Answer:
30.9717685346
Step-by-step explanation:
[tex] \tiny\int_{e}^{{e}^{2}} \left( \sqrt{x + \frac{1}{2} \sqrt{x + \frac{1}{4} \sqrt{x + \frac{1}{8} \sqrt{x + \frac{1}{16} \dots} } } } \: + \sqrt{x - \frac{1}{2} \sqrt{x - \frac{1}{4} \sqrt{x - \frac{1}{8} \sqrt{x - \frac{1}{16} \dots} } } } \right) dx \\ [/tex]
We have the identity
[tex]\left(\sqrt x + \dfrac1{2^n}\right)^2 = x + \dfrac1{2^{n-1}} \sqrt x + \dfrac1{2^{2n}}[/tex]
Take the square root of both sides and rearrange terms on the right to get
[tex]\sqrt x + \dfrac1{2^n} = \sqrt{x + \dfrac1{2^{n-1}} \left(\sqrt{x} + \dfrac1{2^{n+1}}\right)}[/tex]
Decrementing n gives
[tex]\sqrt x + \dfrac1{2^{n-1}} = \sqrt{x + \dfrac1{2^{n-2}} \left(\sqrt{x} + \dfrac1{2^{n}}\right)}[/tex]
and substituting the previous expression into this, we have
[tex]\sqrt x + \dfrac1{2^{n-1}} = \sqrt{x + \dfrac1{2^{n-2}} \sqrt{x + \dfrac1{2^{n-1}} \left(\sqrt x + \dfrac1{2^{n+1}}\right) } }[/tex]
Continuing in this fashion, after k steps we would have
[tex]\sqrt x + \dfrac1{2^{n-k}} = \sqrt{x + \dfrac1{2^{n-(k+1)}} \sqrt{x + \dfrac1{2^{n-k}} \sqrt{x + \dfrac1{2^{n-(k-1)}} \sqrt{\cdots \dfrac1{2^{n-1}} \left(\sqrt x + \dfrac1{2^{n+1}}\right)}}}}[/tex]
After a total of n - 2 steps, we arrive at
[tex]\sqrt x + \dfrac14 = \sqrt{x + \dfrac12 \sqrt{x + \dfrac1{2^2} \sqrt{x + \dfrac1{2^3} \sqrt{\cdots \dfrac1{2^{n-1}} \left(\sqrt x + \dfrac1{2^{n+1}}\right)}}}}[/tex]
Then as n goes to infinity, the first nested radical converges to √x + 1/4. Similar reasoning can be used to show the other nested radical converges to √x - 1/4. Then the integral reduces to
[tex]\displaystyle \int_e^{e^2} \left(\sqrt x - \frac14\right) + \left(\sqrt x + \frac14\right) \, dx = 2 \int_e^{e^2} \sqrt x \, dx = \boxed{\frac43 \left(e^3 - e^{\frac32}\right)}[/tex]
If the expression 400(1.12)y gives the number of fish in a man-made lake y years after fish were placed in the lake, which statement is true?
The expression 400(1.12)^y is an exponential function
The true statement is that the growth rate of the fish in the lake is 12%
How to determine the true statement?The expression is given as:
400(1.12)^y
The above expression is an exponential growth function,
Where:
4000 represents the initial value1.12 represents the growth factorNext, we calculate the growth rate (r) as follows:
r = 1.12 - 1
r = 0,12
Express as percentage
r = 12%
Hence, the true statement is that the growth rate of the fish in the lake is 12%
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Karen runs each lap in 7 minutes. She will run more than 42 minutes today. What are the possible numbers of laps she will run today?
Use N for the number of laps she will run today.
Write your answer as an inequality solved for N.
Answer:
the answer for the question is N=6
Answer:
6 laps they will run
Step-by-step explanation:
I say 6 laps because if you do 42/7 then you will get 6 so 6 laps.
A muffin's label says the muffin has 400 calories per serving, with 2.5 servings per muffin. Kyle eats the entire muffin. What percent of Kyle's daily calories was the muffin, assuming a 2,000-calorie daily diet?
25%
40%
45%
50%
Answer:
50
Step-by-step explanation:
400x2.5
I’m having trouble understanding this can someone help please and thank you
Answer:
A soda costed $3.50.
Step-by-step explanation:
Step 1) We know that a slice of pizza and a soda costs $9 (from Kate's order). With a burger added (Billy's order) it would cost $13.50.
So a burger would cost $4.50:
[tex]13.50-9=4.50[/tex]
Step 2) We also know a burger and a soda costs $8. So we subtract the cost of the burger from Louis' order to find the cost of the soda:
[tex]8-4.50=3.50[/tex]
Cost of soda: [tex]\$3.50[/tex]
4. Jenny is a babysitter and she gets paid $35
for each toddler and $20 for each after school
kid. Write an algebraic expression that
describes her situation. Hint.... You may need
two variables because she is getting paid in two
different ways.
Answer:
35t + 20k
Step-by-step explanation:
Let t = number of toddlers.
