]
= -3x + [(5x-9) ÷ (x+3)]
x² - 3x + [(5x-9) ÷ (x+3)]
4. Divide [(5x-9) ÷ (x+3)]
= 5 + [(-24) ÷ (x+3)]
x² - 3x + 5 + [(-24) ÷ (x+3)]
= x² - 3x + 5 - [24 ÷ (x+3)]
If [tex]a\ \textless \ b[/tex], there are three ordered pairs of positive integers [tex](a,b)[/tex] that satisfy [tex]a^{2}+b^{2}=10(123)^{2}[/tex] If two of these ordered pairs are [tex](39,387)[/tex] and [tex](201,333)[/tex]. What is the third such ordered pair?
Answers
The third ordered pair positive integers that satisfies the equation is (123, 369).
The given parameters;
[tex]a^2 + b^2 = 10(123)^2[/tex]First pair of the equation, = (39, 387)Second pair of the equation = (201, 333)The third ordered pair of the equation can be determined by using general equation of a circle;
[tex]a^2 + b^2 = r^2\\\\a^2 + b^2 = (123\sqrt{10} )^2\\\\a^2 + b^2 = (\sqrt{151290} )^2\\\\a^2 + b^2 = 151290\\\\a^2 = 151290- b^2\\\\ a= \sqrt{151290 - b^2}[/tex]
The radius of the circle is calculated as;
[tex]r^2 = 151290\\\\r = \sqrt{151290} \\\\r = 388.96[/tex]
The value of a can be obtained by randomly choosing numbers less than the radius as values of b.
[tex]b < r\\\\b < 388.96[/tex]
[tex]a = \sqrt{151290 \ - \ (387)^2} \\\\a = 39\\\\(39, \ 387)\\\\a = \sqrt{151290 \ - \ (333)^2}\\\\a = 201\\\\(201, \ 333)\\\\a = \sqrt{151290 \ - \ (369)^2}\\\\a = 123\\\\(123, \ 369)[/tex]
Thus, the third ordered pair positive integers that satisfies the equation is (123, 369).
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Sara has $65 and wants to buy the following items from a store. Board game: $14.95 Building blocks: $25.49 Art kit: $35.79 At the store, all items are one-fourth off of the original price and there is a 5% sales tax. Sara wants to know if she can buy all of these items. Which statement is TRUE?
Answers
The correct statement is Sara can afford to purchase the items because the final price of the items is $60.
The first step is to determine the total cost of the items Sara wants to buy. The total cost can be determined by adding the cost of the items together.
Total cost = $14.95 + $25.49 + $35.79 = $76.23
The second step is to determine the cost of the items after the discount.
Price of the items = total cost x (1 - discount)
= total cost x (1 - 1/4)
total cost x 3/4
$76.23 x 3/4 = $57.17
The third step is to determine the cost of the items after tax.
Cost = (1.05) x $57.17 =$60.03
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Help please
i will give brainliest
Answers
Answer:
1.69
Step-by-step explanation:
What is 18x77 with algorithm
Answers
Answer:
1386
Step-by-step explanation:
First multiply 8 and 77 to get 616
Then multiply 10 and 77 to get 770
Finally, add them all together to get 1386
Answer:
the answer is 1,386. ......A baby is 3 weeks and 4 days old. What it's is age in days?
Answers
Work Shown:
1 week = 7 days
3 weeks = 21 days .... multiply both sides by 3
3 weeks + 4 days = 21 days + 4 days = 25 days
f(x)=x^2-12x+20
a. f(0) = ?
b. f(2) = ?
c. f(x) = 0,x = ?
d. f(x)= 9,x = ?
Answers
Answer:
c . f(x)=0,x=????????????
I need help with this, i did part of it but i'm desperate
Answers
Answer:
Let's add more steps.
step 6: angle AEB measures 360 - 90 -90 - 70 - 39 = 71° Reason, AECF is a quadrangonal and three of its angle measures 90, 90, and (70+39)
step 7: angles AEB and ABE are congruent: reason triangle ABE is isosceles since AE is conruent to AB
step 8 Angle BAE measures 180 -71-71 = 38°: by difference from the sum of the internal angles.
step 9: Angle BAD measures 90 - BAE = 90 - 38 = 52° Reason angles BAD and BAE are complementary.
QED
1. Assume O P || NM.
What are possible lengths for each of the following segments?
LO, ON, LP, and PM
Choose one measure for each segment. You can start with the length of any segment, but remember to consider the
Triangle Proportionality Theorem when choosing the measurements.
2. How did you decide what values to use? Explain how you chose your measurements, and why you know they are
possible measurements for the segments.
