The series converges, since |8/9| = 8/9 < 1.
Let [tex]S_n[/tex] denote the n-th partial sum of the series,
[tex]S_n=\displaystyle\sum_{k=0}^n\left(\frac89\right)^k[/tex]
[tex]S_n=1+\dfrac89+\left(\dfrac89\right)^2+\cdots+\left(\dfrac89\right)^n[/tex]
Multiply both sides by 8/9:
[tex]\dfrac89S_n=\dfrac89+\left(\dfrac89\right)^2+\left(\dfrac89\right)^3+\cdots+\left(\dfrac89\right)^{n+1}[/tex]
Subtract this from [tex]S_n[/tex] to eliminate all but the first term in
[tex]S_n-\dfrac89S_n=1-\left(\dfrac89\right)^{n+1}[/tex]
Solve for [tex]S_n[/tex]:
[tex]\dfrac19S_n=1-\left(\dfrac89\right)^{n+1}[/tex]
[tex]S_n=9-9\left(\dfrac89\right)^{n+1}[/tex]
As [tex]n\to\infty[/tex], the exponential term vanishes, so that
[tex]\displaystyle\lim_{n\to\infty}S_n=\sum_{k=0}^\infty\left(\frac89\right)^k=\boxed{9}[/tex]
Sue weighed 82 kg at the start of the year, but she has lost weight at a consistent rate because she has been diligent with her exercise and diet plan. Sue weighed herself at the end of each month and tracked her progress for the full year using the graph above. How much weight did Sue lose each month? Question 5 options: A) 4 kg B) 2 kg C) 3 kg D) 1 kg
Answer:
Option (B)
Step-by-step explanation:
Sue's weight at the start of the year = 82 kg
Sue weighed herself at the end of every year, so that the straight line on the graph shows the gradual reduction in the weight every month.
Slope of the line will describe the reduction in the weight per month.
Let the two points on the given line are (1, 80) and (11, 60).
Slope of the line will be,
m = [tex]\frac{y_2-y_1}{x_2-x_1}[/tex]
= [tex]\frac{60-80}{11-1}[/tex]
= -2 kg per month
(Here minus sign denotes the reduction of weight per month)
Option (B) will be the answer.
Answer:
2 kg
Step-by-step explanation:
A balloon is flcating 10 feet above the ground. How
far does the balloon need to drop to reach the
ground?
Answer:
10 feet
Step-by-step explanation:
if it is 10 feet above and needs to touch the ground it needs to drop that same amount
Answer:
10ft.
Step-by-step explanation:
if the balloon is floating 10ft from the ground the it can only drop 10ft to reach the ground.
*hope this helps. please give me brainliest, i only need four more *
Estimate 8.25% sales tax on a $39.89 purchase
Turn the percentage into a decimal.
8.25% / 100 = 0.0825
Multiply.
39.89 * 0.0825 ≈ $3.29
Subtract.
39.89 + 3.29 = $43.18
Best of Luck!
Use the equation solver to find the value of x that makes the mathematical statement true. -0.5 0.5 0.87 1.5
Answer:-0.5
Step-by-step explanation:
p: student achieves 90 percent on the geometry final. q: student will receive a passing grade in geometry class. Which statement is logically equivalent to q → p?
Answer:
if a student did not achieve 90 percent on the geometry final, then the student did not pass geometry class
Step-by-step explanation:
The area of a rectangle is at least 10 more than 3 times the width of the rectangle. If the area of the rectangle is at least 250 square units, what are the possible values for the width (w) of the rectangle?
Is 815 divisible by 10?
No.
Here is why:
If you mean equally divided then no because of the little 5 in the ones's place. If that 5 was a 0, it would definitely be divided equally.
Hope this makes sense!
Answer:
no
Step-by-step explanation:
Find a formula for the inverse of the function. f(x) = e^6x − 9
Answer:
[tex]f^{-1}(x)=\frac{1}{6}\ln(x+9)[/tex]
Step-by-step explanation:
So we have the function:
[tex]f(x)=e^{6x}-9[/tex]
To solve for the inverse of a function, change f(x) and x, change the f(x) to f⁻¹(x), and solve for it. Therefore:
[tex]x=e^{6f^{-1}(x)}-9[/tex]
Add 9 to both sides:
[tex]x+9=e^{6f^{-1}(x)}[/tex]
Take the natural log of both sides:
[tex]\ln(x+9)=\ln(e^{6f^{-1}(x)})[/tex]
The right side cancels:
[tex]\ln(x+9)=6f^{-1}(x)[/tex]
Divide both sides by 6:
[tex]f^{-1}(x)=\frac{1}{6}\ln(x+9)[/tex]
And we're done!