Let k = number of kids.
Her pay is
35t + 20k
What numbers could replace each letter to make both of the number sentences true?
N x N= 64
N x P= 56
Caden exercises daily by walking on a treadmill. He sets the machine so that he will walk at a steady rate of 3.6 miles per hour.
a. If t represents time in hours and d represents distance in miles, write an equation that models the relationship between these variables.
b. Use your equation to calculate the distance Caden will walk in 3/4 of an hour.
c. Use your equation to calculate how long it will take for Caden to walk 4.32 miles.
Answer: For this question, the answers are as follows:
a) d = 3.6t
b) 10.8 miles
c) 1.2 hours
Step-by-step explanation:
Part A: The slope of a line is y=mx+b
equation for this part would be : d=mt
plug in the value: d=3.6t
Part B: The value of t is 3 hours
Plug everything in: 3.6 (3)= 10.8 miles
Part C: The Distance given is 4.32 miles.
4.32 miles = 3.6 hours
then divide both sides by 3.6
you get t=1.2 hours
Hope this helped and didn't confuse you!
The manager of a grocery store wants to determine what proportion of people who enter
his store are his regular customers. What size sample should he take so that at 95%
confidence the margin of error will not be more than 0.1?
answer:
74
Step-by-step explanation:
The sample size for the given confidence interval is 96.
What is a confidence interval?A range of estimates for an unknown parameter is called a confidence interval (CI). A confidence level calculates a confidence interval; The most common confidence level is 95 percent. The confidence level is the percentage of corresponding CIs that contains the actual value of the parameter over time.
Given confidence of margin is at 95%
E = 0.1 = margin of error
95% confidence interval of Z = 1.96
sample size = n, which is given by
n = (Z/E)²(p)(1 - p)
n = (1.96/0.1)²(0.5)(0.5)
n = 96.04 = 96 approx
Hence, the sample size is 96.
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10. Use the discriminant to find the values of b for which the equation 4x2 + bx + 1 = 0 has one real solution.
Answer:
b = 4, or -4
Step-by-step explanation:
b^2 - 4*4*1 = 0
b^2 = 16
b = 4, or -4
A t-shirt was on sale for 25% off. The original price was $25.00. How much did you pay for
the t-shirt after the discount was taken off?
Answer:
$18.75
Step-by-step explanation:
25/4 = 6.25
25-6.25=18.75
$18.75
Math: what kind of triangle do you have if it has an obtuse angle which measures more than 90, but less than 80? help please
Answer:
The triangle is obtuse if one angle is between 90º and 180º.
More : Acute < 90º, right = 90º, reflex > 180º (straight 180º, full rotation 360º)
I need help with this one
Step-by-step explanation:
first lets do the bottom rectangular prism
v = l * w * h
l = 9
w = 3
h = 3
can you work out the volume and answer in the comments
Select the correct answer.
Which exponential equation is equivalent to this logarithmic equation?
log_2 x = 24
A. x^24 = 2
B. x^2 = 24
C. 2^24 = x
D. 2^x= 24
Answer:
I think its D
Step-by-step explanation:
8) What is the slope of the line that passes through the points (5.-11) and (-9, 17)?
A)-2
B)
B) - 1
07
D) 2
[tex] \Large{\boxed{\sf Option \ A) \ Slope = -2 }} [/tex]
[tex] \\ [/tex]
Explanation:To find the value of the slope of the line that passes through the points (5 , -11) and (-9 , 17), we will have to use the slope formula.
[tex] \\ [/tex]
[tex] \Large{\left[ \begin{array}{c c c} \underline{\tt Slope \ formula \text{:}} \\~ \\ \tt m = \dfrac{rise}{run} = \dfrac{\Delta y}{\Delta x} = \dfrac{y_2 - y_1}{x_2 - x_1}\end{array} \right] } [/tex]
Where m is the slope of the line.
[tex] \\ [/tex]
To apply the above formula, we first need to identify our values.
[tex] \sf (\underbrace{\sf 5}_{x_1} \ , \ \overbrace{\sf -11}^{y_1} ) \ \ and \ \ (\underbrace{\sf -9}_{x_2} \ , \ \overbrace{\sf 17}^{y_2} ) [/tex]
[tex] \\ [/tex]
Let's substitute these values into the formula:
[tex] \longrightarrow \sf m = \dfrac{17 - (-11)}{-9 - 5} \\ \\ \\ \longrightarrow \sf m = \dfrac{17 + 11}{-14} \\ \\ \\ \longrightarrow \sf m = \dfrac{28}{-14} \\ \\ \\ \longrightarrow \sf m = \dfrac{7 \cdot 4}{7 \cdot (-2)} \\ \\ \\ \longrightarrow \sf m = -\dfrac{4}{2} \\ \\ \\ \longrightarrow \boxed{\boxed{\sf m = - 2}} [/tex]
[tex] \\ [/tex]
[tex] \hrulefill [/tex]
[tex] \\ [/tex]
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Gradient (m) of a Line :
[tex]\underbrace{\rm{ m = \dfrac{y_2 - y_1}{x_2-x_1}}}_{m}[/tex]
Substituting the known values ,
➱ [tex]\rm{m=\dfrac{17-(-11)}{-9-5}}[/tex]
➱ [tex]\rm{m=\dfrac{17+11}{-14}}[/tex]
➱ [tex]\rm{m=\dfrac{28}{-14}}[/tex]
➱ [tex]\rm{m=-2}[/tex]
Henceforth, Gradient = -2
Help me plsssssss Brainless
A school collected 1,008 cans of food The cans were placed in 9 boxes of each box had the same number of cans how many cans were in each box
Answer:
There are 112 cans in each box.
Step-by-step explanation:
You divide 1008 by 9, so that gives you 112.
Hope it helps!
Mangos cost $2.60 for 2 pounds. Pears cost $3.50 for 2.5 pounds. How much would 11 pounds of pears cost?
Answer:
Step-by-step explanation:
Write the ratio 27 feet to 24 yards as a fraction in simplest form. (Hint: 3 feet = 1 yard.)
Answer:
9/8
27 divided by 3 = 9
24 divided by 3 = 8
I hope this helps!
The ratio in the simplest form of 27 to 24 is equal to 9:8 or 9/8.
What is ratio?When comparing two or more numbers, the phrase ratio is used in mathematics. When one quantity is related to another, it can be used to show how big or tiny it is. Division is used to compare two quantities in a ratio. The term "antecedent" in this context refers to the dividend, and "resultant" to the divisor.
From the given information in the question,
To find the simplest form, divide numerator and denominator with the same number.
Numerator = 27 feet
In simple form = 27/3 = 9
Denominator = 24 yards = 72 feet
In simple form = 72/3 = 24
= 9/24
= 3/8 is the simplest form.
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Please click on the picture and help me! I NEED THIS ASAP (BTW THIS IS FROM THE GIVER MOVIE AND BOOK) (20 points if you help me please someone help me!
How many real third roots does -343 have? Explain your reasoning.
Answer: One root
Explanation:
x^3 = -343
x^3 + 343 = 0
x^3 + 7^3 = 0
(x+7)(x^2 - 7x + 7^2) = 0 .... sum of cubes factoring rule
(x+7)(x^2 - 7x + 49) = 0
Set each piece equal to zero and solve for x.
You should find that the first piece x+7 = 0 leads to x = -7 as the only real root.
The other piece x^2-7x+49 = 0 leads to two non-real roots, ie complex roots.
Martin was born in 1996. His grandmother was born fifty-two years earlier. What year was Martin's grandmother born?
A. 2048
B. 2046
C. 1944
D. 1943
Answer:
option C is correct answer.
hope this helps you !
What is renters insurance?
helllllllllllllllllllllp meeeeeeeeee
Answer:
protecting your personal belongings as a renter.
Step-by-step explanation:
your TVs, laptops, furniture, jewelry, etc.
3. Find the multiplicative inverse of -100
Hello.
Let's find the multiplicative inverse of -100.
In order to do that, we need to write -100 as a fraction:
[tex]\bf{\displaystyle\frac{-100}{1}[/tex]
Now, in order to find the multiplicative inverse of -100, we should flop the fraction over:
[tex]\bf{\displaystyle\frac{1}{-100}[/tex]
I hope it helps.
Have an outstanding day. :)
[tex]\boxed{imperturbability}[/tex]
Solve the system using substitution!
4x+4y=-2
y=-x
Worth (50 Points) If answered correctly
Solution:
Given that:
y = -x4x + 4y = -2Substitute the value of y into equation #2.
4x + 4(-x) = -24x - 4x = -20 ≠ -2Since the equation has no variable and is false, this system has no solution.
the Cinde of the radius (2) Evaluate x2 + y2 + 8x+6y=0 (a) (6 (5) 25 ( 4 (d) 5
x2+y2+8x+6y=0
x2+8x+6y+y=0
9x2+7y2
15x2y2
here a subtraction problem that has been solved correctly for 68-59=9 but the choices is 9+68=77 68+9=77 9+59=68 which one is the addition sentence that checks the subtraction problem
Answer:
It's probably 9+59=68
Step-by-step explanation:
=> 9+59=68
68 - 59 gives you 9
So in order to check if the problem was solved correctly you will add the answer plus the subtraction of one number in the two numbers of subtraction
which means between the options 9 + 59 equals to 68 is absolutely the correct answer
.
can u answer plss
and gove explanation plss
Answer:
x=[tex]\sqrt{\frac{7y+8}{5} }[/tex]
Step-by-step explanation:
ABE CDE. Find the length of altitude NE.
Answer:
a four is correct answer
Answer:
A. 4
Step-by-step explanation:
Side AE corresponds to EC, so the scale factor is 3.
12/3=4
Hope this helps :)
Find the circumference of the circle.
The circumference is _____
mi.
Answer:
The answer is 8 pi
Step-by-step explanation:
The formula is radius squared times pi!
I hope this helped!!