3. What is a different method you could use to choose these numbers? Explain your answer.
Answers
1. Assuming OP || NM, considering the triangle proportionality theorem possible measurements for the segments are: ON = 3 units, LO = 1 unit, PM = 6 units, LP = 2 units
2. The possible values are decided by assuming [tex]\frac{ON}{LO} = \frac{PM}{LP} = 3[/tex] using the triangle proportionality theorem.
3. Another method is applying the AA Similarity Theorem to compare corresponding side lengths of the triangles that are proportional to each other.
What is the Triangle Proportionality Theorem?Triangle proportionality theorem states that if a line is parallel to one of the sides of a triangle intersects two of the other two sides, it divides the two sides into proportional corresponding segments.
1. Assuming that OP || NM, we would apply the triangle proportionality theorem to have the following ratio:
[tex]\frac{ON}{LO} = \frac{PM}{LP}[/tex]
Using the above ratio, we can choose the following possible lengths for the segments:
ON = 3 units
LO = 1 unit
PM = 6 units
LP = 2 units
2. The values are decided assuming that the ratio, [tex]\frac{ON}{LO} = \frac{PM}{LP} = 3[/tex]
This means that the ratio of the proportional segments, irrespective of the their measurement should give you an assumed equal ratio of 3:1.
Therefore, since:
[tex]\frac{3}{1} = 3\\\\\frac{6}{2} = 3[/tex]
The measurements, ON = 3 units, LO = 1 unit, PM = 6 units, LP = 2 units are therefore possible measurements.
3. Using the AA Similarity Theorem to prove that ΔLPO ≅ ΔLMN, the corresponding sides of both triangles will be proportional.
LP/LM = LO/LN = PO/MN
We can use this to choose possible measurements while assuming a ratio between the proportional sides of both triangles.
Let's assume that: LP/LM = LO/LN = PO/MN = 1/4
Let,
LO = 1 unit
LN = 4 units
Therefore, ON = LN - LO = 4 - 1 = 3 units.
Same way the other possible measures can be gotten provided a ratio is assumed.
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If △CFE≅△PTR, complete each of the following statements:
CE¯¯¯¯¯¯¯¯ ≅
TR¯¯¯¯¯¯¯ ≅
∠P ≅∠
∠F ≅∠
△FEC ≅△
△RPT ≅△
Answers
Based on the definition of congruent triangles, the statements will be completed as follows:
CE ≅ PRTR ≅ FE∠P ≅ ∠C∠F ≅ ∠T△FEC ≅ △TRP△RPT ≅ △ECFRecall the following about Congruent Triangles:
The three angles of one triangle is congruent to the three corresponding angles in another triangle if both triangles are congruent triangles.The three sides of one triangle is congruent to the three corresponding sides in another triangle if both triangles are congruent triangles.Given that △CFE≅△PTR, thus:
CE corresponds to PR, therefore CE ≅ PRFE corresponds to TR, therefore FE ≅ TRCF corresponds to PT, therefore CF ≅ PT∠P corresponds to ∠C, therefore ∠P ≅ ∠C∠T corresponds to ∠F, therefore ∠T ≅ ∠F∠R corresponds to ∠E, therefore ∠R ≅ ∠EThus, based on the definition of congruent triangles, the statements will be completed as follows:
CE ≅ PRTR ≅ FE∠P ≅ ∠C∠F ≅ ∠T△FEC ≅ △TRP△RPT ≅ △ECFLearn more about congruent triangles on:]
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A group of pupils calculated their average score for a Mathematics
test. If one of them scored 9 marks more, their average score would
be 81. If one of them scored 3 marks less, their average score would
be 78. How many pupils were there in the group?
Answers
Answer: 4
Step-by-step explanation: +9-(-3)=12
81-78=3
so 12/3=4
Jamal wrote that 9 – 8.4 = 6
A) less than
B)greater than
need answer for both drop down menu
Answers
Answer:
See below
Step-by-step explanation:
9 and 8.4 are less than 1 apart, so 9-8.4 must be less than 1. The actual difference is 9-8.4=0.6 which verifies that our answers are true since 0.6<1.
2. If Jordan's section contains 35 rows, how many seats are in her section?
seat(s)
Answers
Answer:
5
Step-by-step explanation:
14. The sun casts a shadow from a flag pole. The height of the flag pole is three times the length of its shadow. The distance between the end of the shadow and the top of the flag pole is 20 feet. Find the height of the flag pole.
Answers
It would be about 6.7 ft
Use what you know about operations and their properties to write three expressions equivalent to the expression shown.
10(2+3)−8⋅3
Answers
1) Simplify 2 + 3 to 5.
10 * 5 - 8 * 3
2) Simplify 10 * 5 to 50.
50 - 8 * 3
3) Simplify 8 * 3 to 24.
50 - 24
4) Simplify.
24