Answer:
D. (X + 3 ) ( X +1 )
process:
x² + 4x + 3
= x² + (3 + 1)x +3
= x² + 3x + x + 3
= X (x+3) + 1 (X+3)
= (X + 3 ) ( X +1 )
A group consists of five men and five women. Three people are selected to attend a conference.
A. In how many ways can three people be selected from this group of ten?
B. In how many ways can three women be selected from the five women?
C. Find the probability that the selected group will consist of all women.
Answer:
a) 120
b) 10
c) 1/12
Step-by-step explanation:
The number of ways that a sample of r can be selected from a population of n is:
nCr = n! / (r! (n−r)!)
a) 3 people selected from a group of 10
₁₀C₃ = 120
b) 3 women selected from a group of 5
₅C₃ = 10
c) Of the 120 committees that can be chose, 10 are all women. So the probability is 10/120 = 1/12.
Convert the line integral to an ordinary integral with respect to the parameter and evaluate it. ModifyingBelow Integral from nothing to nothing With Upper C (y minus z )ds∫C(y−z)ds; C is the helix left angle 12 cosine t comma 12 sine t comma t right angle12cost,12sint,t, for 0 less than or equals t less than or equals 2 pi0≤t≤2π The value of the ordinary integral is nothing.
C is parameterized by
[tex]\vec r(t)=\langle x(t),y(t),z(t)\rangle=\left\langle12\cos t,12\sin t,t\right\rangle[/tex]
for 0 ≤ t ≤ 2π. In the integral, replace y and z as above, and the line element ds is
[tex]\mathrm ds=\sqrt{\left(\dfrac{\mathrm dx}{\mathrm dt}\right)^2+\left(\dfrac{\mathrm dy}{\mathrm dt}\right)^2+\left(\dfrac{\mathrm dz}{\mathrm dt}\right)^2}\,\mathrm dt=\sqrt{145}\,\mathrm dt[/tex]
So the integral is
[tex]\displaystyle\int_C(y-z)\,\mathrm ds=\sqrt{145}\int_0^{2\pi}(12\sin t-t)\,\mathrm dt[/tex]
sin(t) has period 2π, so that term contributes nothing to the integral, leaving us with
[tex]\displaystyle\int_C(y-z)\,\mathrm ds=-\sqrt{145}\int_0^{2\pi}t\,\mathrm dt=\boxed{-2\pi^2\sqrt{145}}[/tex]
What is the length of the line?
square root of 119
square root of 60
13
square root of 17
Answer:
13
Step-by-step explanation:
The line is the hypotenuse of a right triangle that is 5 units high and 12 units wide. The Pythagorean theorem, or the distance formula, tells you the length is ...
length = √(5² +12²) = √(25+144) = √169
length = 13
Penni Precisely buys 100 worth of stock in each of three companies: Alabama Almonds, Boston Beans, and California Cauliflower. After one year, AA was up 20%, BB was down 25%, and CC was unchanged. For the second year, AA was down 20% from the previous year, BB was up 25% from the previous year, and CC was unchanged. If A, B, and C are the final values of the stock, then:______A) A = B = CB) A = B < CC) C < B = AD) A < B < CE) B < A < C
Answer:
E) B < A < C
Step-by-step explanation:
Penni Precisely buys 100 worth of stock in each of three companies: Alabama Almonds, Boston Beans, and California Cauliflower.
Initial Values are: A, B and C
After one year, A was up 20%, B was down 25%, and C was unchanged.
Values after one year:
A=(100%+20%) of $100=$120 worth of stock.B=(100-25%) of $100=$75 worth of stock.C=100% of $100=$100 worth of stock.For the second year, A was down 20% from the previous year, B was up 25% from the previous year, and C was unchanged.
Values after the second year
A=(100%-20%) of $120 =80% of $120=$96 worth of stock.B=(100%+25%) of $75 =125% of $75=$93.75 worth of stock.C=100% of $100 =$100 worth of stock.Therefore, from the final values of the stock, we have that:
B<A<C
The correct option is E.
Solve for y 6(y-1/3) = 4(3y-5)
Answer:
y = 3
Step-by-step explanation:
It can work well to start by eliminating parentheses.
