Find the area bounded by the curves y = x/2 and y =√x.
Answers
Answer:
[tex]\displaystyle \frac{4}{3}[/tex].
Step-by-step explanation:
Start by finding the intersection of the two curves:
[tex]\left\lbrace\begin{aligned}& y = \frac{x}{2} \\ & y = \sqrt{x} \\ & x \ge 0 \\ \end{aligned}\right.[/tex].
[tex]\displaystyle \frac{x}{2} = \sqrt{x}[/tex] while [tex]x \ge 0[/tex].
[tex]\displaystyle \frac{x^{2}}{4} - x = 0[/tex].
[tex]x = 0[/tex] or [tex]x = 4[/tex].
Therefore, these two curves would intersect at two points: [tex](0,\, 0)[/tex] and [tex](4,\, 2)[/tex].
The area bounded between [tex]\displaystyle y = \frac{x}{2}[/tex] and [tex]y = \sqrt{x}[/tex] would be between [tex]x = 0[/tex] and [tex]x = 4[/tex].
Refer to the diagram attached. The graph [tex]y = \sqrt{x}[/tex] is always above the graph of [tex]\displaystyle y = \frac{x}{2}[/tex] over the entire bounded area (except for the two intersections).
Therefore, [tex]\displaystyle \left(\sqrt{x} - \frac{x}{2}\right)[/tex] would represent the vertical distance between the upper and lower curve for any given [tex]x[/tex] over this bounded area (where [tex]0 \le x \le 4[/tex].)
Integrating height over the horizontal variable [tex]x[/tex] over some closed interval would give area. Likewise, the area between the two curves in this question could be found with the following integral:
[tex]\begin{aligned}& \int\limits_{0}^{4} \left(\sqrt{x} - \frac{x}{2}\right)\, dx \\ = \; & \int\limits_{0}^{4} \left(x^{1/2} - \frac{x}{2}\right)\, dx \\ =\; & \left[\frac{2}{3}\, x^{3/2} - \frac{x^{2}}{4}\right]_{x=0}^{x=4} \\ =\; & \frac{2\times 8}{3} - 4 \\ =\; &\frac{4}{3} \end{aligned}[/tex].
Related Questions
What is the equation of the line that is perpendicular to
the given line and has an x-intercept of 6?
4
O y=-3x+8
O x
y=-3x+6
O y= x - 8
Oy= x-6
Answers
Answer:
y= x-6
Step-by-step explanation:
The equation of the line that is perpendicular to the given line and has an x-intercept of 6 is y = [tex]\frac{4}{3}x-8[/tex]
Perpendicular linesPerpendicular lines follows the mathematical principle.
m₁m₂ = -1where
m₁ and m₂ are slopes of the lines.
A line that has a x - intercept of 6 and perpendicular to the given line can be represented as follows.
y = mx + b
where
m = slope
b = y-intercept
(-4, 4)(4, -2)
m = -2 - 4 / 4 + 4 = -6 / 8 = -3 / 4
m₁m₂ = -1
- 3 / 4 m₂ = -1
m₂ = [tex]\frac{4}{3}[/tex]
Therefore,
x-intercept of 6
0 = 6([tex]\frac{4}{3}[/tex]) + b
b = - 8
Therefore, the equation is as follows:
y = [tex]\frac{4}{3}x-8[/tex]
learn more on x-intercept here: https://brainly.com/question/16952211
Evaluate (3x + 5) - (2x + 7) when x = 5.
A. 30 - 34 = -4
B. 20 - 17 = 3
C. 20 - 10 + 7 = 17
D. 30 + 34 = 64
13 of 13
Answers
Answer:
The answer is B. 20 - 17 = 3
Step-by-step explanation:
( 3x + 5) - (2x + 7)...... x = 5
( 3(5) + 5) - ( 2(5) + 7)
( 15 + 5) - (10 + 17)
20 - 17 = 3
Thank you :)
Answer:
The answer is 20 - 17 = 3 [Choice B.]
