30-32 sketch the region enclosed by the given curves and find its area.
31. [tex]y=\cos ^{2} x \sin x, \quad y=\sin x, \quad 0 \leqslant x \leqslant \pi[/tex]
Answers
Check the picture below.
so hmmm for the sake of completion, let's check where they intersect
[tex]cos^2(x)sin(x)=sin(x)\implies cos^2(x)sin(x)-sin(x)=0 \\\\\\ sin(x)[cos^2(x)-1]=0\implies \begin{cases} sin(x)=0\\ cos^2(x)-1=0 \end{cases} \\\\[-0.35em] ~\dotfill\\\\ sin(x)=0\implies x=sin^{-1}(0)\implies \boxed{x=0} \\\\[-0.35em] ~\dotfill\\\\ cos^2(x)-1=0\implies cos^2(x)=1\implies cos(x)=\sqrt{1} \\\\\\ cos(x)=1\implies x=cos^{-1}(1)\implies \boxed{x=\pi}[/tex]
now, notice, in the picture the function that is "above" or the "ceiling" function is the sin(x), so we'll get the area under the curve by using "above" - "below" functions.
[tex]\stackrel{above}{sin(x)}~~ - ~~\stackrel{below}{cos^2(x)sin(x)} \implies sin(x)-[1-sin^2(x)]sin(x) \\\\\\ sin(x)-[sin(x)-sin^3(x)]\implies ~~\begin{matrix} sin(x)-sin(x) \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~ +sin^3(x)\implies sin^3(x)[/tex]
now let's use the triple angle identity of sine
[tex]\stackrel{\textit{triple angle identity}}{sin(3x)=3sin(x)-4sin^3(x)}\implies 4sin^3(x)=3sin(x)-sin(3x) \\\\\\ sin^3(x)=\cfrac{3sin(x)-sin(3x)}{4}\implies sin^3(x)=\cfrac{3}{4}sin(x)-\cfrac{1}{4}sin(3x) \\\\[-0.35em] ~\dotfill\\\\ \displaystyle\int\limits_{0}^{\pi }~sin^3(x)dx\implies \int\limits_{0}^{\pi }~\left[ \cfrac{3}{4}sin(x)-\cfrac{1}{4}sin(3x) \right]dx[/tex]
[tex]\displaystyle \cfrac{3}{4}\int\limits_{0}^{\pi }sin(x)dx-\cfrac{1}{4}\int\limits_{0}^{\pi }sin(3x)dx\implies \left. \cfrac{3}{4} \cdot -cos(x) \right]_{0}^{\pi }-\left. \cfrac{1}{4} \cdot \cfrac{-cos(3x)}{3} \right]_{0}^{\pi } \\\\\\ \cfrac{3}{2}~~ - ~~\cfrac{1}{6}\implies \implies \blacktriangleright \cfrac{4}{3} \blacktriangleleft[/tex]
![30-32 Sketch The Region Enclosed By The Given Curves And Find Its Area. 31. [tex]y=\cos ^{2} X \sin](https://brainimage.s3.us-east-005.backblazeb2.com/answers/attachments/FtoPaefYJN1O4IqmHBEvSKTw4ZELIpUs.png)
Related Questions
In a random sample of 200 items, 5 items were defective. An estimate of the percentage of defective items in the population is Group of answer choices 2.5% 200 5.0% 10.0%
Answers
An estimate of the percentage of defective items in the population if 5 of 200 items were defective is 2.5%.
How to calculate percentage?The percentage of a defective item out of a total item can be calculated as follows:
% defective item = no of defective item/total no of items × 100
According to this question, in a random sample of 200 items, 5 items were defective.
% defective item = 5/200 × 100
% defective item = 2.5%
Therefore, an estimate of the percentage of defective items in the population if 5 of 200 items were defective is 2.5%.
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in 130426 the digit 2 is in the ____ place?
Answers
Answer:
A Tens place.
Step-by-step explanation:
last 2 digits are twenty six.
Find the volume of this triangular prism. Be sure to include the correct unit in your answer. 7 in 5 in 8 in
Answers
explanation
At summer camp, Rob went on a nighttime nature walk with his troop. During the walk, they saw 3 bats, 1 owl, 2 deer, and 6 squirrels.
Rob had never seen a bat before. What percent of the animals they saw on the walk were bats?
Answers
Answer:
25%???
Step-by-step explanation:
I think
4 ÷ 5/7 Give your answer as a mixed number in its simplest form.
Answers
Answer:
5.6
Step-by-step explanation:
4 ÷ 5/7
is 4 divide 5 over 7
which is 5.6
Sandy made several investments. She bought 1000 shares of a company’s stock for $8.60/share, she bought a bond with a face value of $2500 and a coupon rate of 7%, and she invested $5000 into a fund that is expected to grow by 3.5% per year.
The bond Sandy purchased will mature in 10 years. How much interest will she receive semiannually?
How long will it take the fund she invested in to be worth $10,000?
Answers
can someone please solve this crossword puzzle of proofs for me?
Answers
Answers:
Across:
BisectsGivenABDAngle BisectorReflexiveABDAASDown:
GivenCM = MKMidpointNKMAlternate InteriorCMEVerticalASA======================================================
Explanations:
Always start with the given statements and build toward the thing you want to prove. The first line(s) will basically be restating the given facts word for word.
The term "bisect" means to "divide in two equal pieces". Angle ABC is split into the two equal pieces of angles ABD and CBD, hence ABD = CBD. Segment CK is split into the equal pieces of CM and MK, meaning that CM = MK.
The reflexive theorem says any segment is congruent to itself, so BD = BD.
Vertical angles are formed when we intersect two lines to form an X shape. The vertical angles are opposite one another and always congruent.
Alternate interior angles are always congruent when we have parallel lines.
The AAS (angle angle side) congruence theorem says that two triangles are congruent if we have two pairs of congruent angles, and a pair of congruent sides. The congruent sides are not between the angles mentioned. The order matters.
AAS is slightly different from ASA. With ASA, the congruent sides are between the congruent angles.
What’s the answer? I’m confused
Answers
Answer: it would be y/2 if im not mistaken.
Step-by-step explanation: because with x+3/3 the 3 on the top is canceled out leaving x/3 so it would be the same with y+2/2. I hope im right and that this helps. If not i am sorry.
A rhombus, PQRS, has vertices at the points P(6, 2), Q(7,4), R(6, 6) and S(5, 4). PQRS is translated 3 squares left and 2 squares down. The image of PQRS is P'Q'R'S'. Work out the coordinates of the vertices of P'Q'R'S'.
Answers
Answer:
P': (3, 0)
Q': (4, 2)
R': (3, 4)
S': (2, 2)
Step-by-step explanation:
A rhombus, PQRS, has vertices at the points P(6, 2), Q(7, 4), R(6, 6) and S(5, 4). PQRS is translated 3 squares left and 2 squares down.
Rule: (x - 3, y - 2)
1. Using the given rule, find the new coordinates of the new image:
P': (6, 2) ➺ (6 - 3, 2 - 2) ➺ (3, 0)
Q': (7, 4) ➺ (7 - 3, 4 - 2) ➺ (4, 2)
R': (6, 6) ➺ (6 - 3, 6 - 2) ➺ (3, 4)
S': (5, 4) ➺ (5 - 3, 4 - 2) ➺ (2, 2)
New coordinates:
P': (3, 0)
Q': (4, 2)
R': (3, 4)
S': (2, 2)
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What are the slope and y-intercept of the line
represented by the points shown in the table?
O slope: 2, y-intercept: -20
Oslope: 2, y-intercept: 10
Oslope: 5, y-intercept: -20
O slope: 5, y-intercept: 10
Answers
The slope of the line is 2 and y-intercept is -20 and the linear equation can be framed as y = 2x—20 option first is correct.
What is a straight line?A straight line is a combination of endless points joined on both sides of the point.