6(y -1/3) = 4(3y -5)
6y -2 = 12y -20
18 = 6y . . . . . . . . . add 20-6y to both sides
3 = y . . . . . . . . . . . divide by 6
Answer:
y=3
Step-by-step explanation:
6(y−13)=4(3y−5)
(6)(y)+(6)(−13)=(4)(3y)+(4)(−5)(Distribute)
6y+−2=12y+−20
6y−2=12y−20
Step 2: Subtract 12y from both sides.
6y−2−12y=12y−20−12y
−6y−2=−20
Step 3: Add 2 to both sides.
−6y−2+2=−20+2
−6y=−18
Step 4: Divide both sides by -6.
−6y−6=−18−6
y=3
The sum of two numbers is 9.9, and the sum of the squares of the numbers is 53.21. What are the numbers?
Answer:
There are two possibilities:
[tex]x_{1} = 3.5[/tex] and [tex]y_{1} = 6.4[/tex]
[tex]x_{2} = 6.4[/tex] and [tex]y_{2} = 3.5[/tex]
Step-by-step explanation:
Mathematically speaking, the statement is equivalent to this 2-variable non-linear system:
[tex]x + y = 9.9[/tex]
[tex]x^{2} + y^{2} = 53.21[/tex]
First, [tex]x[/tex] is cleared in the first equation:
[tex]x = 9.9 - y[/tex]
Now, the variable is substituted in the second one:
[tex](9.9-y)^{2} + y^{2} = 53.21[/tex]
And some algebra is done in order to simplify the expression:
[tex]98.01-19.8\cdot y +2\cdot y^{2} = 53.21[/tex]
[tex]2\cdot y^{2} -19.8\cdot y +44.8 = 0[/tex]
Roots are found by means of the General Equation for Second-Order Polynomials:
[tex]y_{1} \approx \frac{32}{5}[/tex] and [tex]y_{2} \approx \frac{7}{2}[/tex]
There are two different values for [tex]x[/tex]:
[tex]y = y_{1}[/tex]
[tex]x_{1} = 9.9-6.4[/tex]
[tex]x_{1} = 3.5[/tex]
[tex]y = y_{2}[/tex]
[tex]x_{2} = 9.9 - 3.5[/tex]
[tex]x_{2} = 6.4[/tex]
There are two possibilities:
[tex]x_{1} = 3.5[/tex] and [tex]y_{1} = 6.4[/tex]
[tex]x_{2} = 6.4[/tex] and [tex]y_{2} = 3.5[/tex]
Requiem needs to know if the triangle shown is a right triangle. Which equation could he use to help ?
Answer:
Pythagoras theorem: hypotenuse² = opposite² + adjacent²
Step-by-step explanation:
To know if a triangle is a right angle, we need to have the length of each sides of the triangle. Then we would apply Pythagoras theorem to determine if it is truly a right angled triangle.
Pythagoras theorem is a theorem in the form of an equation which shows the relationship between the sides of a right angled triangle.
let the sides of the right angled triangle be:
opposite =a, adjacent = b and hypotenuse = c
Using Pythagoras theorem
hypotenuse² = opposite² + adjacent²
c² = a² + b²
If the left hand side of the equation = the right hand side of the equation, it is a right angled triangle.
It costs Neil $2.88 to make two dozen muffins. He sold the muffins for $.80 each. If Neil sold six dozen muffins how much profit did he make from the sale?
Hi!
So for this you need an equation. It will be 72 • 0.8 - 2.88 • 3
this is bc 12 x 6 is 72 and 3 bc 2 x 3 is 6.
so after you solve it you should get : $48.96..Hope this helped! :))
In the USA today internet poll 73 percent of readers indicated that they were satisfied with their lives.
1. 73 percent of readers are satisfied with their lives.
A. The population B. The sample
C. a population statistic D. a sample statistic
Answer:
c) Population parameter or Population statistic
Step-by-step explanation:
Population:-
The totality of observation with which we are concerned , whether this number be finite or infinite is called population
Sample :
A sample is subset of a Population
Given data the USA today internet poll is called Population
Given data 73 percent of readers are satisfied with their lives.
