The angles in each of the four quadrants with a common reference to the acute angle 41° are as follows;
For 2nd quadrant; 139°.For 3rd quadrant; 221°.For 4th quadrant; 319°.Which angles in each quadrant have a common reference angle with 41°?It follows from the task content that the angles which have a common reference to the angle 41° in each quadrant he determined.
On this note, the angles can be determined from custom formulas for reference angles in each quadrant as follows;
For 2nd quadrant; θ = (180 - θ) = (180 - 41) = 139°.For 3rd quadrant; θ = (180 + θ) = (180 + 41) = 221°.For 4th quadrant; θ = (360 - θ) = (360 - 41) = 319°.Ultimately, the angles in each quadrant with a common reference alongside 41° are; 139°, 221° and 319°.
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Mr. Charles drinks 8 glasses of water each day. Each glass contains 240ml. How many
millimeters of water does he drink in a week?
Answer:
13440ml a week
Step-by-step explanation:
8x240=1920
1920x7=13440
13440is the amount of mls a week
Answer:
13440ml
Step-by-step explanation:
8 x 240 = 1920ml (per day)
(8 x 240) x 7 = 13440ml (7 days = 1 week)
Is the sequence {3, 6, 12, 6, 3, ...} a function? Explain.
The sequence {3, 6, 12, 6, 3, ...} is not a function
How to determine the function type?The sequence is given as
{3, 6, 12, 6, 3, ...}
In the above sequence, we can see that the sequence has repeated numbers
The repeated numbers are 3 and 6
These numbers are repeated twice
For a sequence of numbers to represent a function, the numbers in the sequence must be distinct
Hence, the sequence is not a function
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Write an equation for the intervals of a parabola with x-intercepts at (2,0) and and (-5,0) that passes through the point (1, -18).
Help is always greatly appreciated.
[tex]\quad \huge \quad \quad \boxed{ \tt \:Answer }[/tex]
[tex]\qquad\displaystyle \tt \rightarrow \: y= 3 {x}^{2} + 9x - 30[/tex]
____________________________________
[tex] \large \tt Solution \: : [/tex]
The values of x where a parabola cuts the x - axis (y = 0) are the roots of the quadratic equation.
I.e -5 and 2 for the given problem.
and the equation can be represented as :
[tex]\qquad\displaystyle \tt \rightarrow \: y = a(x - x1)(x- x2)[/tex]
where, x1 and x2 are the roots of the quadratic equation, a is a constant value (depicting strech in curve)
Now, plug in the values :
[tex]\qquad\displaystyle \tt \rightarrow \: y= a(x- 2)(x - ( - 5))[/tex]
[tex]\qquad\displaystyle \tt \rightarrow \: y = a(x- 2)(x+ 5)[/tex]
[tex]\qquad\displaystyle \tt \rightarrow \: y = a( {x}^{2} + 5x - 2x - 10)[/tex]
[tex]\qquad\displaystyle \tt \rightarrow \: y= a( {x}^{2} + 3x - 10)[/tex]
Now, we need to find the value of a, for that let's use the coordinates of a point lying on the curve (1 , -18)
[tex]\qquad\displaystyle \tt \rightarrow \: - 18 = a( {1}^{2} + 3(1) - 10)[/tex]
[tex]\qquad\displaystyle \tt \rightarrow \: - 18 = a(1 + 3 - 10)[/tex]
[tex]\qquad\displaystyle \tt \rightarrow \: - 18 = a( - 6)[/tex]
[tex]\qquad\displaystyle \tt \rightarrow \: a = ( - 18) \div ( - 6)[/tex]
[tex]\qquad\displaystyle \tt \rightarrow \: a = 3[/tex]
Now, we got all required values. let's plug the value of a in equation, and we will get the required equation of parabola.
[tex]\qquad\displaystyle \tt \rightarrow \: y= 3( {x}^{2} + 3x - 10)[/tex]
[tex]\qquad\displaystyle \tt \rightarrow \: y= 3 {x}^{2} + 9x - 30[/tex]
Answered by : ❝ AǫᴜᴀWɪᴢ ❞
Answer:
[tex]\textsf{Factored form}: \quad f(x)=3(x-2)(x+5)[/tex]
[tex]\textsf{Standard form}: \quad f(x)=3x^2+9x-30[/tex]
Step-by-step explanation:
Factored form of a quadratic function
[tex]f(x)=a(x-p)(x-q)[/tex]
where:
p and q are the x-intercepts.a is some constant.Given x-intercepts:
(2, 0)(-5, 0)Substitute the given x-intercepts into the formula:
[tex]\implies f(x)=a(x-2)(x+5)[/tex]
To find a, substitute the given point (1, -18) into the equation and solve for a:
[tex]\implies -18=a(1-2)(1+5)[/tex]
[tex]\implies -18=a(-1)(6)[/tex]
[tex]\implies -6a=-18[/tex]
[tex]\implies a=3[/tex]
Therefore, the equation of the function in factored form is:
[tex]\boxed{ f(x)=3(x-2)(x+5)}[/tex]
Expand the brackets:
[tex]\implies f(x)=3(x^2+3x-10)[/tex]
[tex]\implies f(x)=3x^2+9x-30[/tex]
Therefore, the equation of the function in standard form is:
[tex]\boxed{f(x)=3x^2+9x-30}[/tex]
The length of a rectangular billboard is 9 meters more than the width. If the perimeter is 42 meters find the dimensions.
Given, the perimeter of the rectangular billboard, P=42 m.
The length of the rectangular billboard is 9 m more than the width.
Let l be the length and w be the width of the rectangular billboard.
Then, the length can be expressed as,
[tex]l=9+w[/tex]The perimeter of a rectangle can be expressed as,
[tex]P=2(l+w)[/tex]Now, put l=9+w and P=42 m in the above equation to find w.
[tex]\begin{gathered} 42=2(9+w+w) \\ \frac{42}{2}=9+2w \\ 21=9+2w \\ 21-9=2w \\ 12=2w \\ \frac{12}{2}=w \\ 6m=w \end{gathered}[/tex]Hence, the length is,
[tex]\begin{gathered} l=9+w \\ l=9+6 \\ l=15\text{ m} \end{gathered}[/tex]Therefore, the length of the rectangular billboard is 15 m and the width of the rectangular billboard is 6 m.
your company is producing special battery packs for the most popular toy during the holiday season. the life span of the battery pack is known to be normally distributed with a mean of 250 hours and a standard deviation of 20 hours. what is the probability that a randomly chosen battery pack lasts longer than 260 hours?
The probability that a randomly chosen battery pack lasts longer than 260 hours is 0.69.
What is a normal distribution?
The normal distribution is a probability distribution that is symmetric about the mean and demonstrates that data that are closer to the mean are more likely to occur than data that are farther from the mean.
Since the life span of the battery pack is known to be Normally distributed, we would apply the formula for normal distribution which is expressed as
z = (x - µ)/σ
Where
x = life spans of battery packs.
µ = mean life span
σ = standard deviation
From the information given,
µ = 250 hours
σ = 20 hours
The probability that a battery pack lasts longer than 260 hours. It is expressed as
P(x > 260) = 1 - P(x ≤ 260)
For x = 260
z = (260 - 250)/20 = 0.5
Looking at the normal distribution table, the probability corresponding to the z score is 0.69
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The probability that a randomly chosen battery pack lasts longer than 260 hours is 0.69.
What is a normal distribution?
The normal distribution is a probability distribution that is symmetric about the mean and demonstrates that data that are closer to the mean are more likely to occur than data that are farther from the mean.
Since the life span of the battery pack is known to be Normally distributed, we would apply the formula for normal distribution which is expressed as
z = (x - µ)/σ
Where
x = life spans of battery packs.
µ = mean life span
σ = standard deviation
From the information given,
µ = 250 hours
σ = 20 hours
The probability that a battery pack lasts longer than 260 hours. It is expressed as
P(x > 260) = 1 - P(x ≤ 260)
For x = 260
z = (260 - 250)/20 = 0.5
Looking at the normal distribution table, the probability corresponding to the z score is 0.69
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Quadrilateral EFGH is inscribed inside a circle as shown below. Write a proof showing that angles H and F are supplementary.
Given
Find
showing that angles H and F are supplementary.
Explanation
Let angle F , which is subtended by minor arc GHE.
so , angle F = half of angle GHE by subtended angle theorem.
similarly , angle H is the angle subtended by major arc GFE.
so , angle H = half of the angle subtended by major arc.
since , total angle around J is 360 degree.
as we know , the sum of angle subtended by minor arc and major arc is 360 degree.
so , angle F + angle H = 1/2(360) = 180 degree.
Final Answer'
Hence , H and F are supplementary angles.
The angle a lies between 0° and 90° and is such that
2 tan²a + sec²a = 5-4 tana
Show that
3 tan²a +4 tana -4 = 0
and hence find the exact value of tan a
If a polynomial f(x) has a remainder of 8 when divided by x-4, what is f(4)?
f(4) =
Answer: [tex]f(4)=8[/tex]
Step-by-step explanation:
The Remainder Theorem states, "If a polynomial [tex]p(x)[/tex] is divided by the binomial [tex]x-a[/tex], the remainder obtained is [tex]p(a)[/tex]."
In this case, we are given the polynomial [tex]f(x)[/tex], and the polynomial [tex]f(x)[/tex] is divided by the binomial [tex]x-4[/tex]. If you were to plug in [tex]4[/tex] for [tex]x[/tex] in [tex]f(x)[/tex], you would get [tex]f(4)=8[/tex] as a result of the binomial theorem.
The point B divides AC in the ratio 2:3.
Work out vector OC in terms of a and b.
Simplify your answer.
It is said that the ratio of point B to point AC is 2: 3.
This implies;
BC/AB = 3/2
BC = 3(b - a)/2
Thus;
AB + BC Equals AC
AC equals b-a plus 3(b-a)/2.
b-a - 3b/2 - 3a/2 = AC
AC = 5b/2 - 5a/2
AC = ⁵/₂(b - a) (b - a)
Using the same vector addition law, we can state that;
OC = AC – AO
OC = ⁵/₂(b - a) - (-a) (-a)
Given that giving OA is the inverse of -a, I used -a for AO.
Thus;
OC = 5b/2 - 5a/2 + a
OC = (⁵/₂)b - (³/₂)a
OC = ¹/₂(5b - 3a) (5b - 3a)
What is vector addition law?
Objects with both direction and magnitude are referred to be vectors. When two vectors have the same magnitude and direction, they are deemed to be one and the same. A line and an arrow are used to symbolize these geometrical objects. This arrow points in the direction of the vector, and the line's length indicates the vector's magnitude. These arrows, therefore, have a starting point and an ending point. Physical quantities like velocity, displacement, and acceleration are represented by vectors.
Notation A vector is frequently written in bold letters, such as an or b.
As seen on the right side, a vector may alternatively be expressed as the letters for its head and tail with an arrow above it.
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What is the remainder when f(x) = x3 + 5x2 –32x – 7 is divided by x – 4? Show your work.
The remainder of f(x) is 385.
What are the 5 steps of long division?
The five steps of long division are given by:
Step 1: Divide
Step 2: Multiply
Step 3: Subtract
Step 4: Bring down
Step 5: Repeat
Given,
f(x) = x3 + 5x2 –32x – 7 is divided by x – 4
To find the Remainder
Now, The solution is:
[tex]x^{2} -9x+98[/tex]
[tex]x-4\sqrt{x^3+5x^2-32x-7}\\[/tex]
[tex]x^{3} - 4x^2[/tex]
- +
-----------------------------
[tex]9x^{2} -32x[/tex]
[tex]9x^{2} +36x[/tex]
- -
-------------------------------
98x -7
98x - 392
- +
----------------------------------
Remainder 385
Hence, The remainder of f(x) is 385.
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On Wednesday, they pick a total of 89 kg of strawberries.
The mean mass of each strawberry is 22 g.
Work out the number of strawberries they picked.
Give your answer correct to the nearest 10
The number of strawberry they picked is about 4050.
What does math division mean?Multiplication is the opposite of division. If 3 groups of 4 add up to 12, then 12 divided into 3 groups of equal size results in 4 in each group. Creating equal groups or determining how many people are in each group after a fair distribution is the basic objective of division.
89 kg = 89000 g conversion
89000=22=4045.45 ≈4045
{unit conversion}
≈4050
The number of strawberry {nearest lo} is about 4050
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A plane (at point P) sees a bridge (at point B) at an angle of depression on 28 degrees. The horizontal distance between the plane and the bridge is 3000 feet. Explain which trigonometric equation can be used to solve for the height of the plane (segment PA). What is the height of the plane? You must show all work and calculations to receive full credit. Round your answer to the nearest foot.
The height of the plane PA is given as; 1591 ft
How to use trigonometric ratios?
There are different trigonometric ratios such as;
cos θ = adjacent/hypotenuse
sin θ = opposite/hypotenuse
tan θ = opposite/adjacent
Now, looking at the given image showing plane and bridge indicated by lines, we can use one of the trigonometric ratios to get the height of the plane PA as;
PA/3000 = tan B
now angle B = 28 degrees from alternate angles postulate. Thus;
PA/3000 = tan 28
PA = 3000 * tan 28
PA = 3000 * 0.5317
PA = 1591 ft
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[tex]tan 28^o = \frac{PA}{3000}[/tex]
tan 28° · 3000= PA
(use a scientific calculator)
1595.128= PA
height = 1595
Albert invested money into the stock market, and the table represents his earnings. What type of function could be used to model his bank account as a function of time? Justify your answer.
Week Balance ($)
1 428
2 620
3 812
4 1,004
This is an exponential function because there is a common difference in the balance between the weeks.
This is an exponential function because there is a common ratio in the balance between the weeks.
This is a linear function because there is a common difference in the balance between the weeks.
This is a linear function because there is a common ratio in the balance between the weeks.
The correct option regarding whether this is an exponential or linear function is given as follows:
This is a linear function because there is a common difference in the balance between the weeks.
When a situation is modeled by an exponential function, and when the situation is modeled by a linear function?The function will be linear or exponential depending if there is a common difference or a common ratio between the terms, as follows:
If there is a common difference between the terms, that is, if the difference between consecutive terms is the same, a linear function models the situation.If there is a common ratio between the terms, that is, if the quotient between consecutive terms is the same, an exponential function models the situation.From the given table, the differences are given as follows:
1004 - 812 = 192.812 - 620 = 192.620 - 428 = 192.The differences are the same, hence it is a linear function and the correct option is:
This is a linear function because there is a common difference in the balance between the weeks.
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Answer:
The correct answer is (C) This is a linear function because there is a common difference in the balance between the weeks.
Step-by-step explanation:
The person up top is correct
a playground is rectangular and the length is 7/8 miles. if the area is 8/20 what is the width?
Answer:
so the width is 16/35 miles
Step-by-step explanation:
Length - 7/8 miles
Area - 8/20 miles
we know from the equation that area = Length x width
we need to rearrange so the width = Area ÷Length
width = 8/20 ÷ 7/8
= 8/20 × 8/7
= 16/35
Determine whether the two figures are similar.
The two triangles are similar.
How to find similar triangles?Similar triangles are triangles that have the same shape, but their sizes may vary.
Two triangles are said to be similar if their corresponding angles are congruent and the corresponding sides are in proportion .
The side lengths of two similar triangles are proportional. That is, if ΔABC is similar to ΔDEF , then the following equation holds:
AB / DE = AC / DF = BC / EF
Hence,
6 / 3 = 10 / 5 = 8 / 4 = 2 / 1
This common ratio is called the scale factor .
The symbol ∼ is used to indicate similarity.
Therefore, the triangles are similar.
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You have exactly 10 coins with a total
value is $1. Three of the coins are
quarters. What are the remaining coins?
The
Answer:
2 dimes and 5 pennies
Step-by-step explanation:
2 dimes (d) is 10 and 5 pennies (p) is 5
75+p+d=100
75+p=80
80+d=100
Answer: 3 quarters, 2 dimes, 1 nickel.
Step-by-step explanation: 3 quarters are worth 75 cents, 2 dimes are worth 20 cents, and 1 nickel is worth 5 cents! Hope this helps
I have missing work may someone help me with my math
2)
[tex]w^2-3,\quad w=4[/tex]When we say that w = 4, it means that we can rewrite the expression and instead of "w", we will put "4", so
[tex]w^2-3\Rightarrow4^2-3[/tex]Now we have a numerical expression to solve, we can easily solve it!
[tex]\begin{gathered} 4^2-3 \\ \\ 4\cdot4-3 \\ \\ 16-3 \\ \\ 13 \end{gathered}[/tex]Therefore, the final answer is
[tex]4^2-3=13[/tex]5)
Here we will do the same thing, the "hard" part is solving the numeric expression, it will be a little bit harder than the 2).
[tex]3(6m-17),\quad m=5[/tex]Again, repeat the same process, rewrite the expression, and instead of "m" you put "5"
[tex]3(6m-17)\Rightarrow3(6\cdot5-17)[/tex]Again, another expression to simplify, remember that we always solve what is inside ( ) first, and we have a multiplication inside ( ) so we must solve it first
[tex]3(6\cdot5-17)=3(30-17)[/tex]Now we solve the multiplication inside ( ) we can do the subctration
[tex]3(30-17)=3\cdot(13)[/tex]The last step is just to solve another multiplication
[tex]3\cdot(13)=39[/tex]Now we simplified everything we can have the final answer:
[tex]3(6\cdot5-17)=39[/tex]6)
Here we have a division, but it's similar with 5) and 2), we have
[tex]\frac{2a}{3}+13,\quad a=15[/tex]No secrets, repeat the process, but here, "a" will turn into "15", then
[tex]\frac{2a}{3}+13\Rightarrow\frac{2\cdot15}{3}+13[/tex]We can do the multiplication at the numerator of the fraction
[tex]\frac{2\cdot15}{3}+13=\frac{30}{3}+13[/tex]Now we can simplify the fraction, 30 divided by 3 is 10, then
[tex]\frac{30}{3}+13=10+13[/tex]Now just do the sum and it's done!
[tex]10+13=23[/tex]Hence the final answer is
[tex]\frac{2\cdot15}{3}+13=23[/tex]ANSWERS:
1) 43
2) 13
3) 67
4) 12
5) 39
6) 23
Solve the following problem using substitution or elimination method.
Two rental car companies are running specials this month. At Johnson’s Rentals, customers will pay $50 to rent a mid-sized car for the first day, plus $2 for each additional day (x). At Martinez Rent-a-Car, the price for a mid-sized car is $30 for the first day and $4 for each additional day (x). How many additional days (x) would it cost the same to rent from either company? What is that cost?
Let x represent the number of additional days you will rent a car and y represent the total cost of renting the car. Write a system to represent the situation, and solve to answer the question.
a) 10 additional days are required for both to cost the same.
b) The cost is 70$.
At Johan's Rental:
Rent= (50+2x)$
At Martinez Rent:
Rent= (30+4x)$
For both the cost same
By using Elimination method,
50+2x = 30+4x
50-30 = 4x-2x
20 = 2x
x = 10
10 additional days are required for both to cost the same.
Cost = 50+2x
= 50+2*10
=50+20
=70.
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Let g(x) = |2x - 10| find g(-3)
Answer:
g(-3) = 16
Step-by-step explanation:
To find g(-3), we need to plus -3 into the given function. So, g(-3) = |2(-3) - 10|. Now we solve using PEMDAS. Two times -3 is -6, then we have g(-3) = |-6-10| which is essentially g(-3) = |-16|. Lastly, we take the absolute value of |-16| which is just 16, therefore g(-3) = 16.
What is (9^6 x7^-9)^-4=
Answer:
[tex] \dfrac{7^{36}}{9^{24}} [/tex]
Step-by-step explanation:
(9^6 x7^-9)^-4=
[tex] = (9^6 \times 7^{-9})^{-4} [/tex]
[tex] = 9^{-24} \times 7^{36} [/tex]
[tex] = \dfrac{7^{36}}{9^{24}} [/tex]
What percent of 933 is 391.83?
Answer:
42%
Tell me if you need expalining!
Step-by-step explanation:
Tea that sells for $2.00 a pound is mixed with a different tea that sells for
$5.00 a pound to produce 40 pounds of a custom blended tea that will
sell for $4.25 a pound. How many pounds of the less expensive tea
should be used in the mixture?
(1) 10
(2) 15
(3) 25
(4) 30
Pls show work pls show work pls show work pls show work
10 pounds of less expensive tea should be used in the mixture.
How to find the less expensive tea price?Inferior tea price = $2.00
Superior tea price = $5.00
Blended tea price = $4.25
Solution:
Price of 40 pounds of a custom blended tea = $4.25 * 40 = $170
Let x be the inferior tea quantity and
(40 - x) be the superior tea quantity.
2x + 5(40 - x) = 4.25 * 40
2x + 200 -5x = 170
-3x = -30
x = 10
Therefore, 10 pounds of less expensive tea should be used in the mixture.
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44+400 divided by (4+62)-24
Answer:
185
Step-by-step explanation:
Answer:
18
Step-by-step explanation:
250
244
18
10
are one of the answers
give this 5 stars if correct
Solve for x Correct Answers PLS show work
Answer:
x = 8
Step-by-step explanation:
The angles shown are vertical angles
Vertical angles are congruent (equal to each other)
This means that 8x + 36 = 5x + 60
Using this equation we can solve for x
8x + 36 = 5x + 60
==> subtract 36 from both sides
8x = 5x + 24
==> subtract 5x from both sides
3x = 24
==> divide both sides by 3
x = 8
Answer:
8x + 36 = 5x + 60
8x = 5x + 24
3x = 24
x = 8
State whether the Law of Detachment, the Law of Syllogism, or the Law of the
Contrapositive was used to draw the conclusion from the given statements. If it's an
invalid conclusion, write invalid.
If you go fishing, then you will need to bring your fishing rod.
If you go fishing, then you will need to bring bait.
Therefore, if you bring your fishing rod, then you will need to bring bait.
Law of Detachment
Law of Syllogism
Law of Contrapositive
Invalid Conclusion
The statement can be described as an Invalid Conclusion.
What is an invalid conclusion?An invalid conclusion arises from a series of statements where the premises are true but the conclusion is not true. The premises in the statements above are all true.
For instance, it is true that if you go fishing you will need a rod. It is also true that if you go fishing you will need bait. But there is no laid down rule that if you bring your fishing rod, you will also need to bring a bait. So, this is an invalid conclusion.
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What is the slope of the line on the graph
Answer:
Slope = (6/1) or 6
Step-by-step explanation:
(0, -1), (1, 5)
(x₁, y₁) (x₂, y₂)
y₂ - y₁ 5 - (-1) 5 + 1 6
m = ------------- = ------------- = ---------- = -------
x₂ - x₁ 1 - 0 1 1
y - y₁ = m(x - x₁)
y - (-1) = 6(x - 0)
y + 1 = 6x - 0
-1 -1
-----------------------
y = 6x - 1
I hope this helps!
Which represents the inverse of the function f(x) = 4x?
Oh(x)=x+4
Oh(x)=x-4
Oh(x) = 3/4x
○ h(x) = 1/4x
Answer:
(d) h(x) = 1/4x
Step-by-step explanation:
You want to know the inverse of the function f(x) = 4x.
Inverse functionThe inverse of y = f(x) will be the solution to x = f(y).
x = f(y)
x = 4y . . . . . . substitute the argument in the function definition
x/4 = y . . . . . divide by 4 to solve for y
The inverse function is ...
h(x) = (1/4)x
A full moon occurs every 30 days. If the last full moon occurred on a Friday, how many days will pass before a full moon occurs again on a Friday?
(I wrote the problem correctly, and its not 30)
Answer:
210 days
Step-by-step explanation:
7 x 30 = 210 days
Answer:
210
Step-by-step explanation:
A Friday occurs every 7 days.
A new moon occurs (approximately) every 30 days.
We need the smallest number that is a multiple of both 7 and 30.
We need the least common multiple (LCM) of 7 and 30.
This is the same as finding the least common denominator for two fractions that have denominators 7 and 30.
First, we find the prime factorizations of the two numbers.
7 = 7
30 = 2 × 3 × 5
The LCM is the product of common and not common factors with the larger exponent.
LCM = 2 × 3 × 5 × 7 = 210
Answer: 210
P.S. This is the correct answer for this problem, but the problem is stated incorrectly since a new moon occurs approximately every 29.5 days, so in reality, the true answer is not 210 days.
chegg a spherical ball is measured to have a radius of 5 mm, with a possible measurement error of 0.1 mm. use the differential to estimate the possible change in volume (in mm3) resulting from the error in measuring the radius.
The possible change in volume that results from the error in measuring the radius is 32. 06 millimeters
How to determine the volume of the sphereIt is important to note that the formula for volume of a sphere is expressed by;
Volume = 4/3 πr³
Given that;
r is the radius of the sphereπ has the value 3.14Based on the information given, we have the measure of the radius to be 5 mm, with a possible measurement error of 0.1 mm
The original value of radius = 5mm
Radius after the measurement error = 5 + 0. 1 = 5. 01mm
Let's substitute the values into the formula
Volume = 4/ 3 (3.14)(5)³
We then expand the bracket
Volume = 4/ 3 (392.5)
Volume = 523. 3 cubic millimeters
For radius of the measurement error = 5.01mm
Volume = 4/ 3 (3.14)(5.1)³
Then, expand the bracket
Volume = 4/3 (416.52)
Volume = 555. 4 cubic millimeters
Change in volume = 555. 4 - 523. 3 = 32. 06 cubic millimeters
Hence, the change in volume is given as 32. 06 cubic millimeters
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use lagrange multipliers to prove that the rectangle with maximum area that has a given perimeter p is a square. let the sides of the rectangle be x and y and let f and g represent the area (a) and perimeter (p), respectively. find the following.
we get x=y =0 which gives 0 perimeter or x=y this implies rectangle must be a square
1. Method of Langrage Multipliers:
To find the extremum values of f(x,y) subject to constraint g(x,y) = k
find all values of x,y and λ, such that :
Δλ(f,x) = λΔg(x,y)
And g(x,y) = k
2. let the two side of the rectangle be x and y
therefore
f(x,y) = xy And g(x,y)= 2(x+y)=p
fₓ=λgₓ => y=2λ ----------------- 1
fy = λgy => x = 2λ----------------2
using equation and 1 and 2
λ=0, but this is not possible because tis implies x = y = 0, which gives 0 perimeter
or
x=y
Hence the rectangle must be a square
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