The slope 'm' of any straight line is given by:
[tex]\rm m =\dfrac{y_2-y_1}{x_2-x_1}[/tex]
Taking two points to find the slope of the line:
(0, -20) and (5, -10)
[tex]\rm m =\dfrac{-10+20}{5-0}[/tex]
m = 2
The straight line equation:
y = mx + c
y = 2x + c
Plug (0, -20)
-20 = c
Thus, the slope of the line is 2 and y-intercept is -20 and the linear equation can be framed as y = 2x—20 option first is correct.
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Find the volume of the figure
Answers
Answer:
[tex]800\pi yd^3[/tex]
Step-by-step explanation:
[tex]A = 10\pi ^2(8)\\= 800\pi yd^3[/tex]
Answer: 2513.27 yd³
Step-by-step explanation:
Volume of a cylinder:
-> This formula means we are finding the area of the base circle and multiplying it be the height
V = πr²h
V = πr²h
V = π(10)²(8)
V ≈ 2513.27 yd³
The volume of the figure is about 2513.27 yd³.
please help me to answer
Answers
Answer:
7.92 cm²
Step-by-step explanation:
Area of Trapezium :
1/2 x (a + b) x h1/2 x (4.8 + 8.4) x 1.20.6 x 13.27.92 cm²What's the circumference of a circle with a diameter of 9 inches?
Use 3.14 for π.
C = [?] inches
Enter the number that belongs in the green box. Do not round your answer.
Hint: C = πd
Answers
Answer:
C = 28.26 in
Step-by-step explanation:
the circumference (C) is calculated as
C = πd ( d is the diameter ) , then
C = 3.14 × 9 = 28.26 in
[tex]\rule{300}{1}\\\dashrightarrow\large\blue\textsf{\textbf{\underline{Given question:-}}}[/tex]
What's the circumference of a circle with a diameter of 9 inches?
[tex]\dashrightarrow\large\blue\textsf{\textbf{\underline{Answer and how to solve:-}}}[/tex]
We are asked to find the circumference of a circle, which we can find using the following formula:-
[tex]\bold{Circumference{(circle)}=\pi d}[/tex]
Where:-π=pi [3.14…]d=diameter (9 inches)How to solve for circumference:-
Replace d with 9 and multiply:-
[tex]\hookrightarrow\sf{C=3.14\times9}[/tex]
[tex]\bigstar[/tex] On simplification,
[tex]\hookrightarrow\sf{C=28.26\:inches}[/tex]
Notice that we are not asked to round our answer, so we leave it like that - C=28.26.
Good luck with your studies.
[tex]\rule{200}{2}[/tex]
7). John had 9 pieces of gum at the
beginning of the day. He now has 8
pieces of gum. How much gum did
he eat during the day? Write the
equation and the solution.
Answers
Answer:
9-1=8
Step-by-step explanation:
he started with 9 but at the end of the day he had 8. meaning he ate 1 peice
Please Help me (i will give brainly ) Is the following graph linear, exponential or quadratic?
Answers
Answer:
Exponential Graph
Step-by-step explanation:
The exponential graph of a function represents the exponential function properties.
Another way to known of if the graph's domain is all real numbers
Answer:
Exponential
Step-by-step explanation:
The listed points are :
(0, 27) (1, 9) (2, 3)The y-values are powers of 3 which decrease. Hence, it is exponential.
11. Find the equation for the circle with center (−2,4) and passing through (5,−3).
Answers
Answer:
In standard form: (x+2)² + (y-4)² = 98
Use the distance formula to find radius:
[tex]\sf d = \sqrt{(x_2 - x_1)^2 + (y_2-y_1)^2}[/tex]
where (x1, y1), (x2, y1) are points
[tex]\rightarrow \sf d = \sqrt{(-2 - 5)^2 + (4-(-3))^2}[/tex]
[tex]\rightarrow \sf d = \sqrt{(-7)^2 + (7)^2}[/tex]
[tex]\rightarrow \sf d = 7\sqrt{2}[/tex]
Equation of circle:
(x-h)² + (y-k)² = r²(x-(-2))² + (y-4)² = (7√2)²(x+2)² + (y-4)² = (7√2)²(x+2)² + (y-4)² = 98Can anyone help solve this step by step?