This is called Population parameter.
given:f(x)=x-7 and h(x)=2x+3 write the rule for f(h(x))
Answer:
Step-by-step explanation:
h(x) = 2x + 3
f(2x+3)
2x+3 - 7
2x - 4
Solve for x
A) 7
B) 10
C) 5
D) -7
Answer:
D) -7
Step-by-step explanation:
[tex] \because \: PQ + QR = PR \\ \therefore \: 2x + 17 + 12 = 22 + x \\ \therefore \: 2x + 29 = 22 + x \\ \therefore \: 2x - x = 22 - 29 \\ \therefore \: x = - 7 \\ [/tex]
Trey is estimating the length of a room in his house. The actual length of the room is 17 m. Trey's estimate is 15 m.
Find the absolute error and the percent error of Trey's estimate. If necessary, round your answers to the nearest tenth.
Answer:
Absolute error = 2m
Step-by-step explanation:
Absolute Error = Measured Value - Actual Value
17m - 15m = 2m
Answer:
2m and 11.8%
Step-by-step explanation:
Absolute error is defined mathematically as;
Error/original value
[Original value - estimated value]
[17-15 ] = 2m
The percentage error would be
[Original value - estimated value]/ original value
[17-15 ] /17 × 100%
0.1176 × 100% = 11.76%
11.8%[ to the nearest tenth]
h(1)=0. h(n)=h(n-1)-9 explicit formula for h(n)=
Answer:
0-9(n-1)
Step-by-step explanation:
We can tell the first term of the sequence is 0 and the common difference is -9. So the explicit formula would be h(n)=0-9(n-1)
The explicit formula for h(n) comes to be h(n)=9-9n.
What is an arithmetic progression?An arithmetic progression is a list of numbers where the difference between consecutive terms is always constant.
[tex]h(n)=h(n-1)-9[/tex]
[tex]h(n)-h(n-1)=-9[/tex]
So the common difference of the arithmetic progression will be -9.
[tex]h(1)=0[/tex] means the first term of the AP is 0.
We know that [tex]n^{th}[/tex] term of an AP is given by:
[tex]h(n)=a+(n-1)d[/tex]
Where a is the first term and d is a common difference.
Fo the given series [tex]a=0[/tex] and [tex]d=-9[/tex]
So, [tex]h(n)=0+(n-1)(-9)[/tex]
[tex]h(n)=9-9n[/tex]
So, the explicit formula for h(n) is [tex]h(n)=9-9n[/tex]
Hence, the explicit formula for h(n) comes to be [tex]h(n)=9-9n[/tex].
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A recipe for 1 loaf of bread calls for 2 cups of flour, 12 tablespoons of water, and 1 teaspoon of salt. The recipe can be scaled up to make multiple loaves of bread. Complete the table that shows the quantities to use for multiple loaves of bread.
Answer:
For 2 loaves of bread, 4 cups of flour, 24 tablespoons of water, and 2 teaspoons of salt are required
For 4 loaves of bread, 8 cups of flour, 48 tablespoons of water, and 4 teaspoons of salt are required
Step-by-step explanation:
Given: A recipe for 1 loaf of bread calls for 2 cups of flour, 12 tablespoons of water, and 1 teaspoon of salt
To find: the missing terms in the box
Solution:
As for 1 loaf of bread, 2 cups of flour, 12 tablespoons of water, and 1 teaspoon of salt are required
So, for 2 loaves of bread, double the quantity.
So, for 2 loaves of bread, 2×2=4 cups of flour, 12×2=24 tablespoons of water, and 1×2=2 teaspoons of salt are required
For 4 loaves of bread, double the quantity used to make 2 loaves of bread.
For 4 loaves of bread, 4×2=8 cups of flour, 24×2=48 tablespoons of water, and 2×2=4 teaspoons of salt are required
See the image attached.
Find the product in simplest form. (3√2) ( √10) 3 √10 3 √20 6 √5 6 √10
Answer:
6 sqrt(5)
Step-by-step explanation:
(3√2) ( √10)
We know that sqrt(a) * sqrt(b) = sqrt(ab)
3 sqrt(2*10)
3 sqrt(20)
3 sqrt(4*5)
We know that sqrt(a) * sqrt(b) = sqrt(ab)
3 sqrt(4) sqrt(5)
3 * 2 * sqrt(5)
6 sqrt(5)
Determine the indicated probability for a binomial experiment with the given number of trials n and the given success probability p. n =8, p = 0.6, P(3 or fewer) Group of answer choices
Answer:
[tex]P(X \leq 3) = 0.1738[/tex]
Step-by-step explanation:
Binomial probability distribution
The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
In which [tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
And p is the probability of X happening.
