Final answer:
Bob's consumer surplus after a tax is $4, and Lisa's is $1, making the total consumer surplus for both after the tax $5.
Explanation:
Consumer surplus is the difference between the value a consumer places on a good and what they actually pay. Before the government levies a tax, Bob's consumer surplus for a movie ticket is the difference between his valuation of $10 and the market price of $5, which is $5. Lisa's consumer surplus is the difference between her valuation of $7 and the market price of $5, which is $2.
After the government implements a $1 tax on movie tickets, increasing the price to $6, Bob's consumer surplus becomes $4 ($10 - $6), and Lisa's consumer surplus is now $1 ($7 - $6). Thus, the total consumer surplus for Bob and Lisa after the tax is implemented is $5 ($4 for Bob and $1 for Lisa).
Write the expression using a single exponent. (6^2)^7
Answer:
[tex]6^{14}[/tex]
Step-by-step explanation:
Use the "Power Law" of exponents that tells us that when you have a base to a power "n" and all that raised to a power "m", it is the same as writing the original base to the single exponent which is the product of n time m:
[tex](b^n)^m=b^{n*m}[/tex]
therefore in your case, the base "b" is 6, the exponent "n" is 2, and the exponent "m" is 7. Then:
[tex](6^2)^7=6^{2*7}=6^{14}[/tex]
Final answer:
To simplify the expression [tex](6^2)^7[/tex] using a single exponent, multiply the inner exponent 2 by the outer exponent 7, which yields 6¹⁴.
Explanation:
To write the expression [tex](6^2)^7[/tex] using a single exponent, you need to apply the rule for raising a power to a power. This rule states that you multiply the exponents together. So for our expression, we have the base number 6 raised to the power of 2, and this result is then raised to the power of 7. To combine them into a single exponent, you multiply 2 by 7, which gives you 14.
Therefore,[tex](6^2)^7[/tex] can be simplified to 6¹⁴. This is because when you raise a power to another power, the powers are multiplied: for example, (a^b)^c = a^(b*c).
A car repair shop offers its customers free coffee while they wait. By the end of each day, the coffee urn, which had started out with 7 1/4 gallons of coffee, was left with 2 1/12 gallons. How many gallons of coffee had been dispensed?
Answer: 5.1667 gallons of coffee had been dispensed
Step-by-step explanation:
Let x represent the number of gallons of coffee that had been dispensed.
By the end of each day, the coffee urn, which had started out with 7 1/4 gallons of coffee, was left with 2 1/12 gallons. This means that the initial number of gallons of coffee was 7 1/4 = 7.25 gallons
The amount left after dispense x gallons is 2 1/12 = 2.0833 gallons.
Therefore,
x +2.0833 = 7.25
x = 7.25 - 2.0833 = 5.1667 gallons
At the end of year X, automobile installment credit accounted for 36% of all outstanding consumer installment credit. At that time automobile finance companies extended $57 billion of credit, or 1/3 of the automobile installment credit. How many billion dollars of consumer installment credit was outstanding at that time?
A. 62
B. 171
C. 475
D. 513
E. 684
Answer:
475 billion dollars
Step-by-step explanation:
Let P be the customer credit
At the end of year X, 36% of P is gotten from automobile installment credit
57 billion credit is 1/3 of the automobile installment credit. This means that the total automobile installment credit 57*3 = 171 billion dollars
36% * P = 171
36/100 * P = 171
36P = 171 *100
P = 17100/36
P= 475 billion dollars
Assume that the terminal side of thetaθ passes through the point (negative 12 comma 5 )(−12,5) and find the values of trigonometric ratios sec thetaθ and sin thetaθ.
Answer:
[tex]\sin \theta = \dfrac{5}{13}[/tex] and [tex]\sec \theta = -\dfrac{13}{12}[/tex]
Step-by-step explanation:
Assume that the terminal side of thetaθ passes through the point (−12,5).
In ordered pair (-12,5), x-intercept is negative and y-intercept is positive. It means the point lies in 2nd quadrant.
Using Pythagoras theorem:
[tex]hypotenuse^2=perpendicular^2+base^2[/tex]
[tex]hypotenuse^2=(5)^2+(12)^2[/tex]
[tex]hypotenuse^2=25+144[/tex]
[tex]hypotenuse^2=169[/tex]
Taking square root on both sides.
[tex]hypotenuse=13[/tex]
In a right angled triangle
[tex]\sin \theta = \dfrac{opposite}{hypotenuse}[/tex]
[tex]\sin \theta = \dfrac{5}{13}[/tex]
[tex]\sec \theta = \dfrac{hypotenuse}{adjacent}[/tex]
[tex]\sec \theta = \dfrac{13}{12}[/tex]
In second quadrant only sine and cosecant are positive.
[tex]\sin \theta = \dfrac{5}{13}[/tex] and [tex]\sec \theta = -\dfrac{13}{12}[/tex]
Need help only have few minutes to answer
how does the graph f(x)=(x-8)^3+4 compare to the paren function g(x)=x^3?
Answer:
Their intercepts are unique.
Explanation:
[tex]\displaystyle x^3 - 24x^2 + 192x - 508 = (x - 8)^3 + 4[/tex]
This graph's x-intercept is located at approximately [6,41259894, 0], and the y-intercept located at [0, −508].
[tex]\displaystyle g(x) = x^3[/tex]
The parent graph here, has both an x-intercept and y-intercept located at the origin.
I am joyous to assist you anytime.
A landscape architect wished to enclose a rectangular garden on one side by a brick wall costing $20/ft and on the other three sides by a metal fence costing $10/ft. If the area of the garden is 122 square feet, find the dimensions of the garden that minimize the cost.
Answer:
The dimensions of the garden that minimize the cost is 9.018 feet(length) and 13.528 feet(width)
Step-by-step explanation:
Let the length of garden be x
Let the breadth of garden be y
Area of Rectangular garden = [tex]Length \times Breadth = xy[/tex]
We are given that the area of the garden is 122 square feet
So, [tex]xy=122[/tex] ---A
A landscape architect wished to enclose a rectangular garden on one side by a brick wall costing $20/ft
So, cost of brick along length x = 20 x
On the other three sides by a metal fence costing $10/ft.
So, Other three side s = x+2y
So, cost of brick along the other three sides= 10(x+2y)
So, Total cost = 20x+10(x+2y)=20x+10x+20y=30x+20y
Total cost = 30x+20y
Substitute the value of y from A
Total cost = [tex]30x+20(\frac{122}{x})[/tex]
Total cost = [tex]\frac{2440}{x}+30x[/tex]
Now take the derivative to minimize the cost
[tex]f(x)=\frac{2440}{x}+30x[/tex]
[tex]f'(x)=-\frac{2440}{x^2}+30[/tex]
Equate it equal to 0
[tex]0=-\frac{2440}{x^2}+30[/tex]
[tex]\frac{2440}{x^2}=30[/tex]
[tex]\sqrt{\frac{2440}{30}}=x[/tex]
[tex]9.018 =x[/tex]
Now check whether it is minimum or not
take second derivative
[tex]f'(x)=-\frac{2440}{x^2}+30[/tex]
[tex]f''(x)=-(-2)\frac{2440}{x^3}[/tex]
Substitute the value of x
[tex]f''(x)=-(-2)\frac{2440}{(9.018)^3}[/tex]
[tex]f''(x)=6.6540[/tex]
Since it is positive ,So the x is minimum
Now find y
Substitute the value of x in A
[tex](9.018)y=122[/tex]
[tex]y=\frac{122}{9.018}[/tex]
[tex]y=13.528[/tex]
Hence the dimensions of the garden that minimize the cost is 9.018 feet(length) and 13.528 feet(width)
Howard opens a savings account with a deposit of 800$. The bank will pay him 3% intrest per year. How much will Howard receive at the end of the year?
Answer:
$824
Step-by-step explanation:
3% (rate of interest) of 800= 24
800 + 24= 824
What will be the effect of paying off an accounts payable balance on the current and the acid-test ratios? Assume that both ratios are greater than 1.
Answer:
Both ratios will increase where the accounts payable balance is paid off.
Step-by-step explanation:
The current ratio is given as
Current ratio = Current asset / current liabilities
Where the current assets are asset that can be converted into cash easily ( including cash and cash equivalents) while the current liabilities are liabilities to be settled in a short term, say 1 year.
Acid test ratio is given as
Acid test ratio = (Current asset - Inventories) / current liabilities
Here, the current assets excludes the assets that are not so easily converted to cash.
From the two formulas stated above, where the accounts payable balance which is an element of the current liabilities is paid off, the current liabilities balance reduces thus resulting in an increase in both ratio.
Hence, current and the acid-test ratios will increase where the accounts payable balance is paid off.
Paying off accounts payable increases both the current and acid-test ratios, assuming they are originally greater than 1, indicating positive financial stability to investors and creditors.
Explanation:The effect of paying off an accounts payable balance on both the current and the acid-test ratios, if they are greater than 1, would be an increase. The current ratio is calculated as current assets divided by current liabilities. When accounts payable (a current liability) is paid off, the denominator of the ratio decreases, leading to an increase in the ratio. Similarly, for the acid-test (or quick) ratio, once again we see a decrease in the denominator after pay off, leading to an increase in the ratio.
Although the net change in an entity’s financial position may seem neutral (decrease in an asset offset by a decrease in liabilities), these ratio increases can be viewed positively by investors and creditors as they imply a greater degree of short-term solvency.
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Which of the following are the vertices of the image of the figure below under the translation (x , y) ---> (x + 4, y - 2)? SELECT ALL THAT APPLY
A. (- 1, 2)
B. (- 9, 6)
C. (- 2, 7)
D. (6, 3)
E. (1, - 4)
F. (- 7, 0)
Answer:
d b f
Step-by-step explanation:
You survey your class to find out the types of pets they have at home. Below are the results: Dog, Dog, Cat, Cat, Cat, Dog, Hamster, Fish, Dog, Fish
a. Make a frequency table for the results.
b. Calculate the relative frequencies of the each animal type.
c. Using the relative frequencies explain which animals are most and least popular. Be specific and explain your reasoning.
Answer:
a) Frequency table:
Category Frequency
Dog 4
Cat 3
Fish 2
Hamster 1
b) Relative frequencies of each animal type
Dog: 4/10 = 0.4Cat: 3/10 = 0.3Fish: 2/10 = 0.2Hamster: 1/10 = 0.1c) Popularity
Dog is the most popular because it has the highest relative frequency.Hamster is the least popular because it has the lowest relative frequency.Explanation:
a. Make a frequency table for the results.
There are four kind of pets: dog, cat, hamster, and fish.
A frequency table shows the number of items for each category (kind of pets).
Count the number of each kind of pet:
Dog: 4Cat: 3Hamster: 1Fish: 2Total: 10With that you build your frequency table.
Frequency table:
Category Frequency
Dog 4
Cat 3
Fish 2
Hamster 1
b. Relative frequencies of each animal type.
The relative frequency is how often an outcome appears divided by the total number of outmcomes.
Here the total number of outcomes is 10 (the ten pets).
So, calculate each relative frequency:
Dog: 4/10 = 0.4Cat: 3/10 = 0.3Fish: 2/10 = 0.2Hamster: 1/10 = 0.1An important feature of the relative frequency is that they must add up 1. Check:
0.4 + 0.3 + 0.2 + 0.1 = 1.c. Using the relative frequencies explain which animals are most and least popular.
Popularity is determined by the frequency with each outcome is repeated. The most popular is the most repeated. The least popular is the least repeated.
Dog is the most popular because it has the highest relative frequency.Hamster is the least popular because it has the lowest relative frequency.Use synthetic division to check that the number k is a lower bound for the real zeros of the function f.
k=-7; f(x)= 5x^3 - 8x^2 + x -5
Answer:
see below
Step-by-step explanation:
The signs on the bottom line alternate, so the value of k is, indeed, a lower bound.
_____
Comment on lower bound for this cubic
The signs of the coefficients alternate, so Descartes' rule of signs will tell you there are zero negative real roots. That is, 0 is a lower bound for real roots. No synthetic division is needed.
Solve the inequality and graph the solution
|2x - 7| > 1
Answer:
Open circle to the right of 4
x > 4
Step-by-step explanation:
Add 7 to both sides
2x > 8
x > 4
Open circle to the right of 4
A theater group charges $12.50 per ticket for its opening play of the season. Production costs for the play are $150. Which function could be used to determine the profit the theater group earns from selling x tickets?
A)
f(x) = 12.50x
B)
f(x) = 12.50x + 150
C)
f(x) = 150x −12,50
D)
f(x) = 12.50x − 150
Option D
Function used to determine the profit the theater group earns from selling "x" tickets is f(x) = 12.50x - 150
Solution:Given that a theater group charges $12.50 per ticket for its opening play of the season
Also given that Production costs for the play are $150
To find: function used to determine the profit the theater group earns from selling "x" tickets
So "x" represents the number of tickets sold
Cost per ticket = $ 12.50
[tex]\text { cost of "x" tickets }=" x " \times \text { cost per ticket }[/tex]
[tex]\text { cost of "x" tickets }= x \times 12.50=12.50x[/tex]
Production costs = 150
Then the profit the theater group earns from selling "x" tickets is:
Profit earned from selling "x" tickets = cost of "x" tickets - production cost
Let f(x) denotes profit earned from selling "x" tickets
f(x) = 12.50x - 150
Thus option D is correct
Mae king earns a weekly salary of $305 plus a 7.5% commission on sales at a gift shop.How much would she make in a work week if she sold 4,300 worth of merchandise
Answer:
Step-by-step explanation:
Mae king earns a weekly salary of $305 plus a 7.5% commission on sales at a gift shop. This means that the total amount that she can earn in a week is not fixed. If in a week, she sold 4,300 worth of merchandise, her commission on this amount of sales will be 7.5 % of 4,300
Commission on sales = 7.5/100× 4300 = 0.075×4300= $332.25
Amount of money made for the week will be the sum of her weekly salary and the commission earned on sales. It becomes
305 + 332.25 = $627.5
On a coordinate plane, 2 solid straight lines are shown. The first line has a negative slope and goes through (negative 4, negative 2) and (0, negative 3). Everything above the line is shaded. The second line has a positive slope and goes through (0, negative 2) and (2, 2). Everything above the line is shaded.
Which number completes the system of linear inequalities represented by the graph?
y > 2x – 2 and x + 4y > ... HELP ME PLZ HURRY
Answer:
-12
Step-by-step explanation:
Answer:
The first One
Step-by-step explanation:
I just got 100% on My Quiz
Three pipes a and b and c can fill a swimming pool in 6 hours. After working on it together for 2 hours,C is closed and A and B finish the remaing work in 7 hours. Find the time taken by alone to fill the swimming pool.
Answer:
The pipe C alone can fill the tank in 14 hours .
Step-by-step explanation:
Given as :
The three pipes a , b , c can fill the pipes in 6 hours
They work for 2 hours
After that c pipe is close and a , b finish remaining work
Now, According to question
In 1 hour pipes ( a + b + c ) fill [tex]\frac{1}{6}[/tex] of the tank
∴ In 2 hour pipes ( a + b + c ) fill [tex]\frac{2}{6}[/tex] = [tex]\frac{1}{3}[/tex] of the tank
Remaining ( 1 - [tex]\frac{1}{3}[/tex] ) = [tex]\frac{2}{3}[/tex] part is filled by pipes a and b in 7 hours
∴ The whole tank is filled by a and b in 7 × [tex]\frac{3}{2}[/tex] = [tex]\frac{21}{2}[/tex] hours
∴ In 1 hour pipes A and b fill the tank in [tex]\frac{2}{21}[/tex] hours
∴ In 1 hour pipes C alone can fill the tank in[tex]\frac{1}{6}[/tex] - [tex]\frac{2}{21}[/tex] hours
Or, In 1 hour pipes C alone can fill the tank in [tex]\frac{9}{126}[/tex] = [tex]\frac{1}{14}[/tex]
Or, In 1 hour pipes C alone can fill the tank in 14 hours
Hence The pipe C alone can fill the tank in 14 hours . Answer
Julie had three wheels from bikes and things that she stacked against the shed. Each wheel fitted so neatly together that Julie took this photo. The radius of the largest sheet is 16 cm and the radius of the middle-sized wheel is 9cm. What is the radius of Julie's smallest wheel?
Answer:
12.5 cm
Step-by-step explanation:
16 - 9 = 7
7 (1/2) = 3.5
9 + 3.5 = 12.5
15 - 3.5 = 12.5
Can someone answer this question?
The inequality is: 5>x≥0
Step-by-step explanation:
We have to write the inequality one by one
The inequality symbols are used to write inequalities.
So,
5 is greater than x will be written as:
5>x
And
0 is less than or equals to x:
0≤x or x≥0
We have to combine the both inequalities so that the variable is not repeated. The inequality symbols have to be written carefully while writing the compound inequalities.
5>x≥0
Keywords: Inequality, Relationships
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To borrow money, you pawn your guitar. Based on the value of the guitar, the pawnbroker loans you $600. One month later, you get the guitar back by paying the pawnbroker $1470. What annual interest rate did you pay?
Answer:
1740%.
Step-by-step explanation:
We have been given that the pawnbroker loans you $600. One month later, you get the guitar back by paying the pawnbroker $1470.
We will use simple interest formula to solve our given problem.
[tex]A=P(1+rt)[/tex], where,
A = Final amount after t years,
P = Principal amount,
r = Annual interest rate in decimal form,
t = Time in years.
1 month = 1/12 year
[tex]1470=600(1+r*\frac{1}{12})[/tex]
[tex]1470=600+\frac{600}{12}*r[/tex]
[tex]1470=600+50*r[/tex]
[tex]1470-600=600-600+50*r[/tex]
[tex]870=50*r[/tex]
[tex]50r=870=[/tex]
[tex]\frac{50r}{50}=\frac{870}{50}[/tex]
[tex]r=17.4[/tex]
Since our interest rate is in decimal form, so we will convert it into percentage by multiplying by 100 as:
[tex]17.4\times 100\%=1740\%[/tex]
Therefore, you paid an annual interest rate of 1740%.
Jill planted two flowers in her garden. The first flower is 2 inches tall, and it is growing 2.25 inches each week. The second plant is 5.75 inches tall, and it is growing 1.5 inches each week. How many weeks will it be until the two plants are the same height?
Answer:it will take the two plants 6 weeks before the heights are the same
Step-by-step explanation:
Jill planted two flowers in her garden.
The first flower is 2 inches tall, and it is growing 2.25 inches each week. Since the growth rate is in an arithmetic progression, we will apply the formula for finding the nth term of the series
Tn = a + (n - 1)d
Tn = the nth height of the first flower
a = the initial height of the first flower
d = the common difference in height of the first flower weekly
n = number of weeks
From the information given,
For the first flower,
a = 2
d = 2.25
Tn ?
n ?
Tn = 2 + (n - 1)2.25
For the second flower,
a = 5.75
d = 1.5
Tn ?
n ?
Tn = 5.75 + (n - 1)1.5
To determine the number of weeks that it will take until the two plants are the same height, we would equate Tn for both flowers. It becomes
2 + (n - 1)2.25 = 5.75 + (n - 1)1.5
2 + 2.25n - 2.25 = 5.75 + 1.5n - 1.5
Collecting like terms
2.25n - 1.5n = 5.75 - 1.5 - 2 + 2.25
0.75n = 4.5
n = 4.5/0.75
n = 6 weeks
To find out how many weeks it will be until the two plants are the same height, set up an equation and solve for x. The plants will be the same height after 5 weeks.
Explanation:To find out how many weeks it will be until the two plants are the same height, we need to set up an equation. Let the number of weeks be represented by x. The height of the first plant can be represented as 2 + 2.25x, and the height of the second plant can be represented as 5.75 + 1.5x. Set these two expressions equal to each other: 2 + 2.25x = 5.75 + 1.5x.
To solve for x, subtract 1.5x from both sides: 2 + 0.75x = 5.75.
Then, subtract 2 from both sides: 0.75x = 3.75.
Finally, divide both sides by 0.75 to solve for x: x = 5.
A stone is a British unit of weight equivalent to 14 pounds. After losing 1 1/2 stone. Jim is now 85% of his original weight. What is Jims current weight?
Answer:
Jim's current weight = 119 pounds
Step-by-step explanation:
1 Stone = 14 pounds
[tex]1\frac{1}{2}[/tex] stone = 1.5 stone
1.5 stone = 1.5 (14 pounds) = 21 pounds
Jim lost 21 pounds
Let X be Jim's Original Weight
Y be his present weight
As per given statement in the Question:
After losing 1.5 stones (21 pounds of weight) Jim now weighs Y
Present weight = original weight - 21
Y = X -21 Equation 1
Also Current Weight = 85 % (Original weight)
Y = 85 % (X) =[tex]\frac{85X}{100}[/tex]
Y=[tex]\frac{85X}{100}[/tex]
put in Equation 1
[tex]\frac{85X}{100}[/tex] = X-21
85X = (X-21) 100
85 X = 100 X -2100
or
2100 = 100 X - 85X
2100 = 15X
or
15 X = 2100
[tex]X=\frac{2100}{15}[/tex]
X= 140 pounds ( Original Weight)
Current Weight = Y = Original weight - 21 (From Equation 1)
Y = X -21
Y = 140 -21
Y = 119 pounds (Current Weight)
Michelle has 8 1/4 pounds of dry cat food for her cat smokey. She places the cat food into 3 containers to use at later date. How much cat food will be in each container
Answer:
Each container contains [tex]2.5 \ pounds \ \ OR \ \ \frac{11}{4} \ pounds \ \ OR \ \ 2 \frac{3}{4} \ pounds[/tex] of cat food.
Step-by-step explanation:
Given:
Amount of cat food = 8 1\4 pounds.
Also 8 1\4 can be rewritten as 8.25 pounds
Number of containers =3
We need to find the amount of cat food in each container.
Amount of cat food in each container can be calculated by Dividing Amount of cat food she has with number of Containers.
Amount of cat food in each container = [tex]\frac{8.25}{3}= 2.75 \ pounds[/tex]
2.75 pounds ca be rewritten as [tex]\frac{11}{4} \ pounds \ \ OR \ \ 2 \frac{3}{4} \ pounds[/tex]
Hence Each container contains [tex]2.5 \ pounds \ \ OR \ \ \frac{11}{4} \ pounds \ \ OR \ \ 2 \frac{3}{4} \ pounds[/tex] of cat food.
Elizabeth brought a box of donuts to share. There are two-dozen (24) donuts in the box, all identical in size, shape, and color. Three are jelly-filled, four are lemon-filled, and seventeen are custard-filled. You randomly select one donut, eat it, and select another donut.
Find the probability of selecting a lemon-filled donut followed by a custard-filled donut.
Answer:
P = 68/552 = 0.123 or 12.3%
Step-by-step explanation:
First, let's calculate the probability of getting a lemon donut. We have only 4 lemon donut among 24 donuts, so probability is:
P(A) = 4/24
Next, as we already ate the lemon donut, we only have 23 donuts now, and among these 23, 17 are custard filled, so probability of choosing one of those is:
P(B) = 17/23
But we want to know the probability that the custard filled donut is choosen after you eat the lemon one so:
P(B|A) = P(A) * P(B)
Replacing:
P(B|A) = 4/24 * 17/23
P(B|A) = 68/552 = 0.123 or 12.3%
Richard walked around 2 rectangular parks. One measures 450 feet by 167 feet and the other measures 234 feet by 156 feet. Part B Richard's brother walked around a different park that is 254 feet by 56 feet.
What is the question?
Answer:
The answer is in the explanation.
Step-by-step explanation:
450x2=900 167x2=334 Then you add those together and you get 1,234 then you do 234x2=468 156x2=312 then you add those two, and you get 780 then you add 1,234+780= 2014. For his brother you do 254x2=508 and 56x2=112 then you add them and you get 620. Then you subtract 2014-620= 1394. That is the final answer. Hope it helps!
How can trigonometry be used to model the path of a baseball?
Trigonometry can be used to model projectile motion, such as the flight of a baseball. Given the angle at which the ball leaves the bat and the initial velocity, you can determine the distance the ball will travel.
Answer:
Trigonometry can be used to model projectile motion, such as the flight of a baseball. Given the angle at which the ball leaves the bat and the initial velocity, you can determine the distance the ball will travel.
Step-by-step explanation:
A state department of public safety added 13x2 + 5x miles of highway to connect CoolTown and Bragville. It cost the department 3x2 – 9x thousand dollars to complete each mile of interstate. Which expression shows the total cost, in thousands of dollars, of creating this highway?
Answer:
x4 + 12x^3 – 45x^2
Step-by-step explanation:
Consider a rectangle that is inscribed with its base on the x-axis and its upper corners on the parabola y=C−x2, with C>0. What are the width and height that maximize the area of this rectangle? What is that maximal area?
The maximum area of the given rectangle will be A = 4(C/3)√(C/3).
What is surface area?The quantity of space enclosing a three-dimensional shape's exterior is its surface area.
In other meaning, if we say side square then it is an area of the square but for a cuboid, there are 6 faces so the surface area will be external to all 6 surfaces area.
As per the given rectangle inscribed in the parabola has been drawn,
The area of rectangle A = (x + x)y
A = 2x(C - x²)
A = 2Cx - 2x³
To find the maximum area, take the first derivative with respect to x.
dA/dx = 2C - 6x² = 0
C - 3x² = 0
x = √(C/3)
Therefore, the area will be as,
A = 2C√(C/3) - 2(√(C/3))³
A = 2C√(C/3) - 2(C/3)√(C/3)
A = 2√(C/3) [C - C/3]
A = 2√(C/3)(2C/3)
A = 4(C/3)√(C/3)
Hence "The maximum area of the given rectangle will be A = 4(C/3)√(C/3)".
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Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges.
[5/1*2]+[5/2*3]+[5/3*4]+...+[5/n(n+1)]+...
Answer: [tex]S_n=5(1-\dfrac{1}{n+1})[/tex] ; 5
Step-by-step explanation:
Given series : [tex][\dfrac{5}{1\cdot2}]+[\dfrac{5}{2\cdot3}]+[\dfrac{5}{3\cdot4}]+....+[\dfrac{5}{n\cdot(n+1)}][/tex]
Sum of series = [tex]S_n=\sum^{\infty}_{1}\ [\dfrac{5}{n\cdot(n+1)}]=5[\sum^{\infty}_{1}\dfrac{1}{n\cdot(n+1)}][/tex]
Consider [tex]\dfrac{1}{n\cdot(n+1)}=\dfrac{n+1-n}{n(n+1)}[/tex]
[tex]=\dfrac{1}{n}-\dfrac{1}{n+1}[/tex]
⇒ [tex]S_n=5\sum^{\infty}_{1}\dfrac{1}{n\cdot(n+1)}=5\sum^{\infty}_{1}[\dfrac{1}{n}-\dfrac{1}{n+1}][/tex]
Put values of n= 1,2,3,4,5,.....n
⇒ [tex]S_n=5(\dfrac{1}{1}-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+......-\dfrac{1}{n}+\dfrac{1}{n}-\dfrac{1}{n+1})[/tex]
All terms get cancel but First and last terms left behind.
⇒ [tex]S_n=5(1-\dfrac{1}{n+1})[/tex]
Formula for the nth partial sum of the series :
[tex]S_n=5(1-\dfrac{1}{n+1})[/tex]
Also, [tex]\lim_{n \to \infty} S_n = 5(1-\dfrac{1}{n+1})[/tex]
[tex]=5(1-\dfrac{1}{\infty})\\\\=5(1-0)=5[/tex]
The number of years a radio functions is exponentially distributed with parameter λ = 1 8 . If Jones buys a used radio, what is the probability that it will be working after an additional 8 years?
Answer:
[tex]P(X>8)=e^{-1}[/tex]
Step-by-step explanation:
The exponential distribution is "the probability distribution of the time between events in a Poisson process (a process in which events occur continuously and independently at a constant average rate). It is a particular case of the gamma distribution". The probability density function is given by:
[tex]P(X=x)=\lambda e^{-\lambda x}, x>0[/tex]
And 0 for other case. Let X the random variable that represent "The number of years a radio functions" and we know that the distribution is given by:
[tex]X \sim Exp(\lambda=\frac{1}{8})[/tex]
We can assume that the random variable t represent the number of years that the radio is already here. So the interest is find this probability:
[tex]P(X>8|X>t)[/tex]
We have an important property on the exponential distribution called "Memoryless" property and says this:
[tex]P(X>a+t| X>t)=P(X>a)[/tex]
Where a represent a shift and t the time of interest.
On this case then [tex]P(X>8|X>t)=P(X>8+t|X>t)=P(X>8)[/tex]
We can use the definition of the density function and find this probability:
[tex]P(X>8)=\int_{8}^{\infty} \frac{1}{8}e^{-\frac{1}{8}x}dx[/tex]
[tex]=\frac{1}{8} \int_{8}^{\infty} e^{-\frac{1}{8}x}dx[/tex]
[tex]=[lim_{x\to\infty} (-e^{-\frac{1}{8}x})+e^{-1}]=0+e^{-1}=e^{-1}[/tex]
Find the area and perimeter of ABC at right. Give approximate (decimal) answers, not exact answers
Answer:
Area of Δ ABC = 21.86 units square
Perimeter of Δ ABC = 24.59 units
Step-by-step explanation:
Given:
In Δ ABC
∠A=45°
∠C=30°
Height of triangle = 4 units.
To find area and perimeter of triangle we need to find the sides of the triangle.
Naming the end point of altitude as 'D'
Given [tex]BD\perp AC[/tex]
For Δ ABD
Since its a right triangle with one angle 45°, it means it is a special 45-45-90 triangle.
The sides of 45-45-90 triangle is given as:
Leg1 [tex]=x[/tex]
Leg2 [tex]=x[/tex]
Hypotenuse [tex]=x\sqrt2[/tex]
where [tex]x[/tex] is any positive number
We are given BD(Leg 1)=4
∴ AD(Leg2)=4
∴ AB (hypotenuse) [tex]=4\sqrt2=5.66 [/tex]
For Δ CBD
Since its a right triangle with one angle 30°, it means it is a special 30-60-90 triangle.
The sides of 30-60-90 triangle is given as:
Leg1(side opposite 30° angle) [tex]=x[/tex]
Leg2(side opposite 60° angle) [tex]=x\sqrt3[/tex]
Hypotenuse [tex]=2x[/tex]
where [tex]x[/tex] is any positive number
We are given BD(Leg 1)=4
∴ CD(Leg2) [tex]=4\sqrt3=6.93[/tex]
∴ BC (hypotenuse) [tex]=2\times 4=8 [/tex]
Length of side AC is given as sum of segments AD and CD
[tex]AC=AD+CD=4+6.93=10.93[/tex]
Perimeter of Δ ABC= Sum of sides of triangle
⇒ AB+BC+AC
⇒ [tex]5.66+8+10.93[/tex]
⇒ [tex]24.59[/tex] units
Area of Δ ABC = [tex]\frac{1}{2}\times base\times height[/tex]
⇒ [tex]\frac{1}{2}\times 10.93\times 4[/tex]
⇒ [tex]21.86[/tex] units square