Answers
Answer:
85.2 m²
Step-by-step explanation:
Perhaps one of the easiest ways to decompose this composite figure is to consider it as a rectangle with a triangle removed from it.
__
The vertical dimension of the missing triangle is 8-4 = 4 m. The horizontal dimension of the missing triangle is 11.4 -8.4 = 3 m. The overall rectangle, including the missing triangle, is 11.4 m wide and 8 m high.
The area of the figure is ...
area = rectangle area - triangle area
= (11.4 m)(8 m) -1/2(4 m)(3 m) = 91.2 m² -6 m² = 85.2 m²
The area of the shape is 85.2 square meters.
_____
Additional comment
The area of a rectangle is the product of its length and width:
A = LW
The area of a triangle is half the product of its base and height:
A = 1/2bh
Question 10 of 10
Find the area of the figure.
5 m
2 m
9 m
O A. 27 m2
O B. 54 m2
O C. 45 m2
O D. 36 m2
Answers
Answer:
45 m2
Step-by-step explanation:
It was the only reasonable answer lol
9x5=45
A push-cart vendor is selling sodas at the beach. partway into the day, he raises his
price per bottle by one quarter. after this price increase, the total dollar amount
collected for the day after selling n bottles at the new price is given by the following
equation:
t = 12 + 1.75n
according to this equation, how many bottles did the vendor sell today before raising the
price?
o a
8
ob.
12
c. 18
d. 21
Answers
Answer:
[tex]8[/tex].
Step-by-step explanation:
Notice that in equation for the total dollar amount collected ([tex]12+ 1.75\, n[/tex]), every additional bottle sold at the new price brings in [tex]1.75[/tex] dollars:
[tex]\begin{aligned}& 12 + 1.75\, n && \text{$n$ bottles at new price} \\ -\; & 12 + 1.75\, (n+1) && \text{$(n+1)$ bottles at new price} \\ =\; & 1.75\end{aligned}[/tex].
Therefore, the per-bottle price after the [tex]\$0.25[/tex] price increase would be [tex]\$1.75[/tex]. The per-bottle price before the price increase would be [tex]\$1.75 - \$0.25 = \$1.50[/tex].
Also notice that when [tex]n = 0[/tex], the total amount collected was [tex]t = 12 + 1.75\, n = 12[/tex]. In other words, the total amount collected was [tex]\$12[/tex] before any bottle was sold at the new price.
Thus, the vendor had collected [tex]\$12\![/tex] by selling at the initial price of [tex]\$1.50[/tex] per bottle. The number of bottles sold at that price would be:
[tex]\begin{aligned}\frac{12}{1.50} = 8\end{aligned}[/tex].
1. SSS
2. SAS
4. ASA
7. HL
8. Not Enough Information X
Answers
Okay so, i need help with this. It’s really important and I need to finish this paper before school tomorrow. Please help i will mark brainliest!!!!!
Answers
Answer:
5x^2 - 4x - 3 = 0
Step-by-step explanation:
For the equation
ax^2 + bx + c = 0
the roots are [-b +/- √(b^2 - 4ac)] / 2a
So comparing the terms to the values in the question:
a = 5
b = -4
c = -3
Evaluate the equation below
Answers
Answer:
55
Step-by-step explanation:
we can get the answer by putting it in the calculator
somebody please help me
Answers
Answer:
is the answer 2 3/4
Step-by-step explanation:
- 1/2 + (3/4 * 4/9) = ?
Answers
Answer:2
Step-by-step explanation:
SOMOENE HELP ASAPPPPP
Answers
53= 29t < 343
Step-by-step explanation:
Answer:
53 = 343t-29
Step-by-step explanation:
53 is equal to (53 =)
t times 343 (343t)
29 less than the quantity (-29)
Good luck! :)
Scott rolled a number cube several times and counted how many times he rolled a 3.
What is the theoretical probability of Scott rolling a 3 if his number cube is fair?
Answers
Using it's concept, it is found that the theoretical probability of Scott rolling a 3 if his number cube is fair is of [tex]\frac{1}{6}[/tex].
What is a probability?A probability is given by the number of desired outcomes divided by the number of total outcomes.
For a theoretical probability, these number of outcomes are taken before any experiment.
In this problem, there are six numbers, one of which is three, hence the probability is given by:
[tex]p = \frac{1}{6}[/tex].
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The perimeter of a rectangle is 90 feet. The width is 15 feet. What is the perimeter?
Answers
=
15
feet
Breadth of the rectangle
=
30
feet.
Explanation:
Let the breadth of the rectangle
=
x
So, the length of the rectangle
=
2
×
x
The perimeter given
=
90
feet
As per the formula for perimeter of a rectangle
Perimeter
=
2
×
(length + breadth )
90
=
2
×
(
x
+
2
x
)
90
=
2
×
(
3
x
)
90
=
6
x
x
=
90
6
x
=
15
So the length of the rectangle
=
15
feet, breadth of the rectangle
=
30
feet.
27) Harry made 4 hits in 9 times at-bat. If she keeps the same success level, how many hits should she make in 18 times at bat?
Answers
Answer:
8 hits
Step-by-step explanation:
The ratio is :
hits : times-at-bat = 4 : 9Now, we have 18 times at bat.
Multiply both sides of the ratio with 2 :
4 x 2 hits : 9 x 2 times at bat8 hits : 18 times at batHarry should make 8 hits in 18 times at bat.
She is serving hot coffee and cold coffee in two rectangular-shaped tanks, each with dimensions width 8.75 inches, length 8.75 inches, height 23 inches.
The hot coffee has a density of 5.8 oz/in3 and the cold coffee has a density of 3.5 oz/ in3
(A) What is the volume of the coffee in each tank?
Show all math work needed to complete this problem.
(B) What is the mass of the coffee in each the tank?
Show all math work needed to complete this problem.
(C) How many tanks can be placed on a table which can support a maximum weight of 800 lb?
Explain why or why not.
Show all math work needed to complete this problem.
Answers
Answer:
Part ADimensions of tank:
width = 8.75 inlength = 8.75 inheight = 23 in⇒ Volume of each tank = width × length × height
= 8.75 × 8.75 × 23
= 1760.9375 in³
Part BGiven:
Hot coffee density = 5.8 oz/in³Cold coffee density = 3.5 oz/in³mass = density × volume
⇒ mass of hot coffee = 5.8 oz/in³ × 1760.9375 in³
= 10213.44 oz (nearest hundredth)
⇒ mass of cold coffee = 3.5 oz/in³ × 1760.9375 in³
= 6163.28 oz (nearest hundredth)
Part CConvert the masses of the tanks from ounces to pounds
1 lb = 16 oz
⇒ mass of hot coffee tank = 10213.44 ÷ 16
= 638.34 lb (nearest hundredth)
⇒ mass of cold coffee tank = 6163.28 ÷ 16
= 385.21 lb (nearest hundredth)
If the table can hold a maximum of 800 lb:
hot coffee: 800 lb ÷ 638.34 lb = 1.25 (nearest hundredth)
cold coffee: 800 lb ÷ 385.21 lb = 2.08 (nearest hundredth)
Therefore, either:
1 tank of hot coffee OR2 tanks of cold coffeecan be placed on a table which can support a maximum weight of 800 lb
A triangle has a base of 8 cm and a height of 12 cm.
What is the area of the triangle?
Enter your answer in the box.
*box* cm2
Answers
Answer: 48 cm^2
Step-by-step explanation:
The equation for the area of a triangle is 1/2(base*height). Plug in the numbers to get an equation of 1/2(8*12). 8*12 = 96. Now divide by 2. 96/2 = 48.
Answer:
48 cm^2
Step-by-step explanation:
a triangle area is hb/2
so 8*12=96
96/2=48 therefore 48 is the answer.