In this question, we have that:
[tex]n = 8, p = 0.6[/tex]
P(3 or fewer)
[tex]P(X \leq 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)[/tex]
In which
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 0) = C_{8,0}.(0.6)^{0}.(0.4)^{8} = 0.0007[/tex]
[tex]P(X = 1) = C_{8,1}.(0.6)^{1}.(0.4)^{7} = 0.0079[/tex]
[tex]P(X = 2) = C_{8,2}.(0.6)^{2}.(0.4)^{6} = 0.0413[/tex]
[tex]P(X = 3) = C_{8,0}.(0.6)^{3}.(0.4)^{5} = 0.1239[/tex]
[tex]P(X \leq 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) = 0.0007 + 0.0079 + 0.0413 + 0.1239 = 0.1738[/tex]
Suppose that you are holding your toy submarine under the water. You release it and it begins to ascend. The graph models the depth of the submarine as a function of time. What is the domain and range of the function in the graph?
Answer:
Domain : 0 ≤ t ≤ 3
Range : -4 ≤ d ≤ 0
Step-by-step explanation:
The graph attached models the depth of submarine as a function of time.
Points on x-axis represent the time and points on y-axis represent increase in height of the submarine.
Domain of a function is represented by the points on x-axis.
Therefore, Domain : 0 ≤ t ≤ 3
Range of function is represented by te points on y-axis.
Therefore, Range : -4 ≤ d ≤ 0
Courtney sold 48 cantaloupes at the farmers' market and had 25 left. Which equation could be used to find x, the number of cantaloupes Courtney had originally?
Answer:
48+25=x
Step-by-step explanation:
Answer:
The answer is D :)
Thank me later.
How do i solve 3/4 x 4 7/12 =
Answer:
2.9375
Step-by-step explanation:
3/4×47/12
3×47/4×12
141/48
2.9375
Answer:
4x12=48
48+7=55
4 7/12 as an improper fraction is 55/12
multiply the numerator with numerator and denominator with denominator
3x55=165
4x12=48
165/48 can be divided by 3
165/3=55__48/3=16
55/16 is simplified
55/16=3 r 7 so the fraction is 3 7/16
Hope this helps
Step-by-step explanation:
what value of x in the solution set of -5×-15>10+20
Answer:
ⁱ ʰᵒᵖᵉ ᵗʰⁱˢ ʷᵒᵘˡᵈ ʰᵉˡᵖ ʸᵒᵘ ᵗᵒ ᵍᵉᵗ ʳᵉᵃˡ ᵃⁿˢʷᵉʳ...
Please answer this correctly without making mistakes
Answer:
10.99 inches
Step-by-step explanation:
7*2=14
diameter=14
circumference=diameter*pi
14*3.14=43.96
quarter circle
43.96/4=10.99
10.99 inches
Answer:
the quarter circle's perimeter = 24.99 inch
Step-by-step explanation:
1. Calculate circumference of a complete circle.
2. Divide by 4 and you have the quarter part. (This is the same as multiplying by 0.25).
3. Just like a pie chart, there are 2 sides from the center to the left and right side edge of the quarter circle. Both are equal to the radius r.
4. Add the numbers found in step 2 and step 3 and you will have found the quarter circle's perimeter.
Given: use 3.14 for pi and r = 7 inch
1. Circumference = 2* pi * r
2. 1/4 * ( 2 * pi * r )
1/4 * ( 2 * 3.14 * 7 ) = 10.99 inch
3 Since r = 7 and we have to sides.
So you just add 2 * 7 = 14 inch
4. Add 10.99 + 14 = 24.99 inch
Darren wanted to see how contagious yawning can be. To better understand this, he conducted a social experiment in which he yawned in front of a large, random crowd and observed how many people yawned as a result. The relationship between the elapsed time ttt, in minutes, since Darren yawned, and the number of people in the crowd, P(t)P(t)P, left parenthesis, t, right parenthesis, who yawned as a result is modeled by the following function: P(t)=5⋅4t10.5
Answer:
10.5 minutes
Step-by-step explanation:
Thinking about the problem
The modeling function is of the form P(t)=A⋅Bf(t), where B=4B=4B, equals, 4 and f(t)=\dfrac{t}{10.5}f(t)=
10.5
t
f, left parenthesis, t, right parenthesis, equals, start fraction, t, divided by, 10, point, 5, end fraction.
Note that each time f(t)f(t)f, left parenthesis, t, right parenthesis increases by 111, the quantity is multiplied by B=4B=4B, equals, 4.
Therefore, we need to find the ttt-interval over which f(t)f(t)f, left parenthesis, t, right parenthesis increases by 111.
Hint #22 / 3
Finding the appropriate unit interval
fff is a linear function whose slope is \dfrac{1}{10.5}
10.5
1
start fraction, 1, divided by, 10, point, 5, end fraction.
This means that whenever ttt increases by \Delta tΔtdelta, t, f(t)f(t)f, left parenthesis, t, right parenthesis increases by \dfrac{\Delta t}{10.5}
10.5
Δt
start fraction, delta, t, divided by, 10, point, 5, end fraction.
Therefore, for f(t)f(t)f, left parenthesis, t, right parenthesis to increase by 111, we need \Delta t=10.5Δt=10.5delta, t, equals, 10, point, 5. In other words, the ttt-interval we are looking for is 10.510.510, point, 5 minutes.
Hint #33 / 3
Summary
The number of people who yawned quadruples every 10.510.510, point, 5 minutes.
ASAP! GIVING BRAINLIEST! Please read the question THEN answer CORRECTLY! NO guessing. Show your work OR give an explaination.
Answer:
A: f(x) = (3/4 x)^2 - 1
Step-by-step explanation:
As you can see in the first image i attached, option A is the function that matches the one in your image. the "- 1" in the function is the y-intercept, and the 3/4 is what stretches your parabola which made it larger. You could also use process of elimination to get rid of option B and D since the y-intercepts do not match. Then you are left with A and C. the 4 in "(4x)^2" would make the parabola shrink, as you can see in the second image.
An excavation crew is digging a tunnel under a bay. The crew has dug 573 meters of the tunnel, which is 34 meters past the halfway point. What will be the length of the tunnel when the crew has finished digging?
Answer:
The total length of the tunnel when the crew has finished digging is 1078 meters
Step-by-step explanation:
total length of the tunnel dug by the crew = 573 meters
let the halfway point of the tunnel = h
if the crew digs 34 meters past the halfway point, then we will this equation below;
h + 34 meters = 573 meters
h = 573 - 34
h = 539 meters
halfway point of the tunnel is 539 meters
Then, the total length of the tunnel when the crew has finished digging = 2h
= 2 x 539 meters
= 1078 meters
Therefore, the total length of the tunnel when the crew has finished digging is 1078 meters
A cellular phone network uses towers to transit calls. Each tower transmits a circular area. on a grid of a city, the equations given represent the transmission boundaries of the towers. Tell which towers, is any, transmit to a phone located at M(3.5, 4.5).
Tower A: x^2 + y^2 = 9
Tower B: (x - 5)^2 + (y - 3)^2 = 6.25
Tower C: (x - 2)^2 + (y - 5)^2 = 4
Answer:
B
Step-by-step explanation:
Notice that
[tex](3.5-5)^2 + (4.5-3)^2 = 4.5[/tex]
Since 4.5 is less than 6.25, tower B transmits to that phone
Answer:
Towers B and C transmit to the phone.
Step-by-step explanation:
First, let see the location of the center for each tower:
Tower A
[tex]A (x,y) = (0,0)[/tex]
Tower B
[tex]B (x,y) = (5,3)[/tex]
Tower C
[tex]C(x,y) = (2,5)[/tex]
Now, the distance between the location of the phone and any of the towers by means of the Pythagorean equation. The phone is under the influence of a tower only if distance is less than transmission boundaries. Then:
Tower A
[tex]d_{A} = \sqrt{(3.5-0)^{2}+(4.5-0)^{2}}[/tex]
[tex]d_{A} \approx 5.701[/tex]
[tex]d_{A} > 3[/tex]
Tower B
[tex]d_{B} = \sqrt{(3.5-5)^{2}+(4.5-3)^{2}}[/tex]
[tex]d_{B} \approx 2.121[/tex]
[tex]d_{B} < 2.5[/tex]
Tower C
[tex]d_{C} = \sqrt{(3.5-2)^{2}+(4.5-5)^{2}}[/tex]
[tex]d_{C} \approx 1.581[/tex]
[tex]d_{C} < 2[/tex]
Towers B and C transmit to the phone.
