In a cubic polynomial function f is defined by f(x) = 4x³ + ax² + bx + k, where a, b, k, are constants and has a local minimum at x = -2 and a local maximum at x = 0, then values of a and b is 12 and 0 respectively. If integrate of f(x)dx = 32 from 0 to 1, then value of k is 27.
The given cubic polynomial function is
f(x) = 4x³ + ax² + bx + k
f'(x) = 12x² + 2ax + b
f''(x) = 24x + 2a
At local maximum f' = 0
f'(0) = 12×(0)² + 2a×(0) + b = 0
b = 0
At local minimum, f' = 0
That is f'(-2) = 0 and f''(-2) > 0
f'(-2) = 12×(-2)² + 2a×(-2) + b = 0
48 - 4a + b = 0
4a - b = 48
a = 12
Therefore, f(x) = 4x³ + 12x² + k
Integrate f(x)dx = 32 from 0 to 1, that is
∫₀¹ f(x)dx = 32
∫₀¹ (4x³ + 12x² + k) dx = 32
[ x⁴ + 4x³ + kx ]₀¹ = 32
(1⁴ - 0⁴) + 4(1³ - 0³) + k(1 - 0) = 32
1 + 4 + k = 32
k = 27
-- The question is incomplete, answering to the question below--
"A cubic polynomial function f is defined by f(x) = 4x³ + ax² + bx + k, where a, b, k, are constants. The function f has a local minimum at x = -2 and a local maximum at x = 0.
A. Find the values of a and b
B. If you integrate f(x)dx = 32 from 0 to 1, what is the value of k?"
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pls help me in this homework id really appreciate it <3
An isosceles triangle has a base that is 3 times the length of its legs. The perimeter of the triangle is 45 cm. What is the length of the base?
Answer:
27cm
Step-by-step explanation:
[tex]s + s + s = p \\3x + x + x = 45 \\ 5x = 45 \\ \frac{5x}{5} = \frac{45}{5} \\ x = 9[/tex]
That's the length of one of its legs
The length of the base = 3x
= 3 x 9
= 45cm
Select the correct answer from each drop-down menu. which if formulas are valid? and are valid if formulas.
Answer:
Step-by-step explanation:
pete runs an ice cream stand that also sells snow cones served in paper cones. the paper cones he usually uses have a diameter of 6 inches and a height of 2 inches, but his supplier is out of them. as a replacement, he purchases paper cones with a diameter of 2 inches and a height of 6 inches. how do the volumes of the original and replacement cones compare?
Answer:
replacement is 1/3 the volume of the original
Step-by-step explanation:
You want to compare the volumes of a cone with diameter 6 in and height 2 in to one with a diameter of 2 in and height of 6 in.
VolumeThe volume of a cone is jointly proportional to the height and the square of the diameter:
V ∝ d²h
RatioThe ratio of the volume of the replacement cone to the original is ...
r = V2/V1 = ((2 in)²(6 in))/((6 in²)(2 in) = 2/6 = 1/3
The replacement cone has 1/3 the volume of the original.
Given function g on the graph, what is the domain of this function?
A part of linear function g is graphed on the grid
Answer: -7[tex]\leq[/tex]x[tex]\leq[/tex]6
Mustafa walks 15 km east and then 10 km south. How far is Mustafa from the starting point? Is in which direction
Answer:
25 kilometres West direction
9/2 * -(8/3) pls answer fast!...step by step explanation plss
Answer:
9 * -8
____ =-72/6
2 * 3
In a series of transformations, which transformation would make two figures similar as opposed to congruent?a. dilationb. reflectionc. rotationd. translation
In a series of transformations, dilation would make two figures similar as opposed to congruent.
Rotation, reflection and translation are known as rigid transformations.
This means they do not change the size or shape of a figure, they simply move it.
These rigid transformations preserve congruence.
Dilation, however, are not rigid transformation, since they change the size of a shape.
Dilation would not change the shape, just the size: the angle measures would be the same, and the ratio of corresponding sides would be equal to the scale factor used in the dilation.