Use the general slicing method to find the volume of the following solid. The solid whose base is the region bounded by the curve y=38cosx and the x-axis on − π 2, π 2, and whose cross sections through the solid perpendicular to the x-axis are isosceles right triangles with a horizontal leg in the xy-plane and a vertical leg above the x-axis. A coordinate system has an unlabeled x-axis and an unlabeled y-axis. A curve on the x y-plane labeled y equals 38 StartRoot cosine x EndRoot starts on the negative x-axis, rises at a decreasing rate to the positive y-axis, and falls at an increasing rate to the positive x-axis. The region below the curve and above the x-axis is shaded. A right triangle extends from the x y-plane, where one leg is on the x y-plane from the x-axis to the curve and is perpendicular to the x-axis, and the second leg is above the x-axis and is perpendicular to the x y-plane. y=38cosx
Answer:
The volume of the solid = 1444
Step-by-step explanation:
Given that:
The region of the solid is bounded by the curves [tex]y = 38 \sqrt{cos \ x}[/tex] and the axis on [tex][-\dfrac{\pi}{2}, \dfrac{\pi}{2}][/tex]
using the slicing method
Let say the solid object extends from a to b and the cross-section of the solid perpendicular to the x-axis has an area expressed by function A.
Then, the volume of the solid is ;
[tex]V = \int ^b_a \ A(x) \ dx[/tex]
However, each perpendicular slice is an isosceles leg on the xy-plane and vertical leg above the x-axis
Then, the area of the perpendicular slice at a point [tex]x \ \epsilon \ [-\dfrac{\pi}{2},\dfrac{\pi}{2}][/tex] is:
[tex]A(x) =\dfrac{1}{2} \times b \times h[/tex]
[tex]A(x) =\dfrac{1}{2} \times(38 \sqrt{cos \ x})^2[/tex]
[tex]A(x) =\dfrac{1444}{2} \ cos \ x[/tex]
[tex]A(x) =722 \ cos \ x[/tex]
Applying the general slicing method ;
[tex]V = \int ^b_a \ A(x) \ dx \\ \\ V = \int ^{\dfrac{\pi}{2} }_{-\dfrac{\pi}{2}} (722 \ cos x) \ dx \\ \\ V = 722 \int ^{\dfrac{\pi}{2}}_{-\dfrac{\pi}{2}} cosx \dx[/tex]
[tex]V = 722 [ sin \ x ] ^{\dfrac{\pi}{2}}_{-\dfrac{\pi}{2}}[/tex]
[tex]V = 722 [sin \dfrac{\pi}{2} - sin (-\dfrac{\pi}{2})][/tex]
[tex]V = 722 [sin \dfrac{\pi}{2} + sin \dfrac{\pi}{2})][/tex]
[tex]V = 722 [1+1][/tex]
[tex]V = 722 [2][/tex]
V = 1444
∴ The volume of the solid = 1444
The College of Business at Tech is planning to begin an online MBA program. The initial start-up cost for computing equipment, facilities, course development, and staff recruitment and development is $350,000.The college plans to charge tuition of $18,000 per student per year. However, the university administration will charge the college $12,000 per student for the first 100 students enrolled each year for administrative costs and its share of the tuition payments.
a. How many students does the college need to enroll in the first year to break even?
b. If the college can enroll 75 students the first year, how much profit will it make?
c. The college believes it can increase tuition to $24,000, but doing so would reduce enrollment to 35. Should the college consider doing this?
Answer:
a).To break even
They need to enroll 87 student
B) Profit=$100000
C) yeah they can think of that.
Profit= $70000
Step-by-step explanation:
Initial start up cost = $350000
Cost from adminstrative bodies
= 12000 for first 100 student
= $1200000
Total cost =$1200000+$350
Totral cost =$ 1550000
Tuition cost
= $18000 per student
To break even
The college needs to enroll
1550000/18000= 86.1
They need to enroll 87 student
B) if they enroll 75 students
Adminstrative cost = 12000*75
Adminstrative cost= 900000
Total cost= 350000+900000
Total cost =$ 1250000
Tuition fee= $18000*75
Tuition fee=$ 1350000
Their profit= 1350000-1250000
Profit=$100000
C).if tuition is $24000 and enrollment number= 35
Adminstrative cost = 35*12000
Adminstrative cost = $420000
Total cost= $770000
Tuition fee= $24000*35
Tuition fee= $840000
Profit= $840000-$770000
Profit= $70000
Joe will rent a car for the weekend. He can choose one of two plans. The first plan has no initial fee but costs $0.80 per mile driven. The second plan has an initial fee of $75 and costs an additional $0.60 per mile driven. How many miles would Joe need to drive for the two plans to cost the same?
Answer:
375 miles
Step-by-step explanation:
m=miles
I set it up as an equation. $75+.60m=.80m
And solved from there
Answer:
375 miles :D
Step-by-step explanation:
11. The shadow of a 10 m high pole is 6 metres. Find
the shadow of a 15 m high pole at the same time
of the day.
Answer:
25m
Step-by-step explanation:
10/6 = length of shadow per metre of pole
so 15 x 10/6 = shadow of 15m high pole
150/6 = 25m
Solve the equation 12x+6y= 24 for x.
Answer: x = -1/2y + 2
Step-by-step explanation:
12x + 6y = 24 To solve for x first start by subtracting 6y from both sides
- 6y -6y
12x = 24 - 6y Now divide each side by 12
x= 2 - 1/2y
Solve for u.
9u+16=7(u+2)
Simplify your answer as much as possible.
U=
Answer:
u=-1
Step-by-step explanation:
Answer:
Isolate the variable by dividing each side by factors that don't contain the variable.
u= -1
A recipe says to use 1/5 cup of flour to make 7/10 serving of waffles. How many cups of flour are in one serving of waffles?
Don’t know the answer
Simplify. 4+5(8−10)2 Enter your answer in the box.
Answer:
-16
Step-by-step explanation:
4 + 5(8-10)2
4 + 5(-2)2
4 + 10(-2)
4 - 20
-16
The answer is: 24
I assume this is a question from k12
4 + 5(8 - 10)^2
4 + 5(-2)^2
4 + 5(4)
4 + 20
= 24
Hope this helps ^^
WILL GIVE BRAINLY FOR ANSWER!! Please help with this question!!! Given the piecewise function: f(x) = 1/2x + 5, x > 2 6, x = 2 x + 4, x < 2 a. Write f' (f prime) as a piecewise function b. Determine if f is differentiable at x = 2. Give a reason for your answer. Photo is attatched.
Answer:
A)
[tex]f'(x) = \left\{ \begin{array}{lIl} \frac{1}{2} & \quad x >2 \\ 0& \quad x =2\\1&\quad x<2 \end{array} \right.[/tex]
B) Continuous but not differentiable.
Step-by-step explanation:
So we have the piecewise function:
[tex]f(x) = \left\{ \begin{array}{lIl} \frac{1}{2}x+5 & \quad x >2 \\ 6& \quad x =2\\x +4&\quad x<2 \end{array} \right.[/tex]
A)
To write the differentiated piecewise function, let's differentiate each equation separately. Thus:
1)
[tex]\frac{d}{dx}[\frac{1}{2}x+5}][/tex]
Expand:
[tex]\frac{d}{dx}[\frac{1}{2}x]+\frac{d}{dx}[5][/tex]
The derivative of a linear equation is just the slope. The derivative of a constant is 0. Thus:
[tex]\frac{d}{dx}[\frac{1}{2}x+5}]=\frac{1}{2}[/tex]
2)
[tex]\frac{d}{dx}[6][/tex]
Again, the derivative of a constant is 0. Thus:
[tex]\frac{d}{dx}[6]=0[/tex]
3)
We have:
[tex]\frac{d}{dx}[x+4][/tex]
Expand:
[tex]\frac{d}{dx}[x]+\frac{d}{dx}[4][/tex]
Simplify:
[tex]=1[/tex]
Now, let's substitute our original equations for the differentiated equations. The inequalities will stay the same. Therefore:
[tex]f'(x) = \left\{ \begin{array}{lIl} \frac{1}{2} & \quad x >2 \\ 0& \quad x =2\\1&\quad x<2 \end{array} \right.[/tex]
B)
For a function to be differentiable at a point, the function must be a) continuous at that point, and b) the left and right hand derivatives must be equivalent.
