Answer:
The height and the radius of the cylinder are 3.67 centimeters and 5.19 centimeters, respectively.
Step-by-step explanation:
The volume ([tex]V[/tex]) and the surface area ([tex]A_{s}[/tex]) of the cone, measured in cubic centimeters and square centimeters, respectively, are modelled after these formulas:
Volume
[tex]V = \frac{h\cdot r^{2}}{3}[/tex]
Surface area
[tex]A_{s} = \pi\cdot r \cdot \sqrt{r^{2}+h^{2}}[/tex]
Where:
[tex]h[/tex] - Height of the cylinder, measured in centimeters.
[tex]r[/tex] - Radius of the base of the cylinder, measured in centimeters.
The volume of the paper drinking cup is known and first and second derivatives of the surface area functions must be found to determine the critical values such that surface area is an absolute minimum. The height as a function of volume and radius of the cylinder is:
[tex]r = \sqrt{\frac{3\cdot V}{h} }[/tex]
Now, the surface area function is expanded and simplified:
[tex]A_{s} = \pi\cdot \sqrt{\frac{3\cdot V}{h} }\cdot \sqrt{\frac{3\cdot V}{h}+ h^{2}}[/tex]
[tex]A_{s} = \pi\cdot \sqrt{\frac{9\cdot V^{2}}{h^{2}} + 3\cdot V\cdot h }[/tex]
[tex]A_{s} = \pi\cdot \sqrt{3\cdot V} \cdot\sqrt{\frac{3\cdot V+ h^{3}}{h^{2}} }[/tex]
[tex]A_{s} = \pi\cdot \sqrt{3\cdot V}\cdot \left(\frac{\sqrt{3\cdot V + h^{3}}}{h}\right)[/tex]
If [tex]V = 33\,cm^{3}[/tex], then:
[tex]A_{s} = 31.258\cdot \left(\frac{\sqrt{99+h^{3}}}{h} \right)[/tex]
The first and second derivatives of this function are require to determine the critical values that follow to a minimum amount of paper:
First derivative
[tex]A'_{s} = 31.258\cdot \left[\frac{\left(\frac{3\cdot h^{2}}{\sqrt{99+h^{2}}}\right)\cdot h - \sqrt{99+h^{3}} }{h^{2}}\right][/tex]
[tex]A'_{s} = 31.258\cdot \left(\frac{3\cdot h^{3}-99-h^{3}}{h^{2}\cdot \sqrt{99+h^{2}}} \right)[/tex]
[tex]A'_{s} = 31.258\cdot \left(\frac{2\cdot h^{3}-99}{h^{2}\cdot \sqrt{99+h^{2}}} \right)[/tex]
[tex]A'_{s} = 31.258\cdot \left[2\cdot h\cdot (99+h^{2}})^{-0.5} -99\cdot h^{-2}\cdot (99+h^{2})^{-0.5}\right][/tex]
[tex]A'_{s} = 31.258\cdot (2\cdot h - 99\cdot h^{-2})\cdot (99+h)^{-0.5}[/tex]
Second derivative
[tex]A''_{s} = 31.258\cdot \left[(2+198\cdot h^{-3})\cdot (99+h)^{-0.5}-0.5\cdot (2\cdot h - 99\cdot h^{-2})\cdot (99+h)^{-1.5}\right][/tex]
Let equalize the first derivative to zero and solve the resultant expression:
[tex]31.258\cdot (2\cdot h - 99\cdot h^{-2})\cdot (99+h)^{-0.5} = 0[/tex]
[tex]2\cdot h - 99 \cdot h^{-2} = 0[/tex]
[tex]2\cdot h^{3} - 99 = 0[/tex]
[tex]h= \sqrt[3]{\frac{99}{2} }[/tex]
[tex]h \approx 3.672\,cm[/tex]
Now, the second derivative is evaluated at the critical point:
[tex]A''_{s} = 31.258\cdot \{[2+198\cdot (3.672)^{-3}]\cdot (99+3.672)^{-0.5}-0.5\cdot [2\cdot (3.672) - 99\cdot (3.672)^{-2}]\cdot (99+3.672)^{-1.5}\}[/tex]
[tex]A''_{s} = 18.506[/tex]
According to the Second Derivative Test, this critical value leads to an absolute since its second derivative is positive.
The radius of the cylinder is: ([tex]V = 33\,cm^{3}[/tex] and [tex]h \approx 3.672\,cm[/tex])
[tex]r = \sqrt{\frac{3\cdot V}{h} }[/tex]
[tex]r = \sqrt{\frac{3\cdot (33\,cm^{3})}{3.672\,cm} }[/tex]
[tex]r \approx 5.192\,cm[/tex]
The height and the radius of the cylinder are 3.672 centimeters and 5.192 centimeters, respectively.
An expression is shown below: 3pf^2 − 21p^2f + 6pf − 42p^2 Part A: Rewrite the expression by factoring out the greatest common factor. (4 points) Part B: Factor the entire expression completely. Show the steps of your work. (6 points)
Hey there! I'm happy to help!
PART A
Let's break down each terms in the expression to find the factors that make it up and see the greatest thing they all have in common
To break up the numbers, we keep on dividing it until there are only prime numbers left.
TERM #1
Three is a prime number, so there is no need to split it up.
3pf²= 3·p·f·f
TERM #2
We have a negative coefficient here. First, let's ignore the negative sign and find all of the factors, which are just 7 and 3. One of them has to be negative and one has to be positive for it to be negative. It could be either way, and when comparing to other, we might want one to be negative or positive to match another part of the expression to find the greatest common factor. So, we will use the plus or minus sign ±, knowing that one must be positive and one must be negative.
-21p²2f= ±7·±3 (must be opposite operations) ·p·p·f
TERM #3
6pf= 2·3·p·f
TERM #4
Since 42 is made up of 3 prime factors (2,3,7), one of them or all three must be negative, because two negatives would make it positive. We will use the plus-minus sign again on all three because it could be just one is negative or all three are, but we don't know. We can use these later to find the greatest common factor when matching.
-42p²= ±2·±3·±7·p·p
Now, let's pull out all of our factors and see the greatest thing all four terms have in common
TERM 1: 3·p·f·f
TERM 2: ±7·±3·p·p·f (7 and 3 must end up opposite signs)
TERM 3: 2·3·p·f
TERM 4: ±2·±3·±7·p·p (one or three of the coefficients will be negative)
Let's first look at the numbers they share. All of them have a three. We will rewrite Term 2 as -7·3·p·p·f afterwards because 3 must be positive to match. With term four, the 3 has to positive so not all three can be negative, so that means that either the 2 or 7 has to be negative, but in the end we they will make a -14 so it does not matter which one because.
Now, with variables. All of them have one p, so we will keep this.
Almost all had an f except the fourth, so this cannot be part of the GCF.
So, all the terms have 3p in common. Let's take the 3p out of each term and see what we have left. In term 4 we will combine our ±7 and ±2 to be -14 because one has to be negative.
TERM 1: f·f
TERM 2: -7·p·f
TERM 3: 2·f
TERM 4: -14·p
The way we will write this is we will put 3p outside parentheses and put what is left of all of our terms on the inside of the parentheses.
3p(f·f+-7·p·f+2·f-14·p)
We simplify these new terms.
3p(f²-7pf+2f-14p)
Now we combine like terms.
3p(f²-7pf-14p)
If you used the distributive property to undo the parentheses you could end up with our original expression.
PART B
Completely factoring means the equation is factored enough that you cannot factor anymore. The only things we have left to factor more are the terms inside the parentheses. Although there won't be something common between all of them, one might have pairs with one and not another, and this can still be factored out, and this can be put into (a+b)(a+c). Let's find what we have in common with the three terms in the parentheses.
TERM 1: f·f
TERM 2: -7·p·f
TERM 3: 2· -7·p (I just put 7 as negative and 2 as positive already for matching)
Term 1 and 2 have an f in common.
Terms 2 and 3 have a -7p in common.
So, we see that the f and the -7p are what can be factored out among all of the terms, so let's take it out of all of them and see what is left.
Term 1: f
Term 2: nothing left here
Term 3: 2
So, this means that all we have left is f+2. If we multiply that by f-7p we will have what was in the parentheses in our answer from Part A, and we cannot simplify this any further. This means that our parentheses from Part A= (f-7p)(f+2). This shows that (f-7p) is multiplied by (f+2)
Don't forget the GCF 3p; that's still outside the parentheses!
Therefore, the answer here is 3p(f-7p)(f+2).
Have a wonderful day! :D
Someone help me please
Give the coordinates of two points that lie on the hyperbola y=2/x
Answer:
(1, 2), (-2, -1)
Step-by-step explanation:
We can choose x = 1 and find y:
y = 2/1 = 2
(x, y) = (1, 2)
We can choose x = -2 and find y:
y = 2/(-2) = -1
(x, y) = (-2, -1)
Simplify -4 • -4 • -4
Answer: -64
Step-by-step explanation: Since we know that -4 x -4 is a positive, it equals 16, then a positive plus a negative equals a negative, so 16 x -4 equals -64
Answer:
-64
Step-by-step explanation:
-4 • -4 • -4
-4*-4 = 16
16*-4
-64
A production line operation is designed to fill cartons with laundry detergent to a mean weight of ounces. A sample of cartons is periodically selected and weighed to determine whether underfilling or overfilling is occurring. If the sample data lead to a conclusion of underfilling or overfilling, the production line will be shut down and adjusted to obtain proper filling.a. Formulate the null and alternative hypotheses that will help in deciding whether to shut down and adjust the production line.: - Select your answer -: - Select your answer -b. Comment on the conclusion and the decision when cannot be rejected. Is there evidence that the production line is not operating properly
Answer:
Step-by-step explanation:
The question is incomplete. The complete question is:
A production line operation is designed to fill cartons with laundry detergent to a mean weight of 32 ounces. A sample of cartons is periodically selected and weighed to determine whether under filling or overfilling is occurring. If the sample data lead to a conclusion of under filling or overfilling, the production line will be shut down and adjusted to obtain proper filling.
A. Formulate the null and alternative hypotheses that will help in deciding whether to shut down and adjust the production line.
B. Comment on the conclusion and the decision when H0 cannot be rejected.
C. Comment on the conclusion and the decision when H0 can be rejected.
Solution:
A) We would set up the hypothesis. Under filling or over filling means two ways. Thus, it is a two tailed test
For null hypothesis,
H0: μ = 32
For alternative hypothesis,
H1: μ ≠ 32
B) if H0 cannot be rejected, it means that there was insufficient evidence to reject it. Thus, it would be concluded that the production line operation filled the cartons with laundry detergent to a mean weight of 32 ounces.
C) There was sufficient evidence to reject the null hypothesis. Thus, it can be concluded that there was under filling or over filling.
There are 9 numbers written, beginning with: 8, 5, 4, 9, 1, ... Finish the sequence.
Answer:
14 -50 -29 -541
Step-by-step explanation:
8-3=5-1=4+5=9-8=1+13=14-64=-50+21=-29-512=-541
i got this by looking up sequence pattern finder in google and clicking on the second option then inserting the numbers you gave hope this helps
[PLEASE HURRY WILL GIVE BRAINLIEST] A square prism was sliced not perpendicular to its base and not through any of its vertices. What is the shape of the cross section shown in the figure?
It appears to be a parallelogram. But without actual numerical data, I don't think it's possible to prove this or not. I could be missing something though.
A team of four boys and five girls is to be chosen from a group of six boys and eight girls. How many different teams are possible?
Answer:
There are a total of 840 possible different teams
Step-by-step explanation:
Given
Number of boys = 6
Number of girls = 8
Required
How many ways can 4 boys and 5 girls be chosen
The keyword in the question is chosen;
This implies that, we're dealing with combination
And since there's no condition attached to the selection;
The boys can be chosen in [tex]^6C_4[/tex] ways
The girls can be chosen in [tex]^8C_5[/tex] ways
Hence;
[tex]Total\ Selection = ^6C_4 * ^8C_5[/tex]
Using the combination formula;
[tex]^nCr = \frac{n!}{(n-r)!r!}[/tex]
The expression becomes
[tex]Total\ Selection = \frac{6!}{(6-4)!4!} * \frac{8!}{(8-5)!5!}[/tex]
[tex]Total\ Selection = \frac{6!}{2!4!} * \frac{8!}{3!5!}[/tex]
[tex]Total\ Selection = \frac{6 * 5* 4!}{2!4!} * \frac{8 * 7 * 6 * 5!}{3!5!}[/tex]
[tex]Total\ Selection = \frac{6 * 5}{2!} * \frac{8 * 7 * 6}{3!}[/tex]
[tex]Total\ Selection = \frac{6 * 5}{2*1} * \frac{8 * 7 * 6}{3*2*1}[/tex]
[tex]Total\ Selection = \frac{30}{2} * \frac{336}{6}[/tex]
[tex]Total\ Selection =15 * 56[/tex]
[tex]Total\ Selection =840[/tex]
Hence, there are a total of 840 possible different teams
BIG Corporation advertises that its light bulbs have a mean lifetime, μ, of 2800 hours. Suppose that we have reason to doubt this claim and decide to do a statistical test of the claim. We choose a random sample of light bulbs manufactured by BIG and find that the mean lifetime for this sample is 2620 hours and that the sample standard deviation of the lifetimes is 650 hours.
In the context of this test, what is a Type II error?
A type II error is (rejecting/failing to reject) the hypothesis that μ is (less than/less than or equal to/greater than/greater than or equal to/not equal to/equal to) ____ when in fact, μ is (less than/less than or equal to/greater than/greater than or equal to/not equal to/equal to) ______.
Answer:
A type II error is failing to reject the hypothesis that μ is equal to 2800 when in fact, μ is less than 2800.
Step-by-step explanation:
A Type II error happens when a false null hypothesis is failed to be rejected.
The outcome (the sample) probability is still above the level of significance, so it is consider that the result can be due to chance (given that the null hypothesis is true) and there is no enough evidence to claim that the null hypothesis is false.
In this contest, a Type II error would be not rejecting the hypothesis that the mean lifetime of the light bulbs is 2800 hours, when in fact this is false: the mean lifetime is significantly lower than 2800 hours.
Will mark as brainliess and thanks for awnsering this simple question
Answer:
x=-2
Step-by-step explanation:
2 times -2=-4+3=-1
The time it takes me to wash the dishes is uniformly distributed between 10 minutes and 15 minutes. What is the probability that washing dishes tonight will take me between 12 and 14 minutes
Answer:
The probability that washing dishes tonight will take me between 12 and 14 minutes is 0.1333.
Step-by-step explanation:
Let the random variable X represent the time it takes to wash the dishes.
The random variable X is uniformly distributed with parameters a = 10 minutes and b = 15 minutes.
The probability density function of X is as follows:
[tex]f_{X}(x)=\frac{1}{b-a};\ a<X<b,\ a<b[/tex]
Compute the probability that washing dishes will take between 12 and 14 minutes as follows:
[tex]P(12\leq X\leq 14)=\int\limits^{12}_{14} {\frac{1}{15-10} \, dx[/tex]
[tex]=\frac{1}{5}\int\limits^{12}_{14} {1} \, dx \\\\=\frac{1}{5}\times [x]^{14}_{12}\\\\=\frac{1}{15}\times [14-12]\\\\=\frac{2}{15}\\\\=0.1333[/tex]
Thus, the probability that washing dishes tonight will take me between 12 and 14 minutes is 0.1333.
Estimate the solution to the system of equations.
Answer:
It's A
Step-by-step explanation:
Trust me i did it in geogebra
An experiment consists of dealing 7 cards from a standard deck of 52 playing cards. What is the probability of being dealt exactly 4 clubs and 3 spades?
Answer: 0.00153
Step-by-step explanation:
Given: An experiment consists of dealing 7 cards from a standard deck of 52 playing cards.
Number of ways of dealing 7 cards from 52 cards = [tex]^{52}C_7[/tex]
Since there are 13 clubs and 13 spades.
Number of ways of getting exactly 4 clubs and 3 spades=[tex]^{13}C_4\times\ ^{13}C_3[/tex]
Now, the probability of being dealt exactly 4 clubs and 3 spades
[tex]=\dfrac{^{13}C_4\times\ ^{13}C_3}{^{52}C_7}\\\\\\=\dfrac{{\dfrac{13!}{4!(9!)}\times\dfrac{13!}{3!10!}}}{\dfrac{52!}{7!45!}}\\\\=\dfrac{715\times286}{133784560}\\\\=0.00152850224271\approx0.00153[/tex]
Hence, the probability of being dealt exactly 4 clubs and 3 spades = 0.00153
6.1.3
What requirements are necessary for a normal probability distribution to be a standard normal probability distribution?
Answer:
μ = 0σ = 1Step-by-step explanation:
A standard normal probability distribution is a normal distribution that has a mean of zero and a standard deviation of 1.
What is 6 1/2 - 2 2/3 =
Answer
3 5/6 or 3.83
Step-by-step explanation:
The dollar value v(t) of a certain car model that is tyears old is given by the following
exponential function:
v(t) = 26,956(0.96)^t
What is the initial cost of the car, and what will the car be worth after 6 years? Round to
the nearest whole number.
initial cost =
value after 6 years =
Please helpppp
Answer: initial cost 26956.00 USD
value after 6 years approx= 21100.00 USD
Step-by-step explanation:
The initial cost is the price of new car , it means t ( time)=0
Substitute t by 0 in our equation and get the initial car's value
v(0)= 26956*0.96^0=26956.00 USD
The value after 6 years: substitute t by 6
v(6)=26956*0.96^6=21100.00 USD
Simplify the following expression:$$(\sqrt{6} + \sqrt{24})^2$$
Answer:
54
Step-by-step explanation:
[tex](\sqrt{6} + \sqrt{24})^2=(\sqrt{6}+2\sqrt{6})^2\\\\=(3\sqrt{6})^2=(3^2)(6)=\boxed{54}[/tex]
One driver drives 25 mph faster than another driver does. They start at the same time and after a certain amount of time, one driver has driven 90 miles, and the other driver has driven 165 miles. What are the speeds of the two drivers?
Answer:
speed of slower driver 30 mph
speed of faster driver = 55 mph
Step-by-step explanation:
let the speed of slower driver b x mph
given
One driver drives 25 mph faster than another driver does
Speed of faster driver = (x+25) mph
we know time = speed / distance
also in same time faster will travel more distance than the slower one.
thus
driver with speed (x+25) mph would have traveled 165 miles
driver with speed x mph would have traveled 90 miles
time for driver with speed (x+25) mph = 165/(x+25)
time for driver with speed x mph = 90/x
Given that
They start at the same time and after a certain amount of time, one driver has driven 90 miles, and the other driver has driven 165 miles.
\time for driver with speed (x+25) mph =time for driver with speed x mph
165/(x+25) = 90/x
165x = 90(x+25)
=> 165x = 90x + 2250
=> 165x -90x = 2250
=> 75x = 2250
=> x = 2250/75= 30
Thus, speed of slower driver 30 mph
speed of faster driver = 30+25 = 55 mph
Help???????????????????????????????????
Answer:
2Explanation:
F(2) means, value of function at x=2.
Here,you can see from the graph,from 0 to 4, it's a straight line and value of y is 2.
Hope this helps...
Good luck on your assignment....
if a to the power x by y is equal to 1 then the value of x is
Answer:
a^x/y=1 x: 0
Step-by-step explanation: w.k.t, a^0=1( any variable raised to 0 is 1)
so, here the exponent is x/y which should have been 0 so that answer was 1.
Which of the following represents the set of possible rational roots for the
polynomial shown below?
2^2+ 5^2 – 8x– 10 = 0
At a high school, 9th and 10th graders were asked whether they would prefer
robotics or art as an elective. The results are shown in the relative frequency
table.
To the nearest percent, what percentage of 10th graders surveyed preferred robotics?
Using the percentage concept, it is found that 51% of 10th graders surveyed preferred robotics, hence option B is correct.
What is a percentage?The percentage of an amount a over a total amount b is given by a multiplied by 100% and divided by b, that is:
[tex]P = \frac{a}{b} \times 100\%[/tex]
In this problem, we have that 33% out of 65% of the students are 7th graders who preferred robotics, hence the percentage is given by:
[tex]P = \frac{33}{65} \times 100\% = 51%[/tex]
Which means that option B is correct.
More can be learned about percentages at https://brainly.com/question/14398287
#SPJ1
Answer:
It's A. 61% The dude above me is wrong.
Step-by-step explanation:
I just took the test
A triangle in the xy-coordinate plane is formed by the points (3, 5), (− 1, 5) , and (3,− 6) . What is the perimeter and area of this triangle?
Answer:
Therefore, the perimeter of the triangle is 26.7 units and the area is 22 square units.
Step-by-step explanation:
Given the vertices of a triangle as: A(3, 5), B(− 1, 5), and C(3,− 6)
Since A and B are on the same y-coordinate, we have that:
AB = 3-(-1)=4 Units
Since A and C are on the same x-coordinate, we have that:
AC=5-(-6)=11 Units
Next, we determine the distance BC using the distance formula.
Given: B(− 1, 5), and C(3,− 6)
[tex]BC=\sqrt{(3-(-1))^2+(-6-5)^2}\\= \sqrt{(4)^2+(-11)^2}=\sqrt{137}$ Units[/tex]
Therefore:
Perimeter of the Triangle
[tex]= 4+11+\sqrt{137}\\ =15+\sqrt{137}$ Units\\=26.7 Units[/tex]
On plotting the triangle, it forms a right triangle such that the:
Base = 4 Units
Height = 11 Units
Therefore:
Area of a triangle [tex]=\dfrac12 *Base*Height[/tex]
Therefore:
Area of the Triangle = 0.5 X 4 X 11
=22 Square Units.
Therefore, the perimeter of the triangle is 26.7 units and the area is 22 square units.
A chemist needs 120 milliliters of a 33% solution but has only 13% and 73% solutions available. Find how many milliliters of each that should be mixed to get the desired solution.
Answer:
40 ml of 73% solution required and 80 ml of 13% solution
Step-by-step explanation:
Let x = amt of 58% solution
It say's the amt of the resulting mixture is to be 120 ml, therefore
(120-x) = amt of 13% solution
A typical mixture equation
0.73x + 0.13(120-x) = 0.33(120)
0.73x + 15.6 - 0.13x = 39.6
0.6x=24
x=40 ml of 73% solution required
and
120 - 40 =80 ml of 13% solution
The polynomial-7.5x^2 + 103 + 2142 models the yearly number of visitors (in thousands) x years after 2007 to a park. Use this polynomial to estimate the number of visitors to the park in 2021.
Answer:
In that year approximately 2114 thousand people visited the park.
Step-by-step explanation:
Since the expression [tex]y(x) = -7.5*x^2 + 103*x + 2142[/tex] models the number of visitors in the park, where x represents the number of years after 2007 and 2021 is 14 years after that, then we need to find "y" for that as shown below.
[tex]y(14) = -7.5*(14)^2 + 103*14 + 2142\\y(14) = -7.5*196 + 1442 + 2142\\y(14) = -1470 + 3584\\y(14) = 2114[/tex]
In that year approximately 2114 thousand people visited the park.
pls help me on this question
Answer:
h < 2
Step-by-step explanation:
Step 1: Distribute
10h + 40 < 60
Step 2: Subtract 40 on both sides
10h < 20
Step 3: Divide both sides by 10
h < 2
Make a matrix A whose action is described as follows: The hit by A rotates everything Pi/4 counterclockwise radians, then stretches by a factor of 1.8 along the x-axis and a factor of 0.7 along the y-axis and then rotates the result by Pi/3 clockwise radians.
Answer:
The required matrix is[tex]A = \left[\begin{array}{ccc}1.07&-0.21\\-0.86&1.35\end{array}\right][/tex]
Step-by-step explanation:
Matrix of rotation:
[tex]P = \left[\begin{array}{ccc}cos\pi/4&-sin\pi/4\\sin\pi/4&cos\pi/4\end{array}\right][/tex]
[tex]P = \left[\begin{array}{ccc}1/\sqrt{2} &-1/\sqrt{2} \\1/\sqrt{2} &1/\sqrt{2}\end{array}\right][/tex]
x' + iy' = (x + iy)(cosθ + isinθ)
x' = x cosθ - ysinθ
y' = x sinθ + ycosθ
In matrix form:
[tex]\left[\begin{array}{ccc}x'\\y'\end{array}\right] = \left[\begin{array}{ccc}cos\theta&-sin\theta\\sin \theta&cos\theta\end{array}\right] \left[\begin{array}{ccc}x\\y\end{array}\right][/tex]
The matrix stretches by 1.8 on the x axis and 0.7 on the y axis
i.e. x' = 1.8x
y' = 0.7y
[tex]\left[\begin{array}{ccc}x'\\y'\end{array}\right] = \left[\begin{array}{ccc}1.8&0\\0&0.7\end{array}\right] \left[\begin{array}{ccc}x\\y\end{array}\right][/tex]
[tex]Q = \left[\begin{array}{ccc}1.8&0\\0&0.7\end{array}\right][/tex]
According to the question, the result is rotated by pi/3 clockwise radians
[tex]R = \left[\begin{array}{ccc}cos(-\pi/3)& -sin(-\pi/3)\\-sin(\pi/3)&cos(\pi/3)\end{array}\right][/tex]
[tex]R = \left[\begin{array}{ccc}1/2&\sqrt{3}/2 \\-\sqrt{3}/2 &1/2\end{array}\right][/tex]
To get the matrix A, we would multiply matrices R, Q and P together.
[tex]A = RQP = \left[\begin{array}{ccc}1/2&\sqrt{3}/2 \\-\sqrt{3}/2 &1/2\end{array}\right] \left[\begin{array}{ccc}1.8&0\\0&0.7\end{array}\right] \left[\begin{array}{ccc}1/\sqrt{2} &-1/\sqrt{2} \\1/\sqrt{2} &1/\sqrt{2}\end{array}\right][/tex]
[tex]A = \left[\begin{array}{ccc}1.07&-0.21\\-0.86&1.35\end{array}\right][/tex]
Use the graph to find estimates of the solutions to the equation x2 + x-6=-2
Answer:
The solutions are the roots of the quadratic. They are found where the graph crosses the x-axis.
Step-by-step explanation:
Let r(t)=〈t2,1−t,4t〉. Calculate the derivative of r(t)⋅a(t) at t=2
Assuming that a(2)=〈7,−3,7〉 and a′(2)=〈3,2,4〉
ddtr(t)⋅a(t)|t=2=______
Answer:
101
Step-by-step explanation:
We are given that
r(t)=[tex]<t^2,1-t,4t>[/tex]
We have to find the derivative of r(t).a(t) at t=2
a(2)=<7,-3,7> and a'(2)=<3,2,4>
We know that
[tex]\frac{d(uv)}{dx}=u'v+v'u[/tex]
Using the formula
[tex]\frac{d(r(t)\cdot at(t))}{dt}=r'(t)\cdot a(t)+r(t)\cdot a'(t)[/tex]
[tex]\frac{d(r(t)\cdot at(t))}{dt}=<2t,-1,4>\cdot a(t)+<t^2,1-t,4t>\cdot a'(t)[/tex]
Substitute t=2
[tex]\frac{d(r(t)\cdot at(t))}{dt}_|t=2=<4,-1,4>\cdot a(2)+<4,-1,8>\cdot a'(2)[/tex]
[tex]\frac{d(r(t)\cdot at(t))}{dt}_|t=2=<4,-1,4>\cdot <7,-3,7>+<4,-1,8>\cdot <3,2,4>[/tex]
[tex]\frac{d(r(t)\cdot at(t))}{dt}_|t=2=28+3+28+12-2+32=101[/tex]
The derivation of the equation will be "101".
Differentiation:Given expression is:
r(t) = 〈t², 1 - t, 4t〉
Let,
a(2) = <7, -3, 7>
a'(2) = <3, 2, 4>
As we know,
→ [tex]\frac{d(uv)}{dx}[/tex] = u'v + v'u
By using the formula, the derivation will be:
→ [tex]\frac{d(r(t).at(t))}{dt}[/tex] = r'(t).a(t) + r(t).a'(t)
= <2t, -1, 4>.a(t) + <t², 1 - t, 4t>.a'(t)
By substituting "t = 2", we get
= <4, -1, 4>.a(2) + <4, -1, 8>. a'(2)
= <4, -1, 4>.<7, -3, 7> + <4, -1, 8>.<3, 2, 4>
= 28 + 3 + 28 + 12 - 2 + 32
= 101
Thus the response above is appropriate.
Find out more information about derivatives here:
https://brainly.com/question/22068446
Find the volume & surface area of a cylinder with radius 4 cm and height 9 cm
Answer:
V= 452.39cm³ (to 2 d.p. )
S.A. = 326.73cm² (to 2 d.p. )
Step-by-step explanation:
Vcylinder = π r² h = π (4)² (9) = 144 π = 452.3893421cm³ = 452.39cm³ (to 2 d.p. )
S.A. cylinder = 2π r h + 2π r² = 2π (4)(9) + 2π (4)² = 104π = 326.725636cm² = 326.73cm² (to 2 d.p. )