[tex]\sqrt[5]{3125e^{\frac{11\pi}{2}i}}\\=\\\sqrt[5]{3125}\sqrt[5]{e^{\frac{11\pi}{2}}i}\\\\=5e^{\frac{11\pi}{10}i}\\\\\boxed{5\left(\cos \left(\frac{11\pi}{10} \right)+i \sin \left(\frac{11\pi}{10} \right) \right)}[/tex]
expressions is equivalent
to a(b-c)-b(a+c)-c(a-b)?
After solving the algebraic expression a(b-c)-b(a+c)-c(a-b) we get the equivalent answer as ₋2(ab ₊ ac).
What is equivalent of algebraic expression?
Expressions that are equivalent do the same thing even when they have distinct appearances. When we enter the same value(s) for the variable, two algebraic expressions that are equivalent have the same value (s).
Given the expression is a(b₋c)₋b(a₊c)₋c(a₋b)
The first step is to multiply the terms.
⇒(a×b ₋ a×c) ₋ (b×a ₊ b×c) ₋ (c×a - c×b)
= (ab ₋ ac) ₋ (ba ₊ bc) ₋ (ca ₋ cb)
Now open the brackets.
= ab ₋ ac ₋ ba ₋ bc ₋ ca ₊ cb
Now comes main step of combining the like terms.
Put similar terms together on both sides of the equation.The two phrases are equal if each of the terms in both of them is the equivalent.
= ab ₋ ba ₋ ac ₋ ca ₋ bc ₊bc
Since, bc is a like term with opposite sign, therefore bc gets cancelled.
= ab ₋ ba ₋ ac ₋ ca
= ₋2ab ₋ 2ac
Take 2 as a common term from both the terms.
= ₋2(ab ₊ ac)
Hence we get the required equivalent of a(b-c)-b(a+c)-c(a-b) as
₋2(ab ₊ ac)
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Solve the following equation:
30= |- 4y + 2|
Suppose [tex]-4y+2\ge0[/tex]. Then by definition of absolute value,
[tex]|-4y+2| = -4y+2 \implies 30 = -4y+2 \implies -4y = 28 \implies \boxed{y=-7}[/tex]
On the other hand, suppose [tex]-4y+2<0[/tex]. Then
[tex]|-4y+2| = -(-4y+2) \implies 30 = 4y - 2 \implies 4y = 32 \implies \boxed{y=8}[/tex]
Recall the definition,
[tex]|x| = \begin{cases} x & \text{if } x \ge 0 \\ -x & \text{if } x < 0 \end{cases}[/tex]
The doctor tells Evelyn that she needs to exercise enough to burn at least 700 calories each day. She prefers practicing yoga or jogging. While practicing yoga she burns 5 calories per minute, and while jogging she burns 8 calories per minute. Let x be the number of minutes she spend practicing yoga and let y be the number of minutes she spends jogging. Write an inequality that models this situation.
The inequality that can be used to model Evelyn situation is
x + y = 60
x + y = 605x + 8y ≥ 700
Inequalityx = number of minutes she spend practicing yogay = number of minutes she spends joggingCalories burnt practicing yoga = 5Calories burnt jogging = 8The inequality
x + y = 60
5x + 8y ≥ 700
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According to the food and nutrition service of the u.s. department of agriculture, in one school year, the national school lunch program (NLP) served 31.2 million school lunches. of these, 16.1 million were free lunches, 3.2 million were reduced-price lunches, and 11.9 million were full-price lunches. the federal government reimburses school districts $2.68 for each free lunch, $2.28 for each reduced-price lunch, and $0.25 for each paid lunch. in addition to cash reimbursements, schools are entitled to receive USDA foods called "entitlement" foods at a value of 19.50 cents for each lunch served. you are the administrator in charge of the school lunch program for your school district. last month, the schools in your district served 27,000 free lunches, 18,000 "reduced-price" lunches, and 54,000 regular-priced lunches. (a) calculate the amount of reimbursement (in $) you expect to receive from the nslp for last month. $____. (b) in addition to the lunch reimbursement, the nslp program pays your district $.035 per one-half pint of milk served with each meal. if each student averaged 1 one-half pint of milk per meal, calculate the total amount of milk reimbursement (in $) you expect for last month. $_____ . (c) the bottom line—what is the total amount of reimbursement (in $) your district will receive for last month? $_______. (d) red tape—the government paperwork you must submit requires that you report the average reimbursement (in $) per meal for both lunch and milk combined last month. calculate this amount. (round your answer to the nearest cent.) $________.
a) The amount of meal reimbursement (in $) you expect to receive from the NSLP for last month is $146,205.
b) The total amount of milk reimbursement (in $) you expect for last month is $103,950.
c) The total amount of reimbursement (in $) your district will receive for last month is $250,155 ($146,205 + $103,950).
d) The average reimbursement (in $) per meal for both lunch and milk combined last month is $2.53 ($250,155/99,000).