Lauren drove the 75 miles to the coast at 75mph, but got stuck in traffic and averaged 25mph on the way back.
-How many total miles did Lauren drive in this round-trip?
-How many total hours did Lauren drive in this round-trip?
-What was the average speed Lauren was driving during this round-trip?
Answer:
Step-by-step explanation:
1) The distance that Lauren drove to the coast is the same as the distance he drove back from the coast. It means that the total miles that Lauren drove in this round-trip is
75 + 75 = 150 miles
2) time = distance/speed
Time spent in driving to the coast is
75/75 = 1 hour
His return speed is 25 mph
Time spent in driving back from the coast is
75/25 = 3 hours
3) Average speed = total distance/total time
Total distance = 75 + 75 = 150 miles
Total time = 1 + 3 = 4 hours
Average speed = 150/4 = 37.5 mph
You deposit $5000 each year into an account earning 4% interest compounded annually. How much will you have in the account in 35 years?
Answer:
$ 402,722.01
Step-by-step explanation:
(2/5 + 3/4) x [ 3 - (1/4 : 1/5)]
Answer & Step-by-step explanation:
(2/5 + 3/4) * [ 3 - (1/4 : 1/5)]
When you see this sign, : , it usually means to divide the numbers because they represent some form of a ratio. So, we will divide 1/4 by 1/5.
First, do all of the operations in the parentheses ().
(2/5 + 3/4)*[ 3 - (1/4 : 1/5)]
(23/20) * [ 3 - (5/4)]
(23/20) * (7/4)
Now, we multiply 23/20 by 7/4.
(23/20) * (7/4)
161/80
This number can not be simplified so we will keep it as it is.
So, the answer to this expression is 161/80
Find the largest prime factor of 309^2 - 147^2.
Answer: 2^4*3^5*19
Step-by-step explanation:
You also need to divided by 2 four times, then divided by 3 five times, then the final one is divided by 19.
The largest prime factor of 309² - 147² is, 19.
What is Greatest common factors?The highest number that divides exactly into two more numbers, is called Greatest common factors.
Given that;
The expression is,
⇒ 309² - 147²
Now, We can simplify as;
⇒ 309² - 147²
⇒ (309 - 147) (309 + 147)
⇒ 162 × 456
⇒ 2 × 3 × 3 × 3 × 3 × 2 × 2 × 2 × 3 × 19
Thus, The largest prime factor of 309² - 147² is,
⇒ 19
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When a foreign object lodged in the trachea (windpipe) forces a person to cough, the diaphragm thrusts upward causing an increase in pressure in the lungs. This is accompanied by a contraction of the trachea, making a narrower channel for the expelled air to flow through. For a given amount of air to escape in a fixed time, it must move faster through the narrower channel than the wider one. The greater the velocity of the airstream, the greater the force on the foreign object. X rays show that the radius of the circular tracheal tube contracts to about two-thirds of its normal radius during a cough. According to a mathematical model of coughing, the velocity v of the airstream is related to the radius r of the trachea by the equation v(r)= k (r 0- r) r ^ 2, 1/2 r 0 less than r less than r0where k is a constant and r0 is the normal radius of the trachea. The restriction on r is due to the fact that the tracheal wall stiffens under pressure and a contraction greater than 1/2r0 is prevented (otherwise the person would suffocate). Determine the value of r in the interval [1/2r0, r0] at which v has an absolute maximum. How does this compare with experimental evidence?
Answer:
Step-by-step explanation:
(a)
[tex]= kr^2(r_0 - r )[/tex]
[tex]= ( kr_0)r^2 - kr^3 \\\\=>v '(r) = ( 2kr_0 )r -3kr^2\\\\= ( -3k)r^2 + ( 2kr_0 )r[/tex]
v has an absolute maximum when v '(r) = 0
v '(r) = 0 =>
( -3k )r2 + ( 2kr0 )r = 0 =>
r = [ -( 2kr0 ) ± sqrt[ ( 2kr0)2 - ( 4 )( -3k )( 0 ) ] ] / [ ( 2 )( -3k ) ]
= [ -2kr0 ± 2kr0 ] / ( -6k)
= 0 or ( -4kr0 / -6k )
= 0 or (2/3)r0
since [tex]r > (1/2)r_0[/tex] in the given interval,[tex]r =(2/3)r_0[/tex], which matches its experimental value.