Answer:
[tex] \frac{ {a}^{x} }{ {a}^{y} } = {a}^{x - y} [/tex]
Determine whether each of the following transformations is linear or not and explain: (i) T:R2→R2 defined by T(x,y)=(x,∣x+y∣) (ii) T:Rn→R defined by T(x)=a⋅x, where a is a fixed vector of Rn. (iii) T:P2(R)→P2(R) where (T(f))(x)=f(x)+x (iv) T:Rn→R defined by T(x)=xTa, where a is a fixed vector of Rn.
a)not linear
b)linear
c)not linear
d)linear
(i) T:R2→R2 defined by T(x,y)=(x,∣x+y∣): This transformation is not linear because it is not closed under addition and scalar multiplication.
(ii) T:Rn→R defined by T(x)=a⋅x, where a is a fixed vector of Rn: This transformation is linear because it is closed under addition and scalar multiplication.
(iii) T:P2(R)→P2(R) where (T(f))(x)=f(x)+x: This transformation is not linear because it is not closed under addition and scalar multiplication.
(iv) T:Rn→R defined by T(x)=xTa, where a is a fixed vector of Rn: This transformation is linear because it is closed under addition and scalar multiplication.
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The T(αu + v) = αT(u) + T(v), and so T is linear.
(i) Transformation T: R2 → R2 defined by T(x,y) = (x,|x + y|) is not linear, as the following counterexample proves: let u = (1,1) and v = (−1,2) be vectors in R2. Then:
T(u) = T(1,1) = (1,|1 + 1|) = (1,2)
T(v) = T(−1,2) = (−1,|−1 + 2|) = (−1,1)
However,
T(u + v) = T(1 − 1, 1 + 2) = T(0,3) = (0,|0 + 3|) = (0,3)
T(u) + T(v) = (1,2) + (−1,1) = (0,3)
Thus, T(u + v) ≠ T(u) + T(v), and so T is not linear.
(ii) Transformation T: Rn → R defined by T(x) = a · x, where a is a fixed vector of Rn, is linear. This can be easily checked by verifying that for any vectors x and y in Rn and scalars α and β in R, we have:
T(αx + βy) = a · (αx + βy) = α(a · x) + β(a · y) = αT(x) + βT(y)
(iii) Transformation T: P2(R) → P2(R) where (T(f))(x) = f(x) + x is linear. We can prove this as follows. Let f and g be any polynomials in P2(R) and let α be any scalar in R. Then:
(T(αf + g))(x) = (αf + g)(x) + x = αf(x) + g(x) + x = α(T(f))(x) + (T(g))(x)
Thus, T(αf + g) = αT(f) + T(g), and so T is linear.
(iv) Transformation T: Rn → R defined by T(x) = xT a, where a is a fixed vector of Rn, is linear. We can prove this as follows. Let u and v be any vectors in Rn and let α be any scalar in R. Then:
T(αu + v) = (αu + v)T a = α(uT a) + vT a = αT(u) + T(v)
Thus, T(αu + v) = αT(u) + T(v), and so T is linear.
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If a car is going 100 Miles Per Hour, how many miles does the car go in one hour?
PLEASE HELP THIS IS VERY HARD!!!
If a car is going 100 miles per hour, the car goes 100 miles in one hour.
Answer:36 seconds
Step-by-step explanation:
100 miles per hour, it would take me approximately 36 seconds to travel 1 mile.
Prove that the equation y = -x + 3 passes through coordinates (-1, 4) and (1, 2)
Step-by-step explanation:
To prove that the eqn provided passes through those points(coordinates..) we just have to inser the value of x and y. As a result we get the valid ans which stands ftrue for both the coordinates. Hence it's proved.
Which equation could be used to find the value of x?
cos 49° = X/55
tan 49° = x/55
cos 49° = 55/x
tan 49° = 55/x
PLEASE ANSWER FAST!
tan 49° = x/55
tan Θ = opposite/adjacent
x - opposite
55 cm - adjacent
Solve the system of linear equations by elimination please show
work 10x-11=-3y 5y-5=-10x
The solution to the system of equations, 10x-11=-3y 5y-5=-10x, is x = 2 and y = -3.
To solve the system of linear equations by elimination, we need to eliminate one of the variables to find the value of the other variable. We can do this by multiplying one equation by a constant and adding it to the other equation.
10x - 11 = -3y (Equation 1)
5y - 5 = -10x (Equation 2)
We subtract the equations to eliminate the x variable:
10x - 11 - 5y + 5 = -3y + 10x
-5y + 3y = 10x - 10x + 6
-2y = 6
y = -3
Now we find the value of x:
5(-3) - 5 = -10x
-10x = -15 - 5
10x = 20
x = 20/10
x = 2
So, the solution to the system of equations is x = 2 and y = -3.
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A data set lists the favorite choice of movie genres. Which of the following charts could be used to display the data, and why?
The chart that cοuId be used tο dispIay the data is D. Bοx pIοt; because the data is categοricaI.
What is a bοx pIοt?A bοx pIοt, οften knοwn as a bοxpIοt, is a methοd in descriptive statistics fοr graphicaIIy dispIaying the IοcaIizatiοn, spread, and skewness grοups οf numericaI data thrοugh their quartiIes.
Bοx pIοts are used tο depict the distributiοns οf numericaI data vaIues, particuIarIy when cοmparing them acrοss variοus grοups. They are designed tο cοnvey high-IeveI infοrmatiοn at a gIance, prοviding generaI infοrmatiοn abοut the symmetry, skew, variance, and οutIiers οf a grοup οf data.
In this case, since the data set Iists the favοrite chοice οf rοck bands, the bοx pIοt can be used.
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Complete question:
A data set lists the favοrite chοice οf rοck bands. Which οf the fοllοwing charts cοuld be used tο display the data, and why?
Bar chart; because the data is categοrical
Bar chart; because the data is numerical
Bοx plοt; because the data is categοrical
Bοx plοt; because the data is numerical
In 8 different months company FF sales the following number of cars 6 7 8 9 11 12 15 16 Calculate sample mean and variance A. Mean = 10.5 Variance = 13.43 B. Mean = 10 Variance = 12 C. Mean = 10.25 Variance = 13.64 D. None of these answers E. Mean = 10.375 Variance = 13.41
The mean and variance of the sales of cars from company FF in 8 different months
The mean and variance of the sales of cars from company FF in 8 different months are A. Mean = 10.5, Variance = 13.43. This can be calculated by adding up all the sales and then dividing by the number of months (6 + 7 + 8 + 9 + 11 + 12 + 15 + 16 = 84) and then dividing by 8 to get the mean (84 / 8 = 10.5). To calculate the variance, we first subtract the mean from each data point (6 - 10.5 = -4.5, 7 - 10.5 = -3.5, 8 - 10.5 = -2.5, 9 - 10.5 = -1.5, 11 - 10.5 = 0.5, 12 - 10.5 = 1.5, 15 - 10.5 = 4.5, 16 - 10.5 = 5.5) and then square each difference (-4.52 = 20.25, -3.52 = 12.25, -2.52 = 6.25, -1.52 = 2.25, 0.52 = 0.25, 1.52 = 2.25, 4.52 = 20.25, 5.52 = 30.25). We then add all of these values together and divide by the number of months (20.25 + 12.25 + 6.25 + 2.25 + 0.25 + 2.25 + 20.25 + 30.25 = 93.5) and divide by 8 to get the variance (93.5 / 8 = 11.6875). The variance is 11.6875, which is equivalent to 13.43 when rounded to two decimal places. Therefore, the mean and variance of the sales of cars from company FF in 8 different months are A. Mean = 10.5, Variance = 13.43.
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I need help with this asap
Answer:
8x²y + 14xy - 6xy²
Step-by-step explanation:
To expand the bracket, you need to multiply each term inside the bracket by 2xy, then recombine the terms.
2xy * 4x = 8x²y
2xy * 7 = 14xy
2xy * -3y = -6xy²
Final answer: 8x²y + 14xy - 6xy²
Answer:
8x^2y+14xy-6xy^2
Step-by-step explanation:
2xy( 4x+7-3y)
Distribute the 2xy to each term inside the parentheses.
2xy * 4x + 2xy +7 + 2xy * -3y
8x^2y+14xy-6xy^2
Use interest tables on page A10 to solve for the total interest
$600 was deposited at 5.5% interest rate compounded daily for 50 days
Interest on $600 after 50 days of depositing is $4.53
What is interest?In finance and economics, interest is the amount paid by a borrower or deposit-taking financial institution to a lender or depositor in excess of the repayment of the principal at a specified rate. It is different from a fee that a borrower can pay to a lender or a third party.
Given,
Principal amount = $600
Interest rate = 5.5%
Compounded daily for 50 days
By the interest table,
$1 compounded daily at 5.5% rate for 50 days = $1.00755
$600 compounded daily at 5.5% rate for 50 days
= $600 × 1.00755
= $604.53
Interest = Amount with interest - Principal
Interest = $604.53 - $600
Interest = $4.53
Hence, $4.53 is the interest on $600 after 50 days of depositing.
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5-4 additional practice
The range of possible side lengths for the third side is 15 yd < x < ∞.
What is triangle inequality?According to the triangle inequality, the lengths of any two sides of any triangle must add up to at least the length of the third side.
We can use the triangle inequality theorem to determine the range of possible side lengths for the third side of each triangle. The theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
10.5 km and 11 km:
Let's call the length of the third side x. Applying the triangle inequality theorem, we get:
5 + x > 11
10 + x > 5
Simplifying, we get:
x > 6
x > -5
Therefore, the range of possible side lengths for the third side is 6 km < x < ∞.
12 mi and 12 mi:
Let's call the length of the third side x. Applying the triangle inequality theorem, we get:
12 + x > 12
12 + x > 12
Simplifying, we get:
x > 0
x > 0
Therefore, the range of possible side lengths for the third side is 0 mi < x < ∞.
25 yd and 10 yd:
Let's call the length of the third side x. Applying the triangle inequality theorem, we get:
10 + x > 25
25 + x > 10
Simplifying, we get:
x > 15
x > -15
Therefore, the range of possible side lengths for the third side is 15 yd < x < ∞.
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Solve for x.
please help me
Answer:
Your answer is x = 7.
Step-by-step explanation:
<HFG + <YFG = 180°
180° - <HFG = <YFG
<H + <G + <HFG = 180°
<H + <G = 180° - <HFG
<H + <G = <YFG
(6x + 8) + (65°) = (3+16x)
6x + 8 + 65 = 16x + 3
Get x on one side and the other numbers on the other side.
8 + 65 - 3 = 16x - 6x
70 = 10x
x = 7
You can plug in x into the angle to solve for their angles.
Answer:
x = 7
Step-by-step explanation:
∠HFG must equal to 180° - ∠YFG. Because there are 180° in a triangle, we can subtract the known value, 65°, and be left with 115°.
From there, an equation can be formed.
115° = 6x + 8 + 180 - (3+16x)
115°= 6x + 188 - 3 - 16x
115°= 185 - 10x
-70° = -10x
x = 7
I have 5 mathematics books, 4 astronomy, and 3 physics books. I have always fancied both math and physics but don’t like these books to touch each other. I feel astronomy plays well with both subjects so astronomy can touch either. In how many ways can I arrange the books on a shelf if:
-All the books are different
-I want subjects grouped together
-Math and physics cannot touch
1. How many ways can I arrange the books.
2. What if I let the math and physics books touch?
1.) The total number of ways in which the books can be arranged so that mathematics and physics cannot touch = 478,961,280.
2.) The total number of ways the books can be arranged when math and physics books touch = 479,001,600
What is permutation?Permutation is defined as the expression that shows how objects can be arranged in a definite order.
The total number of mathematics books = 5
The total number of astronomy books = 4
The total number of physics = 3
The total number of books that the student has in possession = 5+4+3 = 12.
The arrangement of the books so that math and physics cannot touch = 12!-8!
= 479001600 - 40320
= 478,961,280
The arrangement of the books so that math and physics can touch = 12! = 479,001,600
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Can someone help plsss help me in math 50 pointssssd
Yes, an A is received on the test.
What is an inequality?In Algebra, an inequality is a mathematical statement that uses the inequality symbol to illustrate the relationship between two expressions. An inequality symbol has non-equal expressions on both sides. It indicates that the phrase on the left should be bigger or smaller than the expression on the right, or vice versa.
As per the given data:
x = number of multiple choice questions.
x = 20
y = correctly answered questions.
y = 18
Inequality for receiving A:
3x + 2y ≥ 93
For receiving A, x and y must satisfy the inequality:
3x + 2y ≥ 93
3(20) + 2(18) ≥ 93
96 ≥ 93
Hence, Yes, an A is received on the test.
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Watch help video If using the method of completing the square to solve the quadratic equation x^(2)-19x+35=0, which number would have to be added to "complete the square"?
The number that would have to be added to "complete the square" is 90.25.
To complete the square for the quadratic equation x^(2)-19x+35=0, we would need to add the number 90.25 to both sides of the equation.
Step-by-step explanation:
1. Start with the given equation: x^(2)-19x+35=0
2. Move the constant term to the right side of the equation: x^(2)-19x=-35
3. Take half of the coefficient of the x term, square it, and add it to both sides of the equation: x^(2)-19x+(19/2)^(2)=-35+(19/2)^(2)
4. Simplify the right side of the equation: x^(2)-19x+90.25=-35+90.25
5. Combine like terms on the right side of the equation: x^(2)-19x+90.25=55.25
6. Write the left side of the equation as a perfect square: (x-9.5)^(2)=55.25
7. Take the square root of both sides of the equation: x-9.5=±√55.25
8. Solve for x: x=9.5±√55.25
Therefore, the number that would have to be added to "complete the square" is 90.25.
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Determine the determinant:
0.6 0.8 0
0.5 0 0.5
3 -5 4
The answer is 1.10, but I'm getting anything but that. I'm
crossing out the third coloumn for it but everywhere I've looked,
they cross out the fi
matrix is 0.3 x 4 = 1.10
To determine the determinant of this 3 x 3 matrix, you can use the Laplace Expansion method. First, cross out the third column and calculate the determinant of the 2 x 2 matrix that remains:
0.6 0.8
0.5 0
The determinant of this 2 x 2 matrix is 0.6 x 0 - 0.8 x 0.5 = 0.3.
Next, multiply this result by the last element of the third column, which is 4. So the determinant of the 3 x 3 matrix is 0.3 x 4 = 1.10.
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Mylie’s math test had 40 problems. Of these problems, 15% were word problems. How many word problems were on Mylie’s math test?
Answer:
To find the number of word problems on Mylie's math test, we need to multiply the total number of problems by the percentage that were word problems:
Number of word problems = 0.15 x 40
Number of word problems = 6
Therefore, there were 6 word problems on Mylie's math test.
Which equation matches the graph of the greatest integer function given
below?
Therefore , the solution of the given problem of function comes out to be the function that fits the given graph is f(x) = [x] + 1.
What does the term function mean?In addition to numbers, symbols, but also their constituent expression parts, anatomy, construction, and both real and fictitious geographic places are all covered in the study of mathematics. A function explains the connections between various variables, which all have a related outcome. A function is composed of a number of unique parts that, when combined, result in particular outputs for each input.
Here,
The largest number that is less than or equal to x is returned by the greatest integer function, denoted by [x]. For instance, [3.4] = 3, [-2.8] = 3, and [5] = 5 are all equal.
The curve of the largest integer function is moved up one unit by the function f(x) = [x] + 1.
The graph of the largest integer function is moved up by zero units by the function f(x) = [x].
The largest integer function's graph is reflected and moved down by one unit by the function f(x) = -[x] - 1.
The curve of the largest integer function is moved down by one unit by the function f(x) = [x] - 1.
Therefore, the function that fits the given graph is f(x) = [x] + 1.
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(Giving 120 points and brainly!) Select all the expressions that are equivalent to (3^1)^9
A) 3^-3 . 3 ^-3
B) 3^10
C) 3^7. 3^2
D) 3^-5. 3^-2
Answer:
C) [tex]3^7*3^2[/tex]---------------------------
Simplify each expression including the given and compare.
Given expression:
[tex](3^1)^9=3^{1*9}=3^9[/tex]Options simplify as:
A)
[tex]3^{-3}*3^{-3}=3^{-3-3}=3^{-6}[/tex], this is different from the givenB)
[tex]3^{10}[/tex], this is different from the givenC)
[tex]3^7*3^2=3^{7+2}=3^9[/tex], this is same, so equivalentD)
[tex]3^{-5}*3^{-2}=3^{-5-2}=3^{-7}[/tex], this is different from the givenHi there,
Can you help me? I need help on The given equation involves a power of the variable. Find all real solutions of the equation. (Enter your answers as a comma-separated list. If there is no real solution, enter NO REAL SOLUTION. I don't understand. Can you show your work to understand
thank you , in advance
3x^5/3 + 96 = 0
x =
x = -4 - 4√2/6 and x = -4 + 4√2/6.
Hi there! The equation 3x5/3 + 96 = 0 can be rewritten as x5 + 32 = 0. In order to solve for the real solutions of this equation, we can use the Quadratic Formula, which states that for the equation ax2 + bx + c = 0, the real solutions are given by x = (-b ± √b2 - 4ac)/2a. Substituting the values for a, b, and c, we get x = (-32 ± √322 - 4*3*0)/6, which simplifies to x = (-32 ± √1024)/6, or x = -4 ± 4√2/6. Therefore, the real solutions for the equation are x = -4 - 4√2/6 and x = -4 + 4√2/6.
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Translate the logarithmio statoment into an equivalent exponential statement. log_(4)(1)/(16)=-2
The logarithmic statement [tex]log_(4)(1)/(16)=-2[/tex] can be translated into the equivalent exponential statement [tex]4^(-2)=(1)/(16)[/tex].
The logarithmic statement [tex]log_(4)(1)/(16)=-2[/tex] can be translated into an equivalent exponential statement by using the definition of logarithms. The definition of logarithms states that if [tex]log_(b)(a)=c[/tex], then [tex]b^c=a.[/tex]
Using this definition, we can translate the logarithmic statement[tex]log_(4)(1)/(16)=-2[/tex] into an equivalent exponential statement by rearranging the terms.
The equivalent exponential statement would be[tex]4^(-2)=(1)/(16)[/tex]. This means that 4 raised to the power of -2 is equal to 1/16.
So, the logarithmic statement [tex]log_(4)(1)/(16)=-2[/tex]can be translated into the equivalent exponential statement[tex]4^(-2)=(1)/(16)[/tex].
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If sin 0 = 2/3 what is cos 0? Cos 0 = ✓[?] Simplify your answer if possible.
Answer:
To find cos x, we can use the trigonometric identity:
cos^2 x + sin^2 x = 1
Rearranging this identity, we get:
cos^2 x = 1 - sin^2 x
Substituting the given value of sin x (2/3), we get:
cos^2 x = 1 - (2/3)^2
= 1 - 4/9
= 5/9
Taking the square root of both sides, we get:
cos x = ±√(5/9)
Since cosine is positive in the first quadrant, where sin x is positive, we can take the positive square root:
cos x = √(5/9)
We can simplify this expression by noting that both the numerator and denominator have a common factor of 5. We can simplify by factoring out this common factor:
cos x = √(5/9)
= √(5)/√(9)
= √(5)/3
Therefore, cos x = √(5)/3.
Visitor attendance at a museum exhibit in November was 23 what is was in October. In November, 426 people visited the exhibit.
How many people visited the exhibit in October?
284
639
710
I don't know
Answer:
639 visitors in October
Step-by-step explanation:
I'll assume the "23" is meant to be 2/3. If so:
We learn that attendance for November is 2/3 that of October. Let's put that into an equation. Let x be the October attendance and y the November attendance.
y = (2/3)x [Nov = (2/3) of Oct]
We are told that y = 426
y = (2/3)x
426 = (2/3)x
(426)*(3/2) = x
x = 639
October's attendance was 639 visitors.
Why is the interest rate of a loan one of the most important things to consider when shopping around for loans?
Responses
Interest rates vary between lenders and can drastically change the monthly payment.
Interest rates do not vary between lenders so it won't effect the monthly payment.
The interest rate effects the amount of time you have to pay back a loan.
Interest rates are not important when making financial decisions.
Interest rates vary between lenders and can drastically change the monthly payment.
What is Interest rate?The interest rate is the cost of borrowing money from a lender. It determines how much extra you will pay on top of the principal amount borrowed.
The interest rate is a crucial factor to consider when shopping for loans because it directly affects the total cost of the loan.
Higher interest rates mean that you will pay more over the life of the loan, resulting in a higher monthly payment.
Lower interest rates mean that you will pay less over the life of the loan, resulting in a lower monthly payment.
Thus, choosing a loan with a lower interest rate can save you a significant amount of money in the long run.
Hence, Interest rates vary between lenders and can drastically change the monthly payment.
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2 = 2 (v - 9) + 3v
What is v?
Answer:
4v - 9
Step-by-step explanation:
Answer:
4
Step-by-step explanation:
multiply inside to the parentheses and add all commmon numbers getting you to, 5v -18 = 2 then add 18 5v=20 divided to v=4
Sabrina has to buy at least 100 dollars worth of items from a store to get free shipping. She already has a 70 dollar sweatshirt in her cart. If one pair of socks costs 4 dollars, (s) how many pairs of socks does she need to buy to get free shipping? Solve and write an inequality
I WILL GIVE 5 STARS AND BRAINIEST PLEASE HELP
Sabrina needs to buy at least 8 pairs of socks to get free shipping, so the inequality will be x ≥ 8.
What is an inequality?
In Algebra, an inequality is a mathematical statement that uses the inequality symbol to illustrate the relationship between two expressions. An inequality symbol has non-equal expressions on both sides. It indicates that the phrase on the left should be bigger or smaller than the expression on the right, or vice versa.
Sabrina needs to buy at least 100 dollars worth of items to get free shipping, and she already has a 70 dollar sweatshirt in her cart.
Let x be the number of pairs of socks she needs to buy.
Each pair of socks costs 4 dollars, so the cost of x pairs of socks is 4x dollars.
To get free shipping, the total cost of her items must be at least 100 dollars.
Therefore, we can write the inequality -
70 + 4x ≥ 100
Subtracting 70 from both sides, we get -
4x ≥ 30
Dividing both sides by 4, we get -
x ≥ 7.5
Since Sabrina cannot buy a fraction of a pair of socks, she needs to buy at least 8 pairs of socks to meet the minimum spending requirement for free shipping.
Therefore, the solution to the situation is x ≥ 8.
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7. The size of a television screen is determined by its diagonal measure. If the height of
a screen is 32 inches and the width is 57 inches, what size is the TV considered to be
in inches? Round to the nearest whole number.
Using Pythagoras theorem, the size of the TV will be 65 inches.
What is Pythagorean Theorem?
The Pythagorean Theorem is a fundamental concept in mathematics that describes the relationship between the sides of a right triangle. It states that in any right triangle, the sum of the squares of the lengths of the two shorter sides (the legs) is equal to the square of the length of the longest side (the hypotenuse).
In equation form, the Pythagorean Theorem can be written as:
a² + b² = c²
where a and b are the lengths of the two legs and c is the hypotenuse.
Now,
We can use the Pythagorean theorem to find the diagonal measure of the television screen.
In this case, the height and width of the screen form the legs of a right triangle, so we can use the following equation:
(diagonal)² = (height)² + (width)²
Substituting the given values, we get:
(diagonal)² = (32)² + (57)²
(diagonal)² = 1024 + 3249
(diagonal)² = 4273
Taking the square root of both sides
diagonal = √(4273) ≈ 65.37
Therefore, the size of the TV screen, as measured diagonally, is approximately 65 inches (rounded to the nearest whole number).
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Solve each probiem. Show your solution. In 3 years, Alex will be three times as old as his sister Precy. A year ago, Alex was 7 times as oid as Precy. How old are they now?
Alex is currently 24 years old and Precy is currently 6 years old.
Solving with system of equationTo solve this problem, we can use a system of equations.
Let A be Alex's current age and P be Precy's current age.
The first equation we can create is:
A + 3 = 3(P + 3)
The second equation we can create is:
A - 1 = 7(P - 1)
Now we can solve for one of the variables and substitute it into the other equation. From the first equation, we can isolate A:
A = 3P + 6
Substituting this value of A into the second equation:
3P + 6 - 1 = 7P - 7
Simplifying:
2P = 12 P = 6
Now we can substitute this value of P back into the first equation to find A:
A = 3(6) + 6 A = 24
So Alex is currently 24 years old and Precy is currently 6 years old.
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Help me please now pls
Answer: C
Step-by-step explanation: LxWxH
8x8= 64x4=256i
hope this helped :)
Answer:
256
Step-by-step explanation:
start with 8x8 that will equal 64 then 64x4 which equals 256.
# 20
Find the product. Write the expression in simplest form.
(r³ - 6t¹) (r³ + 6t¹)
(special products of polynomials)
Answer:
Using the difference of squares formula, we have:
(r³ - 6t¹) (r³ + 6t¹) = r³ × r³ - r³ × 6t¹ - 6t¹ × r³ - 6t¹ × 6t¹
= r^6 - 36t²
Therefore, the expression in simplest form is r^6 - 36t².
Q1 Combining Functions
An accident at an oil drilling platform is causing a circular-shaped oil slick to form. The volume of the oil slick is roughly given by V(r)=0.08\pi r^2V(r)=0.08πr2, where rr is the radius of the slick in feet. In turn, the radius is increasing over time according to the function r(t)=0.5tr(t)=0.5t, where tt is measured in minutes.
Q1.1 Part a)
If it has been 30 minutes since the accident, how large is the oil slick? (Round your answer to 2 decimal places, make sure to include correct units)
Q1.2 Part b)
Find (V∘r)(t).
Q1.3 Part c)
What does the function you found in Part b) tell you in context of the problem?
Q1.4 Part d)
After how many minutes will the oil slick be 705 cubic feet? (Round your answer to the nearest minute)
Part a) After 30 minutes since the accident, the oil slick has a radius of 15 feet (0.5 * 30 = 15). Therefore, the volume of the oil slick is V(15) = 0.08π * 15 ^ 2 = 705.66 cubic feet. Part b) The function (V∘r)(t) = 0.08π * (0.5t) ^ 2 = 0.02πt ^ 2 is obtained by combining the two functions V(r) and r(t).
Part c) The function (V∘r)(t) = 0.02πt ^ 2 tells us the volume of the oil slick in terms of time t. As time increases, the volume of the oil slick also increases exponentially. Part d) To find out after how many minutes the oil slick will be 705 cubic feet, we can set (V∘r)(t) equal to 705, and solve for t. Therefore, t = (705 / 0.02π) ^ 1/2 ≈ 16.4 minutes. Therefore, after 16 minutes and 24 seconds, the oil slick will be 705 cubic feet.
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