What is reimbursement?Reimbursement is a process by which an entity is compensated for the expenses it incurs under a stated program.
Examples of expenses under the reimbursement policy include business expenses, insurance costs, and overpaid taxes.
Data and Calculations:Annual NSLP lunches = 31.2 million
Lunches Free Reduced Price Full-Price Total
National Quantity 16.1 million 3.2 million 11.9 million 31.2 million
District Lunches 27,000 18,000 54,000 99,000
Reimbursement rate $2.68 $2.28 $0.25 $0.195
Reimbursement $72,360 $41,040 $13,500 $19,305
Total reimbursement = $146,205 ($72,360 + $41,040 + $13,500 + $19,305)
Amount for milk reimbursement = $103,950 ($0.35 x 1.5 x 2 x 99,000)
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AB = 8x + 3, BC =2x + 7 and AC = 80 Find value of AB and BC
Answers:
1) The value of AB is: " 59 units."
2) The value of BC is: "21 units."
_______
3) The value of AB + BC ;
= 59 units + 21 units ;
= [the value of AC ] ;
= {" 80 units ".}.
_______
Step-by-step explanation:
Given the following—and based on the assumption that we are dealing with: "geometric lines"—and/or: "geometric line segments"—we are asked to:
"Find the value of: " AB and BC" .
{Given: " AB = (8x + 3) ; BC = (2x + 7) ; AC = 80 ".}.
----------------------------------------------------------------------------------------------=-
Note: The following "line graphs" are "Not Drawn to Scale."
| (8x + 3) | (2x + 7) |
|<--------------->|<------------------->|
<----·|----------------->|<------------------->| {Note: "Not drawn to scale."}.
A B C
<-----|----------------->|<------------------->|
A C
<-----|<--------------------------------------->|
<-----|----------------------------------------->|
A {"AC = 80 ".} C
<---|------------------------------------------>|
A {"AC = 80 ".}. C
| (8x + 3) ; + (2x + 7) = 80 ;
<-----|----------------->|<--------------------->I
A B C
So: We have to find the value of:
1) AB ; which is: (8x + 3) ; And:
2) BC ; which is: (2x + 7) .
To get these values; we need to find the value for "x".
So: (8x + 3) + (2x + 7) = 80 ;
➝ 8x + 3 + 2x + 7 = 80 ;
➝ Now, Let us combine the "like terms" on the "left-hand side" of the equation:
+8x + 2x + 3 + 7 ;
➝ to get: + 8x + 2x = + 10x ; and:
+3 + 7 = 10 ;
To get: "10x + 10" ;
Now, we can rewrite the equation:
{" 70 ÷ 10 = 7 ."}. " 10x + 10 = 80 ;
_____________________
To solve for "x" ; there are many ways:
_____________________
Method 1) We have: " 10x + 10 = 80 " ;
➝ Now, subtract "10" from each side of the question;
10x + 10 − 10 = 80 − 10 ;
to get: 10x = 70 ;
➝ Now, divide each side of the equation by: "10" ; to isolate "x" on one side of the equation; & to solve for "x" ;
10x /10 = 70 /10 ; to get: " x = 7 "
Method 2)
At the moment above in which we have:
" 10x = 70 " ; we know that "10x ÷ 10 = 1x" ; and then "70 ÷ 10 = 7 ".
{Note that any value; divided by "10" ; is equal to:
{"that value" moved Back by: "One" decimal space.}.
So: " 1x = 7 " ; ↔ " x = 7 ".
_____________
Method 3) :
When we have: " 10x /10 = 70 /10 " ; we have " 1x " on the "left-hand side" of the equation; and: "{ 70 /10 }" = {" 70 ÷ 10 = 7 ."}.
Note that: {" 70 ÷ 10 = 7 "} ; can be determined in many ways;
For instance: {" 70 ÷ 10 = ? "} ;
➝ The zeros for Both the numerator and denominator "cancel out"—
and we have: " [tex]\frac{70}{10}[/tex] " ;
➝ Cancel out each of the two (2) "zeros" ; and we have: " [tex]\frac{7}{1} = 7.[/tex] "
➝ This is assuming that we figure out that: " [tex]\frac{10x}{10}=1x =x[/tex] ."
___________
Now; let's find the correct values for this Brainly Question:
1) AB = 8x + 3 ;
Substitute our calculated value for "x" ; and solve:
➝ AB = 8x + 3 ; ↔ 8(7) + 3 = 56 + 3 = 59 .
➝ BC = 2x + 7 ; ↔ 2(7) + 7 = 14 + 7 = 21 .
Note:
➝ AC = 80 = AB + BC ≟ 59 + 21 ≟ 80 ? Yes!
____________
Answers:
1) The value of AB is: " 59 units."
___
2) The value of BC is: "21 units."
___
3) The value of AB + BC ;
= 59 units + 21 units = {The value of AC } = "80 units".}.
_____
Hope this answer—along with explanations—is helpful to you.
Best wishes in your academic pursuits!
______
The average early morning temperature in Churchill, Manitoba is "-4°C" in October. In January, it is 27°C colder. What is the average early morning temperature in January?
Answer:
-31 degrees
Step-by-step explanation:
-4-27=-31
4. The following set of points belong to a specific function: {(-3,0)(-2,4), (-1,0 ), (0,-6),(1,-8), (2,0),(3,24)} Based on the set of points answers the following questions:
a) What type of function does the set of points produce? Justify your answer.
b) Write an equation for this function based on the set of points that have been given.
The type of function that these sets of points are known to produce is the cubic function.
What is a cubic function?This is a function that is known to be a polynomial of degree 3. This type of polynomial is one that the coefficients are known to be complex numbers.
This is a cubic function given that the x axis is known to go through these points (-3,0), (-1,0), (2,0).
b. Given the sets of pints that we have, we have the equation
f(x) = a(x+3)(x+1)(x-2)
Such that
f = -2 =
4 = a x 1 x -1 x -4
4 = 4a
a = 1
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• What sorts of follow up questions about the statistics might you ask that person in order to obtain the
data needed to make a decision about the validity of that statement? Be sure to use concepts and
vocabulary from this week to guide your response.
The type of follow-up questions that can be asked to obtain the data needed to make a decision about the validity of that statement are:
How was the data obtained?Was the experiment controlled?How large were the control group and the placebo group?Was it a blind/double-blind test?Were extreme outliers removed from the data before running the experiment?What are Follow-Up Questions?This refers to the type of questions that are asked to a person in order to get extra information about a topic.
Hence, we can see that from the complete question, there is the need to perform tests to get results which would involve the procurement of data, controlled experiments, etc, and the follow-up questions that can be used are given above.
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solve (x - 3)^2 = 5 p
please help
Hello!
Let's solve !
⇒ (x - 3)² = 5
⇒ x² - 6x + 9 = 5
⇒ x² - 6x + 4 = 0
⇒ x = -(-6) ± √36 - 4(1)(4) / 2
⇒ x = 6 ± √20 / 2 = 6 ± 2√5 / 2
⇒ x = 3 ± √5
∴ The solutions are x = 3 + √5 and x = 3 - √5.
Answer:
p=(x−3)25
Step-by-step explanation:
Rearrange:
Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :
(x-3)^2-(5*p)=0
STEP1:
1.1 Evaluate : (x-3)2 = x2-6x+9
Equation at the end of step1:
x2 - 6x - 5p + 9 = 0
STEP2:
Solving a Single Variable Equation:
2.1 Solve x2-6x-5p+9 = 0
9. If f(x)=x². g(x)=2x-1. and h(x) =find the following.
Answer:
Please see the attached picture ! :D
-------- HappY LearninG <3 ----------
4. A triangle has vertices A(4, 2), B(3, 7), and C(–4, –5).
a) Explain HOW you would draw the median of the triangle from vertex A. Then draw it on the grid provided.
B). Calculate the length of the median from C to AB
To draw the median of the triangle from vertex A, the mid point of BC must be determined. The median of the vertex A is given at (-1/2, 1). See explanation below.
How you would draw the median of the triangle from vertex A?Recall that B = (3, 7)
and C = (-4, -5).
Note that when you are given coordinates in the format above, B or C = (x, y)Hence the mid point of line BC is point D₁ which is derived as:Connecting D' and A gives us the median of the vertex A. See attached graph.
What is the length of the median from C to AB?
Recall that
A → (4, 2); and
B → (3, 7)
Hence, the Midpoint will be
[tex][(\frac{4+3}{2} )[/tex], [tex](\frac{2+7}{2} )][/tex]
→ [tex](\frac{7}{2}, \frac{9}{2} )[/tex]
Recall that
C → (-4, 5)
Hence,
[tex]\overline{C D_{2} }[/tex] = [tex]\sqrt{[(-4 -\frac{7}{2} })^{2} + (-5-\frac{9}{2} )^{2} ][/tex]
Simplified, the above becomes
= √(586)/2)
= 24.2074/2
= 12.1037
The length of the Median from C to AB ≈ 12
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The circle below is centered at the point (-3, 4) and has a radius of length 3.
What is its equation?
A. (x- 4)2 + (y + 3)2 =
32
0
B. (x + 4)2 + (y- 3)2 =
32
T c. (x- 3)2 + (y + 4)2 = 9
O D. (x+ 3)2 + (y- 4)2 = 9
Answer: [tex](x+3)^2 + (y-4)^2 = 9[/tex]
Step-by-step explanation:
See attached image, this is straight formula substitution.
PLEASE FIND THE COORDINATES!!!
The graph of the given equation is given below.
We have given the equation of the line is f(x)=-1/5x+4f(x)
We have to determine the coordination.
What are the coordinates of the line?line coordinates are used to specify the position of a line just as point coordinates (or simply coordinates) are used to specify the position of a point.
Therefore we get f(x)=y
y=-1/5x+4....(1)
Plug x=0 in (1) then y=4
Plug x=5 in (1) then
y=-1/5(5)+4
=-1+4
=3
Therefore we get the coordinates (0,4) and (5,3)
Therefore the graph is given below,
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Find the amount of simple interest earned for depositing the given principle in an account.
P
=
P
=
$7600
r
=
r
=
8.5%
t
=
t
=
4 months
Answer:
kkrg=u/d
-678/09
=12 ans
In 2011, the maximum amount that you could have contributed to your RRSP (Registered Retirement Savings Plan) was
the lesser of $22,450 or 18% of your earned income from the previous year. How much income do you need to claim a
$22,450 contribution in 2011?
Using proportions, it is found that you need an income of $124,722.22 to claim a $22,450 contribution in 2011.
What is a proportion?A proportion is a fraction of a total amount, and the measures are related using a rule of three.
To claim a $22,450 contribution in 2011, this amount has to be 18% of your income, hence:
0.18x = 22450
x = 22450/0.18
x = $124,722.22.
You need an income of $124,722.22 to claim a $22,450 contribution in 2011.
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Consider the graph of the function f(x) = 10^x
Which statement describes a key feature of function g if g(x) = f(x + 4)
The statement that describes a key feature of g if g(x) = f(x + 4) and f(x) = 10ˣ is "horizontal asymptote of y = 0". (Correct choice: B)
How to analyze a function resulting from a transformation rule
According to the statement, we have an original function f(x) = 10ˣ and a transformation rule for horizontal translation g(x) = f(x + 4). By applying this rule, we have that g(x) = 10ˣ ⁺ ⁴, which have these features:
The y-intercept of g(x) is located at (0, 10000).Horizontal asymptote of g(x) is y = 0.There are no x-intercepts.Therefore, the statement that describes a key feature of g if g(x) = f(x + 4) and f(x) = 10ˣ is "horizontal asymptote of y = 0". (Correct choice: B)
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Use the function below to find F(3).
)(
F(x)=(1/4)^x
Answer:
1/64
Step-by-step explanation:
F(3) means to use 3 in place of x in the given equation.
f(x) = (1/4)^x
f(3) = (1/4)^3
Either use exponents rules:
= 1^3 / 4^3
= 1 / 64
or just use the meaning of exponents
= 1/4 × 1/4 × 1/4
= 1/64
Assume that P(E) = 0.471 and P(F) = 0.595. If E and F are independent, find P(E and F)
The value of the probability P(E and F) is 0.2802
Independent probabilityEvents are known to be independent if the occurrence of one does not affect the other.
Given the following parameters
P (E) =0.471
P(F) = 0.595
If E and F are independent, then;
P(E and F) = P(E)P(F)
P(E and F) = 0.471 * 0.595
P(E and F) = 0.2802
Hence the value of the probability P(E and F) is 0.2802
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A bus can reach from Kathmandu to Lumbini in 15 hours when the speed is 40 km per hour. I how many hours it reaches when the speed of the bus is 50 km per hour.
Answer:
12 hours
Step-by-step explanation:
1) Find the distance using: Distance = speed × time.
Distance = 40 × 15
= 600 km
2) Use the time formula to find the time taken. The formula is Time = distance / speed.
Time = 600 / 50
= 12 hours
Help!!! cant find answers
As the intervals moves from left to right on the graph, the average rate of change increases
How to determine the change in the rate of change?The function is an exponential growth function.
This is known because it has an initial value at (0, 2), and the value increases to the right
The average rate of change of exponential growth functions increases towards the right.
This means that:
The initial statement is "as the intervals moves from left to right on the graph, the average rate of change increases
Because the average rate of change keeps changing, the final statement is:
Since the average rate of change is also changing, the function is changing and approaches infinity
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write the equation of the line that is parallel to the y-axis and passes through the point (7,0)
The equation of the line is x = 7 which is parallel to the y-axis and passes through the point (7,0)
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]
We have:
The line that is parallel to the y-axis and passes through the point (7,0)
As we know the line equation:
Parallel to the y-axis
x = c
Passes through the point (7,0)
x = 7
Thus, the equation of the line is x = 7 which is parallel to the y-axis and passes through the point (7,0)
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Hi- I need help please to get my HS diploma...did not graduate :(
For the inequality y+4x≤x2+2, where is the graph shaded and is the curve solid or dotted?
Answer:
B
Step-by-step explanation:
first for the easy one : the curve line is dotted, if the line points are NOT part of the solution. and it is solid, if the points are included.
you know what the curve line indicates ? it indicates the points for which there is equality.
so, if there is a ">=" or "<=" relation, then that includes equality, and the curve is solid.
if there is only a true inequality ("<", ">"), then the curve is dotted, as the equality is not included.
so, I how this is now really clear.
in our case here (I suspect it really looks like this)
y + 4x <= x² + 2
we have a "<=" sign, so, the curve is ..., well ? ... well ? ...
well ? ... well ? ... solid !
and we still treat inequalities as much as possible as regular equalities.
so,
y <= x² - 4x + 2 (after subtracting 4x from both sides).
and now you see it, right ?
all y values that are smaller or equal to the quadratic expression.
SMALLER ! so, BELOW the curve (as the smaller y values are represented by the lower part of the y-axis).
find 2^-6 as a fraction
Answer:
1/64Step-by-step explanation:
find 2^-6 as a fraction
2^-6 = 0.015625
0.015625 as a fraction
(0.015625/1) × (1000000/1000000) =
15625/1000000
simplified = 1/64.
[tex]\frak{Hi!}[/tex]
[tex]\orange\hspace{300pt}\above2[/tex]
Let's find [tex]\boldsymbol{\sf{2^{-6}}}[/tex] as a fraction
Did you know that if we have a number with negative
exponents, we flop it over? This is how it's done-:
[tex]\boldsymbol{\sf{2^{-6}=\displaystyle\frac{1}{2^6}}}[/tex]. And Now we need to evaluate
[tex]\boldsymbol{\sf{\displaystyle\frac{1}{64}}}[/tex].
done!!
[tex]\orange\hspace{300pt}\above3[/tex]
The speed of light is approximately 299,000,000 meters per second.
What is the speed of the light, in meters per second, using scientific notation?
Answer:
2.99 x 10⁸ meters per second
Step-by-step explanation:
Scientific notation (also called "Standard form") is written in the form of [tex]a \times 10^n[/tex], where [tex]1\leq a < 10[/tex] and [tex]n[/tex] is any positive or negative whole number.
To convert a number into scientific notation, move the decimal point to the left or right until there is one digit before the decimal point.
The number of times you have moved the decimal point is the power of 10 ([tex]n[/tex]).
If the decimal point has moved to the left, the power is positive.
If the decimal point has moved to the right, the power is negative.
To convert the given number to scientific notation
The decimal point for the given number 299000000 is after the last zero:
⇒ 299000000.
Move the decimal point 8 places to the left:
⇒ 2.99000000
Get rid of the redundant zeros:
⇒ 2.99
Multiply by 10 to the power of the number of decimal places moved:
⇒ 2.99 x 10⁸
Therefore, the speed of light using scientific notation is:
2.99 x 10⁸ meters per secondAnalyze the diagram below and complete the instructions that follow.
Find m
A. 15°
B. 30°
C. 75°
D. 150°
Please select the best answer from the choices provided
A
B
C
D
Answer:
I believe the answer to be c 75°
Step-by-step explanation:
a 4 sided shape is 360 total degrees
what is the value of h(2)
a) -2
b) -1
c) 0
d) 3
Answer:
(c) 0
Step-by-step explanation:
The answer to this question requires that you read a graph.
How to read a graphFinding the value of h(2) means finding the value of "y" that corresponds to x=2.
The x-axis of this graph is identified by the "x" at the right side, under the arrow at the end of the x-axis. The y-axis is similarly identified by the "y" to the right of the y-axis at the top of the graph.
Each axis is labeled with numbers.
To find the y-value associated with x=2, first find the vertical line labeled 2 on the x-axis. It is 2 grid squares to the right of the y-axis.
Follow this vertical line until you reach the point where it intersects the line segment(s) representing h(x). At one of these, you encounter a solid dot. At the other point of intersection, you see an open circle (empty dot). The open circle means the point is not included in the function graph.
The value of h(2) is the y-value at the point where the vertical line x=2 meets the defined point on the function graph, the solid dot. The y-value is found by following the horizontal grid line to the y-axis, and reading the y-value there.
The point on this graph where the axes cross, the "origin," has no markings indicating the x- and y-values are zero there. You must infer that from the markings that are shown.
Value of h(2)The point on the graph where x=2 lies on the horizontal line y=0. That is ...
h(2) = 0
Which ordered pair is the best estimate for the solution of the system of equations?
{y=4x−19.4
{y=0.2x−4.2
(−3.5, 4)
(4.9, −3.5)
(4, −3.4)
(4.9, 0)
The ordered pair which is a solution of the system of equations is; (4.9, 0).
Which solution pair is the best estimate of the system solution?The equations given in the task content can be solved as a system as follows, since y = y;
4x -19.4 = 0.2x -4.2
4x -0.2x = 19.4 -4.2
3.8x = 15.2
x = 15.2/3.8
x = 4.9
y = 4(4.9) -19.4
y = 0.2
Hence, the best ordered pair solution is; (4.9, 0).
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Kelli swam upstream for some distance in one hour. She then swam downstream the same river for the same distance in only 6 minutes. If the river flows at 5 km/hr, how fast can Kelli swim in still water?
Choose the most logical value for the variable to represent.
Let x = .
3m = 15
9 + 15m = 12m – 1
3m = 18
9 + 6 – 15 = 15m – 12m
By solving a system of equations, we conclude that Kelli's speed on still water is 6.1km/h
How fast can Kelli swim in still water?Let's say that the speed of Kelli in still water is S. And we know that the speed of the river is 5km/h.
When she swims upstream, her velocity is:
(S - 5km/h)
When she swims downstream, her velocity is:
(S + 5km/h)
With the given information, we can write for a distance D:
(S - 5km/h)*1h = D
(S + 5km/h)*6min = D
First, let's rewrite 6 minutes into hours.
1 hour = 60min
6 min = 6/(60) hours = 0.1 hours
Then the system of equations that we need to solve is:
(S - 5km/h)*1h = D
(S + 5km/h)*0.1h = D
D is the same distance in both cases, then we can write:
(S - 5km/h)*1h = (S + 5km/h)*0.1h
Now we can solve this for S.
S*1h - 5km = S*0.1h + 0.5km
S*1h - S*0.1h = 5km + 0.5km
S*(0.9h) = 5.5km
S = 5.5km/0.9h = 6.1km/h
Kelli's speed on still water is 6.1km/h
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Alexander Efland has a savings account that earns 5.5% interest compounded daily. On May 5, the amount in the account was $28,214.35. How much interest will the money earn in the next 90 days?
Using compound interest, it is found that $390.6 was earned in interest in the next 90 days.
What is compound interest?The amount of money earned, in compound interest, after t years, is given by:
[tex]A(t) = P\left(1 + \frac{r}{n}\right)^{nt}[/tex]
In which:
A(t) is the amount of money after t years. P is the principal(the initial sum of money). r is the interest rate(as a decimal value). n is the number of times that interest is compounded per year.In this problem, the parameters are:
P = 28214.35, r = 0.055, n = 365, t = 3/12 = 0.25.
Hence the amount after 90 days is:
[tex]A(t) = P\left(1 + \frac{r}{n}\right)^{nt}[/tex]
[tex]A(90) = 28214.35\left(1 + \frac{0.055}{365}\right)^{365 \times 0.25}[/tex]
A(90) = $28,604.95
28604.95 - 28214.35 = $390.6 was earned in interest in the next 90 days.
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The arithmetic-geometric mean (AM-GM) inequality: if a,b ≥ 0, then [tex]\frac{a+b}{2} \geq \sqrt{ab}[/tex]. (1)
(a) Let a, b, c, d ≥ 0 and consider the positive real numbers [tex]\sqrt{ab} [/tex] and [tex]\sqrt{cd} [/tex]. Use (1) to prove the four-variable AM-GM inequality:
[tex]\frac{a+b+c+d}{4} \geq \sqrt[4]{abcd}[/tex]. (2)
(b) Let a, b, c ≥ 0 and define [tex]m= \frac{a+b+c}{3}[/tex].
(i) Show that [tex]m=\frac{a+b+c+m}{4}[/tex].
(ii) Hence use (2) to prove the three-variable AM-GM inequality:
[tex]\frac{a+b+c}{3} \geq \sqrt[3]{abc}[/tex].
(a) The inequality follows from the binomial theorem.
[tex](a - b)^2 = a^2 - 2ab + b^2 \ge 0 \implies a^2 + 2ab + b^2 = (a+b)^2 \ge 4ab \\\\ \implies \dfrac{(a+b)^2}4 \ge ab \\\\ \implies \dfrac{a+b}2 \ge \sqrt{ab} ~~~~~~~~ (1)[/tex]
By the same reasoning,
[tex]\dfrac{c+d}2 \ge \sqrt{cd}[/tex]
Adding these results together, we have
[tex]\dfrac{a+b}2 + \dfrac{c+d}2 = \dfrac{a+b+c+d}2 \ge \sqrt{ab} + \sqrt{cd}[/tex]
and since [tex]\sqrt{ab}[/tex] and [tex]\sqrt{cd}[/tex] are both positive, they also satisfy the AM-GM inequality,
[tex]\dfrac{\sqrt{ab} + \sqrt{cd}}2 \ge \sqrt{\sqrt{ab}\times\sqrt{cd}} = \sqrt[4]{abcd}[/tex]
and the 4-variable result follows,
[tex]\dfrac{a+b+c+d}2 \ge 2\times\dfrac{\sqrt{ab}+\sqrt{cd}}2 \ge 2 \sqrt[4]{abcd} \\\\ \implies \dfrac{a+b+c+d}4 \ge \sqrt[4]{abcd} ~~~~~~~~ (2)[/tex]
(b.i) With [tex]m=\frac{a+b+c}3[/tex], we have
[tex]\dfrac{3m}4 = \dfrac{a+b+c}4 \implies m - \dfrac m4 = \dfrac{a+b+c}4 \\\\ \implies m = \dfrac{a+b+c+m}4[/tex]
(b.ii) From (2) it follows that
[tex]\dfrac{a+b+c+m}4 \ge \sqrt[4]{abcm} = \sqrt[4]{abc\times\dfrac{a+b+c+m}4}[/tex]
Using the result from (b.i), this is equivalent to
[tex]\dfrac{a+b+c}3 \ge \sqrt[4]{abc\times\dfrac{a+b+c}3}[/tex]
Take 4th powers on both sides.
[tex]\left(\dfrac{a+b+c}3\right)^4 \ge abc \times \dfrac{a+b+c}3[/tex]
Divide both sides by [tex]\frac{a+b+c}3[/tex].
[tex]\left(\dfrac{a+b+c}3\right)^3 \ge abc[/tex]
Finally, take the cube root of both sides.
[tex]\dfrac{a+b+c}3 \ge \sqrt[3]{abc}[/tex]
as required.