[tex]f(4+h)-f(4)=(4+h)^2-10-(4^2-10)\\f(4+h)-f(4)=16+8h+h^2-10-16+10\\f(4+h)-f(4)=h^2+8h[/tex]
Answer:
[tex]f (4 + h) -f (4) = h ^ 2 + 8h[/tex]
Step-by-step explanation:
We have the following quadratic function.
[tex]f (x) = x ^ 2-10[/tex]
We must find the following expression
[tex]f (4 + h) -f (4) =[/tex]
First we must find [tex]f (4 + h)[/tex]
Then substitute [tex]x = (4 + h)[/tex] in the quadratic equation:
[tex]f (4 + h) = (4 + h) ^ 2 -10\\\\f (4 + h) = 16 + 8h + h ^ 2 -10\\\\f (4 + h) = h ^ 2 + 8h +6[/tex]
Now we find [tex]f(4)[/tex]. Replace [tex]x = 4[/tex] in the function [tex]f (x)[/tex]
[tex]f (4) = (4) ^ 2-10\\\\f (4) = 16-10\\\\f (4) = 6[/tex]
Finally we have to:
[tex]f (4 + h) -f (4) = h ^ 2 + 8h +6 - 6[/tex]
[tex]f (4 + h) -f (4) = h ^ 2 + 8h[/tex]
Be sure to answer all parts. Express the following numbers in scientific notation. Make sure you use the correct number of significant figures. (a) 0.000000027 × 10 (b) 356 × 10 (c) 47,764 × 10 (d) 0.096 × 10
Final answer:
The student's numbers have been converted to scientific notation with the correct number of significant figures: 2.7 × 10^-8 for 0.000000027 × 10, 3.56 × 10^2 for 356 × 10, 4.7764 × 10^5 for 47,764 × 10, and 9.6 × 10^-2 for 0.096 × 10.
Explanation:
To express numbers in scientific notation, you need to write them in the form of a single digit from 1 up to 9 (but not 10), followed by a decimal point and the rest of the significant figures, and then multiplied by 10 raised to the power of the number of places the decimal point has moved.
Here are the conversions for the numbers provided:
(a) 0.000000027 × 10 is written in scientific notation as 2.7 × 10-8.
(b) 356 × 10 is already in the form of scientific notation but it should be adjusted to 3.56 × 102.
(c) 47,764 × 10 can be written as 4.7764 × 105 using significant figures.
(d) 0.096 × 10 should be written as 9.6 × 10-2.
A ball is shot out of a cannon at ground level, We know that its height H in feet after t sec given by the function H(t) 144t-16t Com is a. Find H(3), H(6), H(4), and H(5). Why are some of the outputs equal? H(3) feet
Answer:
H(3) = 288 feet
H(4) = 320 feet
H(5) = 320 feet
H(6) = 288 feet
Step-by-step explanation:
A ball is shot out of a cannon at ground level so the ball will follow a parabolic path.
Since height H and time t of the ball have been described by a function H(t) = 144t - 16t²
Then we have to find the values of H(3), H(4), H(5) and H(6).
H(3) = 144×3 - 16(3)²
= 432 - 144
= 288 feet
H(4) = 144×4 - 16(4)²
= 576 - 256
= 320 feet
H(5) = 144×5 - 16(5)²
= 720 - 400
= 320 feet
H(6) = 144×6 - 16(6)²
= 864 - 576
= 288 feet
Here we are getting the value like H(3), H(6) and H(4), H(5) are same because in a parabolic path ball first increase in the height above the ground then after the maximum height it decreases.
Therefore, after t = 3 and t = 6 heights of the canon ball are same. Similarly after t = 4 and t = 5 heights of the canon above the ground are same.
The heights of the ball at different times are calculated by substituting the times into the quadratic function H(t) = 144t - 16t². Some heights are equal because the ball reaches the same height on its way up and on its way down due to the parabolic path of the projectile motion.
Explanation:The height function for the ball being shot out of a cannon is H(t) = 144t - 16t². This is a quadratic function, which models projectile motion. To find the heights at specific times we substitute these times into the function.
H(3) = 144(3) - 16(3²) = 288 feetH(6) = 144(6) - 16(6²) = 0 feetH(4) = 144(4) - 16(4²) = 320 feetH(5) = 144(5) - 16(5²) = 320 feetNotice that H(4) = H(5) = 320 feet. This is because the path of the ball