This would give us a similar, but not congruent.
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According to the United States Census Bureau, the state of Texas had a population of 16,986,510 in 1990. Census data shows that in 2000, the population of Texas was 20,851,820. Find the percent of change.
Answer:
22.7551745473%
Step-by-step explanation:
I could be very wrong but to find the percentage of change you have to divide the increase by the original number, and then multiply it by 100.
20851820-16986510=3865310
3865310/16986510=0.22755174547
0.22755174547*100=22.755174547
This answer can then be rounded up to 23%
Consider the function below. (If an answer does not exist, enter DNE.)h(x) = 5x3 − 3x5(a) Find the interval of increase. (Enter your answer using interval notation.)(−1,0)∪(0,1)
The interval in which the function h(x) = 5x³ - 3x^5 is increasing is given as follows:
(−1,0)∪(0,1).
How to obtain the intervals in which the function is increasing?The function in this problem is defined as follows:
h(x) = 5x³ - 3x^5.
To obtain the intervals for increase of decrease, the critical points of the function need to be obtained, which are the values of x for which the derivative is of zero.
The derivative of the function in this problem is given as follows:
h'(x) = 15x² - 15x^4.
Hence the critical points of the function are given as follows:
15x² - 15x^4 = 0
15x²(1 - x²) = 0.
Hence:
15x² = 0 -> x = 0.1 - x² = 0 -> x² = 1 -> x = -1, x = 1.Then the signals of the derivative in each interval is given as follows:
x less than -1: negative, as (1 - x²) < 0.x between -1 and 0: positive, as (1 - x²) > 0.x between 0 and 1: positive, as (1 - x²) > 0.x greater than 1: negative, as (1 - x²) < 0.The function is increasing when the derivative is positive, hence the interval is given as follows:
(−1,0)∪(0,1).
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Quetion
a. Write the formula for the area A of a trapezoid. Ue b1 and b2 for the length of the bae, and ue h for the height. A=
b. Solve the formula for h. H=
c. Ue the new formula to find the height h of the trapezoid. The height of the trapezoid i
centimeter
The formula for the area A of a trapezoid [tex]A= \frac{1}{2} (b_{1} + b_{2})h[/tex], the formula for the height of the trapezoid is [tex]h= \frac{2A} {(b_{1} + b_{2})}[/tex] , and the height of the given trapezoid is 6cm.
(a) let us find the formula for the area A of a trapezoid as follows-
As b₁, b₂ being the lengths of the bases and 'h' is the height.
So, the formula for the area A of a trapezoid is given by-
[tex]A= \frac{1}{2} (b_{1} + b_{2})h[/tex]
(b) Let us solve for the formula for the height h of the trapezoid as follows-
As the formula for the area A of a trapezoid is [tex]A= \frac{1}{2} (b_{1} + b_{2})h[/tex]
So, the formula for height h of the trapezoid can be given by -
[tex]h= \frac{2A} {(b_{1} + b_{2})}[/tex]
(c) Let us find the new formula to find the height h of the trapezoid as follows-
Using, [tex]h= \frac{2A} {(b_{1} + b_{2})}[/tex] to find the height h of the trapezoid.
So,
[tex]h= \frac{2A} {(b_{1} + b_{2})}[/tex]
[tex]h= \frac{2(72)} {16 +8}[/tex] --------(substituting the values of b₁ = 16, b₂ = 8, A = 72cm²)
h = 144/24
h = 6 cm
Hence, the height of the trapezoid is 6cm.
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The complete question is -
a. Write the formula for the area A of a trapezoid. Use b1 and b2 for the length of the base, and use h for the height. A=
b. Solve the formula for h. H=
c. Use the new formula to find the height h of the trapezoid. The height of the trapezoid in centimeters.
The cost C in dollars of manufacturing x bicycles at a production plant is given by the function shown below.
C(x) = 5x²-1500x + 125,000
a. Find the number of bicycles that must be manufactured to minimize the cost.
b. Find the minimum cost.
a. How many bicycles must be manufactured to minimize the cost?
bicycles
Given function:
C(x) = 5x² - 1500x + 125000It has a positive leading coefficient, hence the parabola opens up. It means it has the minimum point at vertex.
The x-coordinate of the vertex is:
x = - b/(2a) = - (-1500)/(2*5) = 150The minimum cost is:
C(150) = 5(150)² - 1500(150) + 125000 = 12500Answer:
a) The minimum number of bicycles is the x-coordinate of the vertex, 150,b) The minimum cost is the value of C when x is 150, this is $12500.5 3/5÷2 2/3 in its simpilest form
Answer: 2.1
Step-by-step explanation:
Calculator
There are no results for an airplane can carry a maximum of 36600 pound of cargo,passenger and luggage.On on the plane's next flight,it will carry 7,320 pounds of cargo.If each passenger with luggage weighs 220 pounds,what is the greatest number of passengers the plane can carry on its next fligh a 33 passenger b 133passenger c 166 passengers d 199 passengers
Answer:
b. 133 passengers
Step-by-step explanation:
The anount of weight taken by the passengers can be up to [tex]36600-7320=29280[/tex] lbs.
Dividing this by [tex]220[/tex] lb per passenger, we get there can be up to [tex]\lfloor 29280/220 \rfloor=133[/tex]
Describe all the
positive integers that would make (32)x greater than 39. Explain your reasoning
The positive integers that will make (32)x greater than 39 must be greater than 39/32
How to find positive integers that will make (32)x greater than 39A whole number that is greater than zero is what is meant by the term "positive integer."
All counting numbers are included in the set of positive integers (that is, the natural numbers)
In the problem, the given expression is (32)x greater than 39 rewriting gives
(32)x > 39
solving for x
x > 39/32
the integer must be greater than 39/32
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a van full of storm chasers is driving along a straight road at 94 kilometers per hour in hopes of filming a tornado. the tornado, 7 kilometers away, is traveling along the same road and moving directly toward them, traveling at 69 kilometers per hour. if neither the van nor the tornado changes course, how long will it be before they meet?
The time taken to traveled the van and the tornado before they meet as per the speed of the van and the tornado is equal to 2.6 minutes.
As given in the question,
Speed of the van along the straight road = 94 kilometers per hour
After travelling distance of 7 kilometers
Speed of the tornado along the same road = 69 km/hour
Total relative speed of van and tornado = 94 + 69
= 163km/hour
Distance travelled between two different speed = 7km
Time used to traveled before they meet = distance / speed
= 7 / 163
= 0.043 hours
= 2.58minutes
= 2.6minutes
Therefore, the time required to traveled the van and the tornado before they meet is equal to 2.6 minutes.
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12. Juan is a programmer going to school at
night to earn a higher degree. He starts the
month with $500 in his savings account. His
income is $4750. Each month he currently
spends $750 on food; $400 on clothes; $530
on video games and movies. His classes cost
$500 each month. His essential expenses
(rent, medical costs, insurance, tax) total
$2800. How much does Juan have for his Net
Cash Flow and Ending Cash at the end of the
month?
[A]Net Cash Flow: -$230
Ending Cash: $730
[B] Net Cash Flow: $2,570
Ending Cash: $3,070
[C] Net Cash Flow: -$230
Ending Cash: $270
[D] Net Cash Flow:-$270
Ending Cash: $230
[E] Net Cash Flow: $270
Ending Cash: -$770
Juan's Net Cash Flow at the end of the month is -$230 and his Ending Cash is $270.
To calculate Juan's Net Cash Flow and Ending Cash, you need to subtract his expenses from his income.
Juan's income is $4750 per month. His essential expenses, including rent, medical costs, insurance, and tax, total $2800 per month. He also spends $750 on food, $400 on clothes, and $530 on video games and movies, for a total of $2680 in non-essential expenses.
Therefore, his total expenses are $2800 + $2680 = $5480.
His Net Cash Flow is his income minus his expenses, which is $4750 - $5480 = -$730.
His Ending Cash is his Net Cash Flow plus his starting balance of $500, which is -$730 + $500 = $270.
Therefore, the correct answer is option C: Net Cash Flow: -$230, Ending Cash: $270.
Write an equation of the line that passes through P(3,3) and Q(1,-1).Write the equation in slope intercept form
Answer:
y = 2x-3
Step-by-step explanation:
Find slope from the two points (which is 2)
and then plug in one of the points (either (3,3) or (1,-1)) to find b in y=mx+b.
Combine everything, and there's your equation
Check for Reasonableness What is a good estimate for 380% of 60? Explain.
what two terms are used to refer to a search in which each value in a sequence is examined until a target value is found or the end of the sequence is reached?
This problem can be solved using divide and conquer techniques. An method known as "divide and conquer" works by breaking down a large task into smaller ones.
An algorithmic pattern is known as Divide and Conquer. In algorithmic approaches, the idea is to take a dispute involving a large amount of input, divide the input into smaller pieces, solve the problem on each of the smaller pieces, and then combine the piecewise solutions into a comprehensive solution. The Divide and Conquer is the algorithmic name of the approach that was used to solve the issue.
The three steps below are used in a dispute utilizing the Divide and Conquer method.
Create a group of subproblems from the main issue.
Conquer: Recursively and individually solve each subproblem.
Combine: Combine the solutions to the individual subproblems to arrive at the overall solution.
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mason stands on the 5^\text{th}5 th 5, start superscript, start text, t, h, end text, end superscript step of a vertical ladder. the ladder has 151515 steps, and the height difference between consecutive steps is 0.5\text{ m}0.5 m0, point, 5, start text, space, m, end text. h(n)h(n)h, (, n, )models the height above the ground of mason's feet (in \text{m}mstart text, m, end text) after moving nnn steps (if mason went down the ladder, nnn is negative.) which number type is more appropriate for the domain of hhh?
Since the stages are simply whole values taken one after the other, they should be whole numbers or integers.
What in mathematics is a whole number?
Complete numbers those figures that contain zero and natural integers. not a decimal or fraction. {0, 2, 3, 4, 5 6, 7, 8, 9, 10, 11 …} Integer. a negative number, zero, or a counting number. Whole numbers are made up of all the ordinal numbers and 0.
Natural numbers are the term used to describe counting numbers in mathematics. Therefore, the whole number can be described as a combination of all natural numbers and 0. Along with zero, all positive integers are also considered whole numbers.
He is currently on step 5, there are 15 steps total, therefore he is actually 10 steps away from the top of the ladder. If we walk down the ladder with N being negative, he is -5 steps from the ground.
{ N | N ∈ ℤ , -5 ⩽ N ⩽ 10 }
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32 minutes and 2 seconds - 12 minutes and 15 seconds
32 minutes and 2 seconds - 12 minutes and 15 seconds
0 days 0 hours 32 minutes 2 seconds
- 0 days 0 hours 12 minutes 15 seconds
_________________________________
= 0 days 0 hours 19 minutes 47 seconds
[tex]=\fbox{19 minutes and 47 seconds or 19.783333 minutes}[/tex]
2. Which of the following best represents the
range of the function?
a. All real numbers.
b. y ≤ 3
c. y = 3
d. x ≤ 3
Many businesses and organizations insist that official communications use professional-looking fonts, such as times new roman. fonts characterized as less professional, such as comic sans, should be avoided. a student would like to determine if font choice makes a difference in the average grade on a typed response in english classes. there are currently four english classes available from which the student can collect data, and each class has a similar writing assignment. at the start of the assignment, students are randomly assigned a font, either times new roman or comic sans, and then complete the assignment. the teachers grade the assignments, and the mean grade for each font is found.Method A: At the start of the assignment, students choose a font, either Times New Roman or Comic Sans, and then complete the assignment.Method B: At the start of the assignment, students are randomly assigned one of two fonts, either Times New Roman or Comic Sans, and then complete the assignment.In both methods, the teachers grade the papers received, and the mean grade for each font is determined.Which method describes an experiment?a. Method A is an experiment because the students choose their own font.b. Method B is an experiment because the students are randomly assigned a font.c. Both methods describe experiments because both fonts are used on the assignment.d. Neither method describes an experiment because a third font should be included for comparison.
Calculate the double integral. {8x}/{1 + xy} dA ; R = 0, 8 0, 1
The value of the double integral ∫∫ (8x)/(1 + xy) dA ; R = [0, 8] x [0, 1] is 11.77
In this question we need to calculate the double integral. {8x}/{1 + xy} dA ; R = 0, 8 0, 1
i.e., to find ∫∫ (8x)/(1 + xy) dA ; R = [0, 8] x [0, 1]
First we integrate the function (8x)/(1 + xy) as a function of y, treating the variable x as a constant.
G(x) = ∫_[0, 8] ((8x)/(1 + xy) ) dy
G(x) = 8 ln(8x + 1)
Now we calculate the integral of the previous result as a function of x.
i.e., ∫_[0, 1] G(x) dx
= ∫_[0, 1] [8 ln(8x + 1)] dx
= 9 ln(9) - 8
= 11.77
Therefore, the solution to the double integral (8x)/(1 + xy) dA ; R = [0, 8] x [0, 1] is 11.77
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Meteorologists in Texas want to increase the amount of rain delivered by thunderheads by seeding the clouds. Without seeding, thunderheads produce, on average, 300 acrefeet. The meteorologists randomly selected 29 clouds which they seeded with silver iodide to test their theory that average acrefeet is more.
What conclusion can they reach based on the above output when they set the significance level to be 0.10?
a.The data does not provided statistical evidence that the average acrefeet from seeded clouds is more than 300.
b.The data does provide statistical evidence that the average acrefeet from seeded clouds is more than 300.
c.The data does not provided statistical evidence that the sample average acrefeet from seeded clouds is more than 300.
d.Two of the above are correct.
e.We can not draw conclusions based on the above statistical output because the conditions are not met due to the extreme out
There is no statistical support in the data for the claim that seeded clouds typically cover more than 300 acres per foot by average acrefeet.
Since the significance level is set at 0.10, the meteorologists can conclude that the data does not provide statistical evidence that the average acrefeet from seeded clouds is more than 300.
This is because the calculated p-value is greater than the significance level. This means that there is not sufficient evidence to reject the null hypothesis, which states that the average acrefeet from seeded clouds is no greater than 300.
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suppose that a box contains 6 cameras and that 5 of them are defective. a sample of 2 cameras is selected at random ( without replacement). define the random variable x as the number of defective cameras in the sample. write the probability distribution for x .write your answer in fraction form or round to 3 decimal places.
If x is the number of defective cameras in the 2-size sample, that means that:
P(x = 0) = 0 (since there are no 2 cameras that can be selected without replacement that are not defective)
P(x = 1) = 5/15 = 1/3 (since there are 15 total ways to select a distinct sample and only 5 of these would have the non-defective one in them)
P(x = 2) = 10/15 = 2/3 (since there are 15 total ways to select a distinct sample and 10 of these would NOT have the non-defective one in them)
find the length of the curve. r(t) = 4t i + 8t3/2 j + 9t2 k, 0 ≤ t ≤ 1
The length of the curve [tex]r(t) = 4t i + 8t^{3/2} j + 9t^{2}k, 0\leq t\leq 1[/tex] is 13.
The given curve is defined by the parametric equation [tex]r(t) = 4t i + 8t^{3/2} j + 9t^{2}k, 0\leq t\leq 1[/tex].
The length of a curve is given by the formula
⇒ [tex]|| r(t) || = \int\limits^a_b {\sqrt{{(x')^{2}} + (y')^{2} + (z')^{2} } } \, dt[/tex]
where the variables x', y', z' are defined as
[tex]x' = \frac{d(4t)}{dt} = 4 \\y' = \frac{d(8t^{3/2})}{dt} = 12t^{1/2} \\x' = \frac{d(9t^{2})}{dt} = 18t\\[/tex]
Hence, substituting the values in the formula for the length
⇒ [tex]|| r(t) || = \int\limits^0_1 {\sqrt{{(4)^{2}} + (12t^{1/2})^{2} + (18t)^{2} } } \, dt[/tex]
⇒ [tex]|| r(t) || = \int\limits^0_1 {\sqrt{{16 + 144t + 324t^{2}} } \, dt[/tex]
The given definite integral is in the form of
[tex]\int\limits^a_b {\sqrt{P(x)}} \, dx = \int\limits^a_b {\sqrt{{ax^{2} + bx + c}} } \, dx[/tex]
where P(x) is a polynomial of degree 2.
Writing the polynomial P(x) as
⇒ [tex]\sqrt{P(x)} = \sqrt{a}\sqrt{{(x + b_1)^{2} + c_1}} dx[/tex]
⇒ [tex]\int\limits^a_b {\sqrt{P(x)}} \, dx = \sqrt{a}\int\limits^a_b {\sqrt{{(x + b_1)^{2} + c_1}}} \, dx[/tex]
where [tex]b_1, c_1[/tex] are defined as
⇒ [tex]b_1= \frac{a}{2b} \\c1 = \frac{c}{a} - b_1^{2}[/tex]
Therefore, the solution will be
⇒ [tex]\int\limits^a_b {\sqrt{{(x + b_1)^{2} + c_1}}} \, dx = \frac{(b_1 + x)\sqrt{P(x)} + c_1log{(b_1 + x + \sqrt{P(x)}}}{2}[/tex]
Hence, solving the integral
⇒ [tex]|| r(t) || = \int\limits^0_1 {\sqrt{{16 + 144t + 324t^{2}} } \, dt[/tex]
we get the value as 13.
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Shota built a time travel machine, but he can't control the duration of his trip. Each time he uses the machine he has a 0. 80. 80, point, 8 probability of staying in the alternative time for more than an hour. During the first year of testing, shota uses his machine 202020 times. Assuming that each trip is equally likely to last for more than an hour, what is the probability that at least one trip will last less than an hour? round your answer to the nearest hundredth. P(\text{at least one} < 1\text{ hour})=.
The Probability that at least one trip will last less than one hour =7/8
What is probability?It refers to the ratio of favorable outcome to the total possible outcome.
The possibility of happening an event is probability.
How to calculate probability of the given question?Probability of staying in the alternative time for more than an hour is 8
each trip is likely to last for more than one hour so probability of at least one trip will last more than an hour =1/8
Now, probability of at least one trip will last less than one hour = 1-1/8
7/8
hence, the result is 7/8
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two planes flying at the same altitude are heading toward jwd airport. the paths of these planes form a right triangle. the jet is flying due east toward jwd at 450 mph. the turboprop is approaching jwd from the south at 275 mph. when each is 100 miles from the airport how fast is the distance between the planes changing?
The speed of the distance between the two planes is increasing at 325 mph as they approach JWD Airport.
The two planes form a right triangle, with the jet flying due east at 450 mph and the turboprop approaching from the south at 275 mph. The hypotenuse of the triangle is the distance between the planes, and the length of the hypotenuse is changing as the planes approach the airport. The rate of change of the length of the hypotenuse is equal to the sum of the rates of change of the two sides of the triangle, which are the speeds of the jet and the turboprop. As they are both 100 miles from the airport, the speed of the distance between the two planes is equal to the sum of the two speeds, or 450 mph + 275 mph = 325 mph. Therefore, the distance between the two planes is changing at a rate of 325 mph.
Rate of change of distance between the planes = Rate of change of jet + Rate of change of turboprop
= 450 mph + 275 mph
= 325 mph.
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