Answer:
There are 60 seconds in a minute, so to convert meters per minute to meters per second, we need to divide by 60.
90 meters per minute ÷ 60 = 1.5 meters per second
Therefore, the car's speed is 1.5 meters per second.
Step-by-step explanation:
Step-by-step explanation:
There are 60 in a minute, so to convert meters per minute to meters per second, we need to divide by 60
90 meters per minute ÷ 60 = 1.5 meters per second
The Water Department checks the city water supply on a regular basis for
contaminants such as trihalomethanes (THMs). The Water Department takes
200 samples and estimates that the concentration of THMs in your drinking
water is 3 ppb (parts per billion), with a standard deviation of 0. 3 ppb.
Assuming the samples were random and unbiased, how much confidence
can you have in this data?
We can be 95% confident that the true mean concentration of THMs in your drinking water lies within the range [2.9584 ppb, 3.0416 ppb]. This was calculated using a confidence interval formula for the mean.
The confidence interval for the mean concentration of THMs can be calculated using the formula x ± z * (s/√n), where x is the sample mean, z is the z-score for a given confidence level, s is the sample standard deviation and n is the sample size.
In this case, we have x = 3 ppb, s = 0.3 ppb, and n = 200. To calculate the confidence interval at a certain confidence level (e.g. 95%), we need to find the corresponding z-score. For a 95% confidence level, the z-score is approximately 1.96.
Substituting these values into the formula above gives us:
3 ± 1.96 * (0.3/√200) = [2.9584, 3.0416]
So we can be 95% confident that the true mean concentration of THMs in your drinking water lies within the range [2.9584 ppb, 3.0416 ppb].
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A rectangular portrait is 1 yard wide and 2 yards high. It costs $10.17 per yard to put a gold frame around the portrait. How much will the frame cost?
Answer:
The perimeter of the portrait is $2\times(1+2)=6$ yards.
To find the cost of the frame, we need to calculate the length of the frame needed, which is the same as the perimeter of the portrait. So, the length of the frame needed is 6 yards.
The cost of the frame will be the cost per yard multiplied by the length of the frame:
$6~\text{yards} \times $10.17/\text{yard} = $61.02$
Therefore, the frame will cost $61.02.
What is the length of the hypotenuse?
8ft 6ft c=?
Answer:10ft
Step-by-step explanation:
we must use pythagorean theorem
[tex]which \ is \ a^{2} +b^{2} =c^{2} \\6^{2} + 8^{2} =36+64=100=10^{2}\\c=10--- > hypotenuse[/tex]
Suppose X and Y are independent and X∼Bernoulli(1/2) and Y∼Bernoulli(1/3). Determine a) the PMF of X+Y b) the CDF of X+Y c) the range of X+Y d) the expectation of X+Y e) the variance of X+Y f) the expectation of X⋅Y
The PMF of X+Y is given byP(X + Y = k) = P(X = 0,Y = k) + P(X = 1,Y = k-1) + P(X = 1,Y = k) = (1/2)(1/3)^k + (1/2)(1/3)^(k-1) + (1/2)(1/3)^(k-1)for k = 0, 1, 2.
The CDF of X+Y can be determined as below:P(X + Y = 0) = (1/2)(1/3)^0 = 1/2P(X + Y ≤ 1) = P(X = 0,Y = 0) + P(X = 1,Y = 0) + P(X = 0,Y = 1) = 1/2 + (1/2)(1/3) + (1/2)(1/3) = 5/6P(X + Y ≤ 2) = P(X = 0,Y = 0) + P(X = 1,Y = 0) + P(X = 0,Y = 1) + P(X = 1,Y = 1) + P(X = 0,Y = 2) = 1/2 + (1/2)(1/3) + (1/2)(1/3) + (1/2)(1/3)(2/3) + (1/2)(1/3)^2 = 11/18P(X + Y ≤ 3) = 1, as the range of X+Y is 0, 1, 2.
The expectation of X+Y can be determined asE(X + Y) = E(X) + E(Y) = 1/2 + 1/3 = 5/6
The variance of X+Y can be determined asVar(X + Y) = Var(X) + Var(Y) = (1/2)(1/2) + (1/3)(2/3) = 7/18
The expectation of X⋅Y can be determined as below: E(X⋅Y) = P(X = 1,Y = 1) = (1/2)(1/3) = 1/6
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The prime factorization of 6. Use exponents when appropriate and order the factors from least to greatest
The prime factorization of 6 is the expression of 6 as a product of its prime factors.
A prime factor is a prime number that is a factor of the given number. To find the prime factorization of 6, we need to express 6 as a product of its prime factors. Since 6 can be evenly divided by 2 and 3, which are both prime numbers, the prime factorization of 6 is 2 x 3. We can also write this as 2^1 x 3^1, using exponents to show that there is only one 2 and one 3 in the prime factorization of 6. Finally, since 2 and 3 are both prime, we order the factors from least to greatest as 2 x 3.
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Rewrite x/√16+x^2 by using x= 4 tan θ The answer should not include a radical.
The equation x/√16+x² can be rewritten by using x= 4 tan θ as x/4.
The given expression is x/√16+x^2 and we need to rewrite it by using x= 4 tan θ.
We know that 1 + tan2θ = sec2θ ⇒ 1 = sec2θ − tan2θ ... (1)
x = 4tanθ⇒ x/4 = tanθ ... (2)
Let's substitute (2) in (1) and get:
sec2θ − tan2θ = sec2θ − x²/16 ... (3)
Now, let's simplify the denominator and numerator of x/√16+x² using trigonometry.
We have:
x/√16+x²= x/[4sec2θ]
Using (2), we know that x = 4 tanθ, so substituting this in (3) we get:
sec2θ − x²/16 = sec2θ − (4tanθ)2/16= sec2θ − tan2θ= 1 (from equation (1))
Therefore,
x/√16+x²= x/[4sec2θ] = x/[4√(sec2θ − tan2θ)] (from equation (3))= x/[4√1]= x/4
Hence, x= 4 tan as x/4 can be used to rewrite x/16+x².
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khomo and Peter bought a house for R575 000 as an investment. khomo payed R245 000 and Peter payed the rest . they sold the house 5 years later and made a profit of R 234 500 if they share the profit in the same ratio as their respective investment, how much profit will Peter receive
Peter will receive R134,417.50 of the profit.
What is Investment ?
Investment refers to the purchase of goods that are not consumed today but are used in the future to create wealth. In financial terms, investment typically involves buying assets such as stocks, bonds, real estate or other financial instruments with the expectation of earning a profit or income from them.
Khomo paid R245,000 for the house, so his investment ratio is:
245,000 / 575,000 = 0.426
Peter's investment ratio is:
1 - 0.426 = 0.574
If they share the profit in the same ratio as their respective investment, then Peter will receive:
0.574 x 234,500 = 134,417.5
Therefore, Peter will receive R134,417.50 of the profit.
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the area of base of a cylindrical tank is 38.5m². Find the perimeter of base
Answer: We know that the area of the base of a cylinder is given by the formula:
A = πr²
where A is the area of the base, π is the mathematical constant pi (approximately equal to 3.14), and r is the radius of the base.
To find the perimeter of the base, we need to know the circumference of the circle that forms the base of the cylinder. The formula for the circumference is:
C = 2πr
where C is the circumference and r is the radius.
To find the radius of the base, we can rearrange the formula for the area:
A = πr²
r² = A/π
r = √(A/π)
Substituting the given value for the area of the base, we get:
r = √(38.5/π) ≈ 3.5 m
Now, we can use the formula for the circumference to find the perimeter of the base:
C = 2πr = 2π(3.5) ≈ 22 m
Therefore, the perimeter of the base is approximately 22 meters.
Step-by-step explanation:
For the following exercises, find the domain, vertical asymptotes, and horizontal asymptotes of the functions. 10. f(x)=x−14 11. f(x)=5x+22 12. f(x)=x2−9x 13. f(x)=x2+5x−36x 14. f(x)=x3−273+x 15. f(x)=x3−16x3x−4 16. f(x)=x3+9x2+14xx2−1 17. f(x)=x2−25x+5 18. f(x)=x−6x−4 19. f(x)=3x−14−2x For the following exercises, find the x - and y-intercepts for the functions. 20. f(x)=x2+4x+5 21. f(x)=x2−xx 22. f(x)=x2+11x+30x2+8x+7 23. f(x)=x2−10x+24x2+x+6 24. f(x)=3x2−1294−2x2
10. Domain: all real numbers; Vertical Asymptotes: x=14; Horizontal Asymptotes: none
11. Domain: all real numbers; Vertical Asymptotes: none; Horizontal Asymptotes: y=5x+22
12. Domain: all real numbers; Vertical Asymptotes: x=0 and x=9; Horizontal Asymptotes: none
13. Domain: all real numbers; Vertical Asymptotes: x=-5; Horizontal Asymptotes: y=0
14. Domain: all real numbers; Vertical Asymptotes: none; Horizontal Asymptotes: y=x3
15. Domain: all real numbers; Vertical Asymptotes: none; Horizontal Asymptotes: y=x3-16x3
16. Domain: all real numbers; Vertical Asymptotes: none; Horizontal Asymptotes: y=x2-1
17. Domain: all real numbers; Vertical Asymptotes: none; Horizontal Asymptotes: y=0
18. Domain: all real numbers; Vertical Asymptotes: x=-6; Horizontal Asymptotes: y=0
19. Domain: all real numbers; Vertical Asymptotes: none; Horizontal Asymptotes: y=3x-14
20. x-intercepts: -4 and 1; y-intercepts: 5
21. x-intercepts: 0 and 1; y-intercepts: 0
22. x-intercepts: -8 and -3; y-intercepts: 7
23. x-intercepts: -1 and 6; y-intercepts: 0
24. x-intercepts: -2 and 0; y-intercepts: -1294
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Find the measurements of the numbered angles of this circle
98 Degrees and 82 degrees are the measurements of the numbered angles of this circle.
What is a circle?
With no sides or edges, a circle is a figure with a round shape. A circle can be characterized in geometry as a closed shape, a two-dimensional shape, or a curved shape.
The collection of all points in the plane that make up a circle are all equally spaced from a certain point known as the "centre," making the form a closed two-dimensional shape. The symmetry line of reflection is formed by each line that traverses the circle. Moreover, it possesses rotational symmetry around the center for each angle.
Measure of angle 1 = (33 + 131)/2
= 82 degrees
Measure of angle 2 = 180 - 82
= 98 Degrees
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Jim takes out a mortgage for 30 years at an interest rate of 2. 49% and his monthly repayments are $986. 50. What is the principal loan amount? Round your answer to the nearest ten thousand dollars. Do NOT round until you have calculated the final answer
The principal mortgage amount is $220,000.
To compute the principal loan amount, we need to apply the compound interest formulation for a loan:
M = P [ i(1 + i)^n ] / [ (1 + i)^n – 1]
Where M is the monthly reimbursement, P is the primary mortgage amount, i is the month-to-month interest charge, and n is the range of months.
First, we need to calculate the month-to-month interest charge:
i = 2.49% / 12 = 0.02075
Subsequent, we want to calculate the range of months:
n = 30 years x 12 months/year = 360 months
Now we are able to plug in the given values:
$986.50 = P [ 0.02075(1 + 0.02075)^360 ] / [ (1 + 0.02075)^360 – 1]
Solving for P, we get:
P = $221,114.07
Rounding to the nearest ten thousand dollars, the principal mortgage amount is $220,000.
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Use properties to rewrite the given equation. Which equations have the same solution as 2.3p – 10.1 = 6.5p – 4 – 0.01p? Select two options. 2.3p – 10.1 = 6.4p – 4 2.3p – 10.1 = 6.49p – 4 230p – 1010 = 650p – 400 – p 23p – 101 = 65p – 40 – p 2.3p – 14.1 = 6.4p – 4
We can say that after answering the offered question The terms in the equation given equation can be streamlined and rearranged to produce the following equation: 23p – 101 = 64p – 40
What is equation?An equation is a mathematical statement that proves the equality of two expressions connected by an equal sign '='. For instance, 2x – 5 = 13. Expressions include 2x-5 and 13. '=' is the character that links the two expressions. A mathematical formula that has two algebraic expressions on either side of an equal sign (=) is known as an equation. It depicts the equivalency relationship between the left and right formulas. L.H.S. = R.H.S. (left side = right side) in any formula.
Using the equality properties, we can rewrite the preceding equation as follows:
2.3p – 10.1 = 6.5p – 4 – 0.01p
2.3p – 10.1 = 6.49p – 4
We can alter the equations using the equality characteristics to look for equations that have the same solution. The answers to the next two equations are the same as those for the previous one:
2.3p – 10.1 = 6.4p – 4
The following equation can be created by adding 0.09p to both sides of the given equation:
2.3p - 10.1 + 0.09p = 6.5p - 4 - 0.01p + 0.09p
2.39p – 10.1 = 6.4p – 4
23p – 101 = 65p – 40 – p
The terms in the given equation can be streamlined and rearranged to produce the following equation:
23p – 101 = 64p – 40
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Publishing scientific papers online is fast, and the papers can be long. Publishing in a paper journal means that the paper will forever in libraries. The British Medical Journal combines the two: it prints short and readable versions, with longer versions available online. The journal asked a random sample of 104 of its recent authors several questions. One question was "Should the journal continue using this system?" In the sample, 72 said "yes." Do the data give good evidence that more than 67% of authors support continuing this system? Interpret the p-value in the context of the problem. If an error has been committed, explain which type of error could it be.
There is no significant evidence that more than two-thirds (67%) of authors support continuing this system.
Define null hypothesis?This hypothesis is either rejected or accepted, depending on whether the population or sample under investigation is viable.
In other words, the null hypothesis is an assertion that the sample observations are the result of chance. It is asserted that surveyors who are interested in the data have made this allegation. It has the prefix H0.
Let p be the proportion of authors who support continuing the system.
Then hypotheses are:
H0: p=0.67
Ha: p>0.67
For calculating the test statistic:
z = p(s)-p/√ [p0(1-p)/N]
where, p(s) = 72/104 ≈ 0.692, p is 0.67 and N is the sample size =104
Then Z ≈ 0.477
The test statistic's p-value is 0.317.
Since 0.317>0.05 and we're assuming a significance level of 0.05, we are unable to reject the null hypothesis.
p-value 0.317 is the likelihood that the sample is chosen from the distribution assumed under null hypothesis, that is where the fraction of authors favouring the new publication scheme is at most 0.67.
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Alex had an income of $7,000. He got 9% of his income from tips. How much of his income
was not from tips?
well, Not from tips that'd be 100% - 9% = 91%, so 91% of $7000.
[tex]\begin{array}{|c|ll} \cline{1-1} \textit{\textit{\LARGE a}\% of \textit{\LARGE b}}\\ \cline{1-1} \\ \left( \cfrac{\textit{\LARGE a}}{100} \right)\cdot \textit{\LARGE b} \\\\ \cline{1-1} \end{array}~\hspace{5em}\stackrel{\textit{91\% of 7000}}{\left( \cfrac{91}{100} \right)7000}\implies \text{\LARGE 6370}[/tex]
if a = 2.4 units, b = 4 units, c = 6 units, and d = 8 units, what is the volume of the figure above?
A) 162.56
B) 111.36
C) 256
D) 130.56
I neeeeeed help, this is due tomorrow!!!!
10 points to whoever gives the correct answer!
Write in expanded form
A bronze statue has a volume of 36 cm³ and a mass of 311.4g. Work out the density of the statue. Give your answer to 2 d.p.
Answer:
8.65 g/cm³
Step-by-step explanation:
You want the density of a bronze statue with a volume of 36 cm³ and a mass of 311.4 g.
DensityDensity is the ratio of mass to volume:
ρ = mass/volume = (311.4 g)/(36 cm³) ≈ 8.65 g/cm³
The density of the statue is about 8.65 g/cm³.
write 150ml as a percentage of 2litres
Answer:
7.5%
Step-by-step explanation:
1 L = 1000 ML2 L = 2000 MLWhat is a percentage?A percentage is a ratio or a number expressed in the form of a fraction of 100. Percentages are often used to express a part of a total.
In order to express 150 mL as a percentage of 2 liters (2000 mL), you need to divide 150 by 2000 and then multiply the result by 100 to get the percentage.
(150 ÷ 2000) × 100 = 7.5%.Therefore, 150 mL is 7.5% of 2 liters.
Fill in the gaps below:
a. from A to B
b. from B to C
c. from A to C
Answer:
a. from A to B, a rotation by 90 degrees anti-clockwise about (0,0).
b. from B to C, a translation by vector (-3/4, -1/4).
c. from A to C, a rotation by 60 degrees, anti-clockwise about (0,0).
Hope this helps, I'm sorry if it doesn't. If you need more help, ask me! :]
Oliver earns $25.50 per hour and saves 25% of his income. Oliver wants to save at least $160
a. Write this situation as an inequality.
b. What is the minimum number of hours Oliver needs to work?
Answer: a. 25.50(h)(25%) >= 160
b. 26
Step-by-step explanation:
a. Let the number of hours Oliver works be h hours.
25.50(h)(25%) >= 160
b. 25.50(h)(25%) >= 160
h >= 25.1 (cor. to 3 sig. fig)
The minimum number of hours Oliver needs to work is 26 hours.
Banks and Credit Unions. (choose all that apply)
A. . Are very similar. They offer the same products and services. What makes them different is the way in which they are legally organized.
B. . Have similar interest rates. However, credit unions often have higher savings rates (better) and lower loan rates (better) than banks.
C. . Are different because credit unions cannot issue ATM and credit cards, making it inconvenient to access your funds.
D. . Are very different. Most people cannot join a credit union
A. Differently organized; B. Different rates; C. Inconvenient access; D. Limited membership.
To calculate the interest rate on a loan, you will need to figure out the amount of interest you will need to pay over the life of the loan. First, you will need to determine the loan amount, the interest rate, and the loan term. Then, you will need to calculate the total interest by multiplying the loan amount by the interest rate and the loan term. Finally, you will need to divide the total interest by the loan amount and then multiply that by 100 to get the annual percentage rate (APR). This will give you the interest rate you will need to pay over the life of the loan.
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ABCD is a trapezium.
EBC is a straight line.
F is the point on AB so that DFE is a straight line.
Angle BEF = 45°
Angle FBC = 100°
Work out the value of angle ADF.
Give a reason on the same line of each
stage of your working.
+
F
Answer: angle ADF =
E
45°
100°
B
Diagram not drawn to scale
C
Total marks: 4
Answer:
35 Degrees
Step-by-step explanation:
Given in figure.
Happy to help :)
The solution is, the value of the angle ADF = 35° .
Here, we have,
from the given diagram, we get,
ABCD is a trapezium.
EBC is a straight line.
F is the point on AB so that DFE is a straight line.
Angle BEF = 45°
Angle FBC = 100°
now, we have,
angle FBE = 180 - 100
= 80 degrees
now, from triangle BEF , we get,
Angle EFB = 55 degrees
again, as AFB is vertically opposite angle,
so, angle AFB = Angle EFB = 55 degrees
finally from triangle ADF , we get,
Angle ADF = 180 - 90 - 55
= 90 - 55
= 35 degrees
so, the value of the angle ADF = 35° .
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Helphelphelphelphelphelp
Answer: -1
Step-by-step explanation:
An easy way to figure out the slop of a linear graph is the formula:
(y1-y2)/(x1-x2)
In order to use this, find two spots on the line that run exactly through a point, in this example (0,20) and (20,0).
Plug in the xs and ys into the formula, which gets you
(20-0) / (0-20)
When simplified, the result is -1 and that is your slope
If the exponential model f(x)=8(9)x is written with the base e, it will take the form A0ekx. What is A0 and what is k?
Answer:
429
Step-by-step explanation:
15. The expression √x is equivalent to 14i√3. What is the value of x?
a.-588
b. -588i
c.588
d. 588i
16. An expression is shown below.
x^5 + 6x^3 +8x
Which value of x makes the expression equal to 0?
a. -2i²
b. -2i
c. 4i
d. 4i²
(15) The value of x is -588 (option A).
(16) The value of x makes the expression equal to 0 is -2i. (option B).
What is the value of x?The value of x is calculated by solving the equation as follows;
(15) We have the equation:
√x = 14i√3
Squaring both sides, we get:
x = (14i√3)^2
x = -588
(16) Factoring the expression, we get:
x(x⁴ + 6x² + 8)
Setting each factor equal to 0, we get:
x = 0 or x⁴ + 6x² + 8 = 0
Let's solve for the second equation:
Let y = x²
Then we have:
y² + 6y + 8 = 0
Factorizing, we get:
(y + 2)(y + 4) = 0
So y = -2 or y = -4
Since y = x², this means:
x² = -2 or x² = -4
Taking the square root of both sides, we get:
x = ±√(-2) or x = ±√(-4)
Simplifying using imaginary numbers, we get:
x = ±i√(2) or x = ±2i
Thus, for the expression √x is equivalent to 14i√3, the value of x is -588. And for the expression x⁵ + 6x³ + 8x, x = -2i.
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what is (3-4i)+(3+4i)
Answer:
6
Step-by-step explanation:
.
Answer:
6
Step-by-step explanation:
look above
simplify: 1÷2+2 3÷4-3÷8
The expression 1÷2+2 3÷4-3÷8 when simplified is 9/4
How to simplify the expressionFrom the question, we have the following parameters that can be used in our computation:
1÷2+2 3÷4-3÷8
Express the terms properly as fractions
So, we have the following
1/2 + 2 3/4 - 3/8
Express each term as proper/improper fraction
So, we have the following representation
1/2 + 11/4 - 3/8
Express the denominator as the same
So, we have
4/8 + 22/8 - 3/8
Evaluate the like terms
18/8
So, we have
9/4
Hence, the solution is 9/4
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NO EXPLANATION JUST ANSWER!
The surface area of a cube with a volume of 4913 cubic yards is 972 square yards.
What is surface area?Surface area is the measurement of the total area of a 3D object. It is the sum of all the individual faces of a 3D object, including the area of each side of the object. It is a way to measure the total exposed area of an object, including any protrusions or indentations.
The surface area of a cube is the sum of the areas of the six faces of the cube. The formula for the surface area of a cube is 6 × s2, where s is the length of the side of the cube.
In this case, the volume of the cube is 4913 cubic yards, so the length of the side of the cube is ∛4913 = 17. Therefore, the surface area of the cube is 6 × 172 = 972 square yards.
To find the surface area of a cube, the first step is to determine the length of the side of the cube. This is done by finding the cube root of the volume (in this case, 4913). Once the length of the side is known, the surface area of the cube is simply 6 times the length of the side squared.
In conclusion, the surface area of a cube with a volume of 4913 cubic yards is 972 square yards.
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Pleaseeeee hurryyyyy
Julia had a bag filled with gumballs. There were 1 watermelon, 2 lemon-lime, and 3 grape gumballs. What is the correct sample space for the gumballs in her bag?
A. Sample space = watermelon, lemon-lime, lemon-lime, grape, grape, grape
B. Sample space = lemon-lime, watermelon, grape
C. Sample space = 1, 2, 3, 4, 5, 6
D. Sample space = 1, 2, 3
Answer:
Step-by-step explanation:
Answer:
c
Step-by-step explanation:
dudjeisosidheis9sjdhxisjdgskkjehud
Demonstrate and explain how to evaluate the derivative for each of the following definite integrals using the fundamental theorem of calculus. ?A)d/xd∫ 4x(2(6cos(t)+7) 4)dtB)d/xd∫ x3(4sin(t 3−3))dt
The derivative for each of the definite integrals using the fundamental theorem of calculus is
a) d/dx ∫ 4x(2(6cos(t)+7)⁴)dt is 4(2(6cos(b)+7)⁴) - 4(2(6cos(a)+7)⁴)
b) d/dx ∫ x³(4sin(t³−3))dt is 12(b²sin(b³-3) - a²sin(a³-3))
The fundamental theorem of calculus tells us that the derivative of the definite integral of a function f(x) with respect to x is equal to the function evaluated at the upper limit of integration minus the function evaluated at the lower limit of integration. In other words, if we have an integral of the form ∫f(x)dx evaluated from a to b, then
d/dx ∫f(x)dx = f(b) - f(a)
Let's apply this to the first integral, A).
A) d/dx ∫ 4x(2(6cos(t)+7)⁴)dt
We begin by recognizing that the function inside the integral is a function of t, not x. However, we want to take the derivative with respect to x. This means that we need to use the chain rule to differentiate the integrand with respect to x.
Using the chain rule, we have
d/dx [4x(2(6cos(t)+7)⁴)] = 4(2(6cos(t)+7)⁴)(d/dx [4x])
= 4(2(6cos(t)+7)⁴)(4)
= 32(2(6cos(t)+7)⁴)
Now, we can apply the fundamental theorem of calculus. Let's say we are evaluating the integral from a to b. Then,
d/dx ∫ 4x(2(6cos(t)+7)⁴)dt = [4b(2(6cos(t)+7)⁴)] - [4a(2(6cos(t)+7)⁴)]
= 4(2(6cos(b)+7)⁴) - 4(2(6cos(a)+7)⁴)
This is the final answer for A).
Now, let's move on to integral B).
B) d/dx ∫ x³(4sin(t³−3))dt
Again, we need to use the chain rule to differentiate the integrand with respect to x.
d/dx [x³(4sin(t³−3))] = (d/dx [x³])(4sin(t³−3))
= 3x²(4sin(t³−3))
Now, we apply the fundamental theorem of calculus. Let's say we are evaluating the integral from a to b. Then,
d/dx ∫ x³(4sin(t³−3))dt = [3b²(4sin(t³−3))] - [3a²(4sin(t³−3))]
= 12(b²sin(b³-3) - a²sin(a³-3))
This is the final answer for B).
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