Let's first determine if the function is continuous at the point. Remember that a function is continuous at a point if and only if:
[tex]\lim_{x \to n^-} f(n)= \lim_{x \to n^+}f(n)=f(n)[/tex]
Let's find the left hand limit of f(x) at it approaches 2.
[tex]\lim_{x \to 2^-}f(x)[/tex]
Since it's coming from the left, let's use the third equation:
[tex]\lim_{x \to 2^-}f(x)\\=\lim_{x \to 2^-}(x+4)[/tex]
Direct substitution:
[tex]=(2+4)=6[/tex]
So:
[tex]\lim_{x \to 2^-}f(x)=6[/tex]
Now, let's find the right-hand limit:
[tex]\lim_{x \to 2^+}f(x)[/tex]
Since we're coming from the right, let's use the first equation:
[tex]\lim_{x \to 2^+}(\frac{1}{2}x+5)[/tex]
Direct substitution:
[tex](\frac{1}{2}(2)+5)[/tex]
Multiply and add:
[tex]=6[/tex]
So, both the left and right hand limits are equivalent. Now, find the limit at x=2.
From the piecewise function, we can see that the value of f(2) is 6.
Therefore, the function is continuous at x=2.
Now, let's determine differentiability at x=2.
For a function to be differentiable at a point, both the right hand and left hand derivatives must be equivalent.
So, let's find the derivative of the function as x approahces 2 from the left and from the right.
From the differentiated piecewise function, we can see that as x approaches 2 from the left, the derivative is 1.
As x approaches 2 from the right, the derivative is 1/2.
Therefore, the right and left hand derivatives are not the same.
Thus, the function is continuous but not differentiable.
ben has bundled cable television and internet service. he pays $89.92 each month for the service. how much will he pay per year for cable and internet
Answer:
$1,079.04
Step-by-step explanation:
multiply: 89.92x12=1078.04
Solve each calculation. Be sure to report your answer with the correct number of decimal places. 7.365 ms 0.250 ms + 10.3 ms ms 125.010 cL 4.1023 cL – 0.30 cL cL
Answer:
1.17.9
2. 120.61
Step-by-step explanation:
Answer:
17.9
120.61
Step-by-step explanation:
21.406 in standard form
21.406 in standard form
Answer:
21406 X 10^3
What is the molar mass of UF6??
Answer:
I think it's 352. 02 g/mol
Step-by-step explanation:
googled it :)
Answer:
352.02 g/mol
In chemistry, the molar mass of a chemical compound is defined as the mass of a sample of that compound divided by the amount of substance in that sample, measured in moles.
Is the expression (5 • 2) x equal to the expression 10 x
Answer:
Yes
Step-by-step explanation:
(5*2)x = 10x
5*2 = 10 so
10x = 10x
They are equal.
Answer:
Yes
Step-by-step explanation:
5 x 2 is 10, and so 5(2)x is 10x
What is the answer to this?
Answer:
−6y+24
Step-by-step explanation:
Answer:
-4
Step-by-step explanation:
parenthesis come first (PEMDAS)
2×8=16
2×3y=6y
9+16-6y-1
Add like terms
9+16=25
25-6y-1
Subtract like terms
25-1=24
-6y=24
Multiple each sides by using fractions
1/(-)6×24/1=4
Consider the Line L(t) = <2+t,5-4t>. Then L intersects:
The x- axis at the point (3.25,0) when t = 5/4
The y- axis at the point (0, 13) when t = -2
The parabola y = x^2 at the points ___ and ___ when t = ____ and ____
Answer:
The parabola [tex]y=x^{2}[/tex] at the points [tex](-2+\sqrt{17},21-4\sqrt{17})[/tex] and [tex](-2-\sqrt{17},21+4\sqrt{17})[/tex] when [tex]t=t_{1}=-4+\sqrt{17}[/tex] and [tex]t=t_{2}=-4-\sqrt{17}[/tex]
Step-by-step explanation:
We have the following line written in parametric form :
[tex]L(t)=(2+t,5-4t)[/tex] with [tex]t[/tex] ∈ IR.