Step-by-step explanation:
Put x = 5 into the formula (3x + 5) - (2x + 7)
[3(5)+5] - [2(5)+7]
= [15 + 5] - [10 + 7]
=20 - 17
= 3
Round 529.5635 to the nearest
hundredth.
Answers
Answer: 529.56
Explanation:
Answer:
529.5600
Step-by-step explanation:
How do you read decimals? One way to read decimals is with a place value chart. Notice that the place names are similar to the place names for whole numbers, but the names go in the opposite direction. This is because the values of the places decrease as you move from left to right. Think about how this applies to money: a dime, which is worth one-tenth of a dollar, is worth more than a penny, which is worth one one-hundredth of a dollar.
write 1,200 in scientific notation
Answers
Answer:
1.2 x 10^3 Hope it helps :)
Step-by-step explanation:
1200= 1.2 x 10^3
Guided Practice
Assume that 4.5% is an annual interest rate. Find the interest rate for an account that is compounded quarterly and monthly.
A.
1.125%; 0.375%
B.
18%; 54%
C.
0.9%; 0.45%
Answers
Answer:
A.
1.125%; 0.375%
Step-by-step explanation:
Quarterly interest rate is 4.5 / 4 = 1.125%.
Monthly rate is 4.5 / 12 = 0.375%.
g(x) = (x - 9)/(sqrt(x) - 3) a) Make two tables, one showing the values of 8 for =8.9,8.99 and 8.999 and one showing values of 8 for X =9.1,9.01 , and 9.001. (b) Make a conjecture about the value of lim x 9 x-9 sqrt x -3
Answers
10 points!! Pls help with 5 and 6
Answers
Answer:
I only know number 5 so here it is
the solution for the left is: 2x6=12
the solution for the right is: 5x3+4x1=15+4=19
and the answers are: 1.12 . 2.19
PLS HELP WILL MARK BRAINLIEST. IF GOTTEN RIGHT SHOW WORK
Answers
9514 1404 393
Answer:
12 feet
Step-by-step explanation:
You seem to have a few of these, so let's find a generic solution.
The vertex form of a quadratic equation is ...
f(x) = a(x -h)² +k
When we expand that, we get ...
f(x) = a(x² -2hx +h²) +k = ax² -2ahx +ah² +k
Comparing this to your standard form quadratic ...
h(t) = -16t² +vx +s
we can see that ...
a = -16-2ah = vah² +k = sSolving for h and k, we get ...
(-2)(-16)h = v
h = v/32
and
k = s -ah² = s -(-16)(v/32)²
k = s +v²/64
So, the generic solution for these problems is ...
h = time to maximum height = v/32
k = maximum height = s +v²/64
__
This problem asks for the maximum height, which will be ...
s +v²/64 = 3 +24²/64 = 3 +9 = 12 . . . . feet
what is the slop of 2x+3y-9=0
Answers
Answer:
Step-by-step explanation:
We need to write this equation in y=mx+b where m is the slope.
3y= -2x +9
y= -2/3 +9
So now that it is in the form of y=mx+b, we can find the slope. -2/3 is the slope of the equation. Hope that helps!
?????????help pls??????????
Answers
Answer:
distributive property
Step-by-step explanation:
i think its distributive property
Is this correct i am not sure
Answers
Answer:
YesStep-by-step explanation:
Collinear points are the points on the same line.
We see that points E, B and F are part of same line n.
Your answer is correct.
Answer:
EBF points are coplaner
Step-by-step explanation:
Coplaner points are the points which are ColinearThey are situated in one line .When is x? greater than x??
A. Always
B. Never
C. when x < 0
D. when x > 1
Answers
[tex] \text{{\bf{C.}} \red{when x < 0}}[/tex]
PLS HELP ME SOLVE THIS
The pH level, P, of a liquid can be determined from the hydronium ion (H3O+) concentration, c, using the equation P = –log(c). If the pH level of grape juice is 3.3, what is the hydronium ion concentration?
A. 5.01 x 10^–4
B. 1.19 x 10^–2
C. 1.99 x 10^3
D. 1.53 x 10^5
Answers
Logarithm involves the use of exponents to represent numbers.
If the pH level of grape juice is 3.3, the hydronium ion concentration is [tex]5.01 \times 10^{-4[/tex]
Given that:
[tex]P = -\log(c)[/tex]
To calculate c when [tex]P = 3.3[/tex]
We first substitute 3.3 for P in [tex]P = -\log(c)[/tex]
So, we have:
[tex]3.3 = -\log(c)[/tex]
Divide both sides by -1
[tex]-3.3 = \log(c)[/tex]
Apply law of logarithm
[tex]10^{-3.3)} = c[/tex]
Rewrite as:
[tex]c = 10^{-3.3}[/tex]
Using a calculator, we have:
[tex]c = 0.000501[/tex]
Rewrite as
[tex]c = 5.01 \times 10^{-4[/tex]
Hence, the hydronium ion concentration is [tex]5.01 \times 10^{-4[/tex]
Read more about logarithms at:
https://brainly.com/question/8657113
slope=-2 passes thru point 5,-3, write the equation of the line
Answers
Step-by-step explanation:
hope this helps. the process is shown above
simplify the following equation:
10 +6/2+3
Answers
Answer:
16 ...................
When graphed in the (x,y) coordinate plane, at what point do the lines x - y = 12
and y= 2 intersect?
Answers
Answer:
(14, 2).
Step-by-step explanation:
x - y = 12
y = 2
Solving this system:
Plug y = 2 into the first equation:
x - 2 = 12
x = 14.
Therefore the point of intersection is at (14, 2).
Haihai, can you help me please (•◡•) /
Answers
Answer:
its A!!! :) good luck with the rest of your assignment!
Please help quickly it is super late rn
Answers
Step-by-step explanation:
-4^(-5)/ 2^(-3) =
2³/(-4)^5 =
8/(-1024) =
- 1/128 (d)
What is the probability that a resident reports high satisfaction, given that the resident is a renter?
O 13.7%
O 20.0%
O 36.6%
O 68.2%
Answers
Answer:
The probability that a resident reports high satisfaction while the resident is a renter is Option(a) [tex]13.7\%[/tex]
Step-by-step explanation:
Given:
Levels of satisfaction
High Medium Low Total
Owners [tex]26[/tex] [tex]34[/tex] [tex]18[/tex] [tex]78[/tex]
Renters [tex]15[/tex] [tex]12[/tex] [tex]5[/tex] [tex]22[/tex]
Total [tex]41[/tex] [tex]46[/tex] [tex]23[/tex] [tex]110[/tex]
The Resident is a renter.
Step 1:
To find the probability that a resident reports high satisfaction.
P(Renter- higher satisfaction)= Higher Satisfaction by render ÷ Total level of satisfaction
[tex]=\frac{15}{110}[/tex]
[tex]=0.137[/tex]
[tex]13.7\%[/tex] is the probability that a resident reports high satisfaction.
Therefore, Option (a) is a correct answer.
Learn more about Probability, refer:
https://brainly.com/question/13604758https://brainly.com/question/10734660Answer:
A) 13.7%
Step-by-step explanation:
edge 2023
The difference of F and 1/3 is 1/12
Answers
Answer:
Step-by-step explanation:
[tex]F -\frac{1}{3} = \frac{1}{12}\\F= \frac{1}{12} +\frac{1}{3}\\F= \frac{1}{12} - \frac{4}{12}\\F= \frac{1-4}{12}\\F= \frac{-3}{12}=-\frac{1}{4}[/tex]
HELP ME PLEASE!!!!!!
Answers
9514 1404 393
Answer:
-2x +4
Step-by-step explanation:
We can write the polynomial p(x) in terms of the factors (x+2) and (x-3) as ...
p(x) = (x+2)(x-3)q(x) +ax +b
Where ax+b is the remainder from division by x^2 -x -6 = (x+2)(x-3). The values of 'a' and 'b' can be found from ...
p(-2) = 8 = -2a +b
p(3) = -2 = 3a +b
Subtracting the first equation from the second gives ...
(3a +b) -(-2a +b) = (-2) -(8)
5a = -10
a = -2
Then the first equation tells us ...
8 = -2(-2) +b
4 = b
So, the remainder from division by (x^2 -x -6) is (-2x +4).
If a:b=b:3=3 then find the value of a.
Answers
Answer:
b:3=3
⇒b=1
a:b=3
⇒a:1=3
⇒a=1/3
Step-by-step explanation:
6×3/8
8×4/5
8×3/4
5×4/5
2×2/3
multiplication of. proper fraction by proper fraction
Answers
Step-by-step explanation:
2.25
6.4
4
1.333
Hope is will help you
6.4
4
1.333
Hope my answer is correct
Caden's mother gave him a $20 bill to get groceries. If he spent $14.39, how much change should he ge! back?
Answers
Answer:
$5.61
Step-by-step explanation:
Subtract the amount spent by the amount given to the cashier.
$20 - $14.39 = $5.61
Answer:
If he spent 14.39, and he only had a 20 dollar bill, Caden would have to give the 20 dollar bill meaning we suubtract 14.39 from 20.
20-14.39 = 5.61
Caden should get $5.61 back.
Step-by-step explanation:
Plz help!! Select the correct answer from each drop-down menu.
The given equation has been solved in the table.
Step
Statement
1
3r – 10 = -16
2.
3.1 – 10 + 10 = -16 + 10
3
3r = -6
4
3.1
3
5
-2
Use the table to complete each statement.
In step 2, the
In step 4, the division
property of equality was applied.
property of equality was applied.
Answers
Answer:
Addition Property of Equality
Division Property of Equality
Step-by-step explanation:
3x - 10 = -16
~Add 10 to both sides
3x = -6
~Divide 3 to both sides
x = -2
As you can see from the table given, we are using the addition property of equality in step 2 and division property of equality in step 4.
Best of Luck!
Open Response
How many times less is the
value of the 4 in 356,740 than
the 4 in 763,450? Explain.
Answers
Answer:
10 times less
Step-by-step explanation:
In the number 356, 740, the 4 is in the tenths place.
In 763,450, the 4 is in the hundredths place.
100/10 = 10
11. Solve: (6t - 9) + 8 = 7 + 2(t - 1)
Answers
[tex] \large \mathfrak{Solution : }[/tex]
let's solve for t :
6t - 9 + 8 = 7 + 2(t - 1)6t - 1 = 7 + 2t - 2 6t - 2t = 5 + 14t = 6t = 6 ÷ 4t = 3/2 or 1.5Answer:
t = [tex]\frac{3}{2}[/tex]
Step-by-step explanation:
Given
(6t - 9) + 8 = 7 + 2(t - 1) ← distribute parenthesis on both sides
6t - 9 + 8 = 7 + 2t - 2 , that is
6t - 1 = 5 + 2t ( subtract 2t from both sides )
4t - 1 = 5 ( add 1 to both sides )
4t = 6 ( divide both sides by 4 )
t = [tex]\frac{6}{4}[/tex] = [tex]\frac{3}{2}[/tex]
HELP ASAP!!!! I’m very confused I will mark Brainlist.
Answers
Answer:
Step-by-step explanation:
vertical stretch by 2
so 2f(x)
reflection in x-axis -2f(x)
transition 3 units down -2f(x)-3
f(x)=x²
g(x)=-2x²-3
71. Evaluate a + a3 - 4 * 4 when a = 2
Answers
Answer:
= -8
Step-by-step explanation:
a = 2
2 + (2×3)- 16
= 2 + 6 - 16
= -8
Answer:
-6
Step-by-step explanation:
substitute the value of the variable into the equation and simplify
how should 13,079 be written in in expanded form
Answers
Answer:
10,000 + 3,000 + 70 + 9
Step-by-step explanation:
It will be 10,000 + 3,000 + 70 + 9
That is how expanded form works.
Hope it helps :)
If you have any questions ask me.
Answer:
10000. 3000. 70. 9
Step-by-step explanation:
