Let y = x + 22 and dy = dx, so the integral becomes
[tex]\displaystyle \int_3^9 \frac{\mathrm dx}{(x+21)\sqrt{x+22}} = \int_{25}^{31} \frac{\mathrm dy}{(y-1)\sqrt{y}}[/tex]
Now let z = √y, so that z ² = y. Then 2z dz = dy, and the integral becomes
[tex]\displaystyle \int_3^9 \frac{\mathrm dx}{(x+21)\sqrt{x+22}} = \int_{\sqrt{25}}^{\sqrt{31}} \frac{2z}{(z^2-1)z} \\\\ = \int_5^{\sqrt{31}} \frac{2}{z^2-1}\,\mathrm dz[/tex]
Expand the integrand into partial fractions:
[tex]\dfrac{2}{z^2-1} = \dfrac1{z-1}-\dfrac1{z+1}[/tex]
Then we have
[tex]\displaystyle \int_3^9 \frac{\mathrm dx}{(x+21)\sqrt{x+22}} = \int_5^{\sqrt{31}}\left(\frac1{z-1}-\frac1{z+1}\right)\,\mathrm dz \\\\ = \left(\ln|z-1|-\ln|z+1|\right)\bigg|_5^{\sqrt{31}} \\\\ =\left[\ln\left|\frac{z-1}{z+1}\right|\right]\bigg|_5^{\sqrt{31}} \\\\ =\ln\left(\frac{\sqrt{31}-1}{\sqrt{31}+1}\right) - \ln\left(\frac{4}{6}\right) \\\\ =\ln\left(32-2\sqrt{31}\right) - \ln\left(\frac23\right) \\\\ =\boxed{\ln\left(48-3\sqrt{31}\right)}[/tex]
QUESTION 2
A board is 86 cm. in lenght and must be cut so that one piece is 20 cm. longer than the other piece
Find the lenght of each piece.
A26 cm and 60 cm
b. 33 cm and 53 cm
C 30 cm and 56 cm
d. 70 cm and 16 cm
One piece will be length x and the other piece will be 20 cm longer, so it will be x + 20 cm long.
Added together the length of these two boards will equal 86 cm. So you can write an equation:
x + (x + 20) = 86
Remove the parentheses and add the two x's together to get:
2x + 20 = 86
Subtract 20 from both sides:
2x = 66
Divide both sides by 2 and you have:
x = 33
The short piece is 33 cm and the other piece is 20 cm longer or 33 + 20 = 53 cm.
The height of a triangle is 4 yards greater than the base. The area of the triangle is 70 square yards. Find the length of the base and the height of the triangle.
9514 1404 393
Answer:
base: 10 yardsheight: 14 yardsStep-by-step explanation:
Let b represent the length of the base. Then (b+4) is the height and the area of the triangle is ...
A = 1/2bh
70 = 1/2(b)(b+4)
b² +4b -140 = 0 . . . . . multiply by 2, put in standard form
(b +14)(b -10) = 0 . . . . factor
b = 10 . . . . the positive solution
The base of the triangle is 10 yards; the height is 14 yards.
Evaluate the expression 2x-xy if x= -1/2 and y=8
Answer:
3
Step-by-step explanation:
2x-xy
2(x)-(x)(y)
2(-1/2)-(-1/2)(8)
-1 - (-4)
-1 + 4
3
Answer:
3
Step-by-step explanation:
2x-xy
Let x = -1/2 and y = 8
2(-1/2) - (-1/2)(8)
Multiply
-1 - (-4)
Subtracting a negative is like adding
-1+4
Add
3
What type of polynomial is: sqrt 7x^4 - x^3
Answer:
Quartic cause highest degree is 4
apter 3 If a driver uses of a tank of gas every day, what fraction of a tank will he use in a) 3 days? b) 1 week? 21
You need to re copy and paste it, I can’t see the full possible answers/question.
M is the midpoint of HA. The coordinate of His
(3x + 5,3y) and the coordinate of A is (x-1, -y). What is
the coordinate of M?
Step-by-step explanation:
ANSWER:(2x+2,y)
father of economics
I need you guy’s help answer thanks so much
Answer:
D
Step-by-step explanation:
inverse just means opposite so if its shown dividing it first then adding, then the inverse is multiplying then subtracting
I need help ASAP thank you
Answer:
√9 × √6
√54
√27 × √2
Step-by-step explanation:
We can obtain the answer to the question given above as illustrated below:
3√6
Recall
a√b = √(a×a×b)
Thus,
3√6 = √(3×3×6)
3√6 = √(9 × 6)
Recall
√(a × b) = √a × √b
√(9 × 6) = √9 × √6
Therefore,
3√6 = √9 × √6
Recall
√9 × √6 = √(9 × 6)
√9 × √6 = √54
Thus,
3√6 = √54
Recall
√54 = √(27 × 2)
√54 = √27 × √2
Therefore,
3√6 = √27 × √2
Therefore,
3√6 = √9 × √6 = √54 = √27 × √2
What's the value of n if 13+19= n+25
.
.
..
.
.............
Answer:
[tex]13+19=n+25[/tex]
[tex]n+25=13+19[/tex]
Add 13+19= 32
[tex]n+25=32[/tex]
Now Subtract 25 from both sides
[tex]n+25-25=32-25[/tex]
[tex]n=7[/tex]
OAmalOHopeO
What are the common factors of 24 and 60
2 and 3 are the common factor of 24 and 60
What is the value of x when y equals 66?
y=0.985897x+0.194185
Answer:
x = 66.74715005725
Step-by-step explanation:
First you bring over the added variable. 0.194185, and subtract it from 66. Then you divide your difference by 0.985897. This gives you 66.74715005725
Now there is a square city of unknown size with a gate at the center of each side. There is a tree 20 b from the north gate. That tree can be seen when one walks 14 bu from the south gate, turns west and walks 1775 bu. Find the length of each side of the city.
Answer:
The length of each side of the city is 250b
Step-by-step explanation:
Given
[tex]a = 20[/tex] --- tree distance from north gate
[tex]b =14[/tex] --- movement from south gate
[tex]c = 1775[/tex] --- movement in west direction from (b)
See attachment for illustration
Required
Find x
To do this, we have:
[tex]\triangle ADE \sim \triangle ACB[/tex] --- similar triangles
So, we have the following equivalent ratios
[tex]AE:DE = AB:CB[/tex]
Where:
[tex]AE = 20\\ DE = x/2 \\ AB = 20 + x + 14 \\ CB = 1775[/tex]
Substitute these in the above equation
[tex]20:x/2 = 20 + x + 14: 1775[/tex]
[tex]20:x/2 = x + 34: 1775[/tex]
Express as fraction
[tex]\frac{20}{x/2} = \frac{x + 34}{1775}[/tex]
[tex]\frac{40}{x} = \frac{x + 34}{1775}[/tex]
Cross multiply
[tex]x *(x + 34) = 1775 * 40[/tex]
Open bracket
[tex]x^2 + 34x = 71000[/tex]
Rewrite as:
[tex]x^2 + 34x - 71000 = 0[/tex]
Expand
[tex]x^2 + 284x -250x - 71000 = 0[/tex]
Factorize
[tex]x(x + 284) -250(x + 284)= 0[/tex]
Factor out x + 284
[tex](x - 250)(x + 284)= 0[/tex]
Split
[tex]x - 250 = 0 \ or\ x + 284= 0[/tex]
Solve for x
[tex]x = 250 \ or\ x =- 284[/tex]
x can't be negative;
So:
[tex]x = 250[/tex]
Which proportion would you use to solve the following problem?
A map has a scale of 1 cm : 5 km. Determine how far apart two cities are if they are 4 cm apart on the map.
A.
B.
C.
D.
Answer:
20 km
Step-by-step explanation:
We can use ratios to solve
1 cm 4 cm
-------- = ------------
5 km x km
Using cross products
1 * x = 4 * 5
x = 20
20 km
Answer:
1/5 = 4/x
Step-by-step explanation:
Each cm on the map represents 5 km. If it shows 4 cm apart of the map, you can use the proportion 1 cm : 5 km = 4 cm : x km.
GIVING 25 POINTS AND BRAINIER IF ANSWERED!
What is one thing you would not do when finding the question in a word problem?
A. Look for a problem similar to the word problem you are trying to solve.
B. The question may not be directly stated.
C. So you can understand what the facts are in the word problem.
D. To define your strategy or game plan to solve the word problem.
Answer:
B. The question may not be directly stated.
Can you please help me
9514 1404 393
Answer:
1/63
Step-by-step explanation:
There are various ways the question "how much larger" can be answered. Here, we choose to answer it by telling the difference between the two fractions:
4/9 -3/7 = (4·7 -9·3)/(9·7) = 1/63
The larger fraction is 1/63 unit larger than the smaller fraction.
Sierra used the following steps to copy angle ABC:
1. Draw a ray with the endpoint D.
2. Place the compass on point B. Swing an arc that intersects rays BA and BC that creates point E on BA and Fon BC.
3. Keeping the compass at the same width, place it on point D, and swing an arc that creates point G on the ray.
4. Open the compass to the distance between points A and E.
5. Keeping the compass at the same width, place it on point G and swing an arc that intersects the arc created from point D.
6. Label the intersection of the arcs created from point D with point H, and draw a ray from point D through point H.
In which step did the error occur? Explain. (1 point)
Step 2. She did not set the proper compass length.
O Step 2. She drew an arc Instead of a circle.
Step 3. She placed the compass at an incorrect point.
Step 4. She measured the incorrect length.
The process of copying an angle through construction involve ensuring
that the distance between the two rays forming the angle is the same or
equal in the original and copy of the angle at a given equal distance from
the vertex of both angles
The step where the error occur is step 4. She measured the incorrect length
The reason for the above selection is as follows:
The steps to copy an angle are as follows;
Draw the initial line, l, (the working line) of the copy of the angle as a ray with an endpoint, DIn the given angle ∠ABC, with the compass placed at point B, construct an arc that intersect the two sides of the given line (BA at point E, and BC at point F) by opening the compass to a radius, rPlace the compass at the point D on the line l and with the radius r from above, draw an arc (D, r) that intersects the line l at the point GPlace the compass at point E on line AB, and construct an arc passing through point F, therefore having radial length EFPlace the compass at point G and construct an arc with radial length EF that intersects the arc (D, r) at point H and draw a ray from the point D passing through point H to complete the constructionA critical step in the copying process involves the finding the distance
between the initial line and the terminal line EF at a given distance DG =
AE from the angle vertex
The step where the error occur is in step 4. by opening the compass
between points A and E which are points on the same line and placing the
compass at point G on line l to draw an arc intersecting the arc created
from point D at H, thereby making the radial distance between the two lines
forming the angle, in the copied angle at a distance E from the vertex,
equal to AE rather than EF
Therefore, the error occurred in step 4. She measured the incorrect length
Learn more about the steps to copy an angle here:
https://brainly.com/question/4292471
A physician orders Diflucan oral suspension 200 mg now followed by 100 mg daily for a total of two weeks. How many 35 mL bottles of Diflucan 10 mg/ml suspension are needed to fill this prescription?
Answer:
Step-by-step explanation:
Total amount = 200 mg + (13 days)×(180 mg)/day = 2620 mg
One 35 mL bottle contains (35 mL)×(10 mg)/mL) = 350 mg of medication.
(2620 mg)×(1 bottle)/(350 mg) ≈ 7.5 bottles
8 bottles are needed.
find the surface area of the prism HURRY HURRY
Answer:
should be 192
Step-by-step explanation:
Solve -5/2 +3/4÷1/6 using the order of operations left to right.
Answer: -21/2
Concept:
Usually, when doing mathematics operation, it goes by the PEMDAS order:
ParenthesisExponentMultiplicationDivisionAdditionSubtractionHowever, since the question asks to solve the order of operation from left to right, then we would be doing [addition] --> [Divison]
Solve:
Given
(-5/2) + 3/4 ÷ 1/6
Operate with addition
=(-10/4) + 3/4 ÷ 1/6
=(-7/4) ÷ 1/6
Operate with division
=(-7/4) × 6/1
=-42 / 4
=-21 / 2
Hope this helps!! :)
Please let me know if you have any questions
(4m-n-3). 12n13
8m 20
Answer:
24n^7/m^16
Step-by-step explanation:
Just simplify and rewrite what ever you get simplify again 2 times and now make the calculation as you see every big number is a whole number and is a pair.
Carmen Abdul and David sent a total of 78 text messages over their cell phones during the weekend . Abdul sent 10 fewer messages then Carmen . David sent two times as many messages as Abdul how many messages did they each send?
Answer:
Carmen:27
Abdul:17
David=34
Step-by-step explanation:
Carmen+Abdul+David = 78
Carmen-Abdul=10
David=2Abdul
Carmen=Abdul+10
Carmen+Abdul+David = Abdul+10+Abdul+2Abdul=78
4Abdul=68
Abdul = 68/4=17
Carmen = 17+10=27
David = 2 * 17 = 34
27+17+34=78
Which table represents a linear function?
Answer:
the the 3rd one Is the one
A job can be done by 30 men in 15 days. How long would it take 10 men to do the same job?
Answer:
45 days
Step-by-step explanation:
Based on the giving information, 30*15/10.
Reduce the fraction by canceling the greatest common factor: 3*15 = 45
At a sale this week, a desk is being sold for $312. This is a 35% discount from the original price.
What is the original price?
Answer:
$480
Step-by-step explanation:
Let original price be x
{Original price - discount amount = sale price}
So, x - (x * 35/100) = 312
x - 35x/100 = 312
100x - 35x / 100 = 312
65x/100 = 312
65x = 31200
x = 31200/65
x = $480
So I think the original price of desk was $480
The original price of the desk is $480.
What is discount?A discount is the reduction of either the monetary amount or a percentage of the normal selling price of a product or service.
Given that, a desk is being sold for $312.
Let the original price of the desk be x.
Here, x-35% of x =312
x- 35/100 ×x=312
x-0.35x=312
0.65x=312
x=312/0.65
x=480
Therefore, the original price of the desk is $480.
To learn more about the discount visit:
https://brainly.com/question/3541148.
#SPJ2
14feet 23feet round each measurement to the nearest ten
A restaurant operator in Accra has found out that during the lockdown,if she sells a plate of her food for Ghc 20 each,she can sell 300 plates but for each Ghc5 she raises the price,10 less plates are sold. A.Draw a table relating 5 different price levels with their corresponding number of plates sold.
B.Use the table to find the slope of the demand
C.Find the equation of the demand fraction.
D.Use your equation to determine the price in Ghc if she sells one plate of food to maximize her revenue
The demand equation illustrates the price of an item and how it relates to the demand of the item.
The slope of the demand function is -1/2The equation of the demand function is: [tex]R(x) = (300 - 10x) \times (20 + 5x)[/tex]The price that maximizes her revenue is: Ghc 85From the question, we have:
[tex]Plates = 300[/tex]
[tex]Price = 20[/tex]
The number of plates (x) decreases by 10, while the price (y) increases by 5. The table of value is:
[tex]\begin{array}{cccccc}x & {300} & {290} & {280} & {270} & {260} \ \\ y & {20} & {25} & {30} & {35} & {40} \ \end{array}[/tex]
The slope (m) is calculated using:
[tex]m = \frac{y_2 - y_1}{x_2 - x_1}[/tex]
So, we have:
[tex]m = \frac{25-20}{290-300}[/tex]
[tex]m = \frac{5}{-10}[/tex]
[tex]m = -\frac{1}{2}[/tex]
The equation of the demand is as follows:
The initial number of plates (300) decreases by 10 is represented as: (300 - 10x).
Similarly, the initial price (20) increases by 5 is represented as: (20 + 5x).
So, the demand equation is:
[tex]R(x) = (300 - 10x) \times (20 + 5x)[/tex]
Open the brackets to calculate the maximum revenue
[tex]R(x) =6000 + 1500x - 200x - 50x^2[/tex]
[tex]R(x) =6000 + 1300x - 50x^2[/tex]
Equate to 0
[tex]6000 + 1300x - 50x^2 =0[/tex]
Differentiate with respect to x
[tex]1300 - 100x =0[/tex]
Collect like terms
[tex]100x =1300[/tex]
Divide by 100
[tex]x =13[/tex]
So, the price at maximum revenue is:
[tex]Price= 20 + 5x[/tex]
[tex]Price= 20 + 5 * 13[/tex]
[tex]Price= 85[/tex]
In conclusion:
The slope of the demand function is -1/2The equation of the demand function is: [tex]R(x) = (300 - 10x) \times (20 + 5x)[/tex]The price that maximizes her revenue is: Ghc 85Read more about demand equations at:
https://brainly.com/question/21586143
simplify the following without using calculator
√50 + √98
Answer:
The exact answer is 12*√2, if you need approximately answer it is 12*1.4= 16.8
Step-by-step explanation:
√50 + √98 = √(25*2)+√(49*2)= √25√2+√49*√2= 5√2+7*√2=12*√2
Y= 3x-1
2x+6=y substitution method
Answer: (7, 20)
Concept:
There are three general ways to solve systems of equations:
EliminationSubstitutionGraphingSince the question has specific requirements, we are going to use substitution to solve the equations.
Solve:
Given equations
y = 3x - 1
2x + 6 = y
Substitute the y value since both equations has isolated [y]
2x + 6 = 3x - 1
Add 1 on both sides
2x + 6 + 1 = 3x - 1 + 1
2x + 7 = 3x
Subtract 2x on both sides
2x + 7 - 2x = 3x - 2x
[tex]\boxed{x=7}[/tex]
Find the value of y
y = 3x - 1
y = 3(7) - 1
y = 21 - 1
[tex]\boxed{y=20}[/tex]
Hope this helps!! :)
Please let me know if you have any questions
please help !!!!
i would really appreciate it
Answer: A
Step-by-step explanation: x=-2, y=3, z=-3
Answer:
A. -2, 3, -3
Step-by-step explanation:
x = 7 - 2y + z
y = 21 + 6x + 2z = 21 + 6×(7 - 2y + z) + 2z =
= 21 + 42 - 12y + 6z + 2z = 63 - 12y + 8z
13y = 63 + 8z
y = (63 + 8z)/13
2x + 2y - 3z = 11
2×(7 - 2y + z) + 2×(63 + 8z)/13 - 3z = 11
2×(7 - 2×(63 + 8z)/13 + z) + 2×(63 + 8z)/13 - 3z = 11
14 - 4×(63 + 8z)/13 + 2z + 2×(63 + 8z)/13 - 3z = 11
-2×(63 + 8z)/13 - z = -3
-2×(63 + 8z) - 13z = -39
-126 - 16z - 13z = -39
-29z = 87
z = -3
y = (63 + 8×-3)/13 = (63 - 24)/13 = 39/13 = 3
x = 7 - 2×3 + -3 = 7 - 6 - 3 = -2