In order to find the intersection between [tex]L(t)[/tex] and the parabola [tex]y=x^{2}[/tex] we know that ''[tex]2+t[/tex]'' is the x-coordinate of the line and ''[tex]5-4t[/tex]'' is the y-coordinate of the line. Now, to solve this problem we need to find the values of ''[tex]t[/tex]'' in which the intersection occurs. We can do this by replacing the components ''[tex]x[/tex]'' and ''[tex]y[/tex]'' of [tex]L(t)[/tex] in the equation of the parabola ⇒
[tex]L(t)=(2+t,5-4t)[/tex] = ( x component , y component ) = ( x , y ) ⇒
In the parabola : [tex]y=x^{2}[/tex] ⇒ [tex]5-4t=(2+t)^{2}[/tex]
Solving the equation we find that :
[tex]t^{2}+8t-1=0[/tex]
Using the quadratic formula with
[tex]a=1[/tex] , [tex]b=8[/tex] and [tex]c=-1[/tex]
We find that the two possible values for t :
[tex]t_{1}=\frac{-b+\sqrt{b^{2}-4ac}}{2a}[/tex] and [tex]t_{2}=\frac{-b-\sqrt{b^{2}-4ac}}{2a}[/tex]
are [tex]t_{1}=-4+\sqrt{17}[/tex] and [tex]t_{2}=-4-\sqrt{17}[/tex]
This values [tex]t_{1}[/tex] and [tex]t_{2}[/tex] are the values of the parameter t where the line intersects the parabola so we can find the points by replacing the values of the parameter in the equation [tex]L(t)[/tex] :
[tex]L(t_{1})=(-2+\sqrt{17},21-4\sqrt{17})[/tex] and
[tex]L(t_{2})=(-2-\sqrt{17},21+4\sqrt{17})[/tex]
The final answer is
The parabola [tex]y=x^{2}[/tex] at the points [tex](-2+\sqrt{17},21-4\sqrt{17})[/tex] and [tex](-2-\sqrt{17},21+4\sqrt{17})[/tex] when [tex]t=t_{1}=-4+\sqrt{17}[/tex] and [tex]t=t_{2}=-4-\sqrt{17}[/tex]
Given the formula below, solve for x. y - y1 = m(x-x1)
Answer:
y-3 = 2/9 (x-8)
Step-by-step explanation:
3n-5=-8(6+5n) in distributive property
Answer:
n=-1
Step-by-step explanation:
3n-5=-8(6+5n)
3n-5=-48-40n
add 40n to both sides
43n-5=-48
n=-1
Step-by-step explanation:
Solve the linear equation. 3x = 24 x = ?
Answer:
3x =24
x = 24 -3
x = 21 (ans)
Answer:
X=8
Step-by-step explanation:
3x=24
divide each side by 3 to isolate x
x=8
If the probability that your DVD player breaks down before the extended warranty expires is 0.034, what is the probability that the player will not break down before the warranty expires
Answer:
0.966
Step-by-step explanation:
Given that:
Probability of DVD player breaking down before the warranty expires = 0.034
To find:
The probability that the player will not break down before the warranty expires = ?
Solution:
Here, The two events are:
1. The DVD player breaks down before the warranty gets expired.
2. The DVD player breaks down after the warranty gets expired
In other words, the 2nd event can be stated as:
The DVD does not break down before the warranty gets expired.
The two events here, have nothing in common i.e. they are mutually exclusive events.
So, Sum of their probabilities will be equal to 1.
[tex]\bold{P(E_1)+P(E_2)=1}\\\Rightarrow 0.034+P(E_2)=1\\\Rightarrow P(E_2)=1-0.034\\\Rightarrow P(E_2)=\bold{0.966}[/tex]
What is wrong with this problem. 2(n +3) = 2n +3
Steps to solve:
2(n + 3) = 2n + 3
~Distribute left side
2n + 6 = 2n + 3
~Subtract 6 to both sides
2n = 2n - 3
~Subtract 2n to both sides
n = -3
Best of Luck!
What is the quotient of 29,596÷28?
Answer:
√ 96
log
( 7 )
6 ! /3 !
Step-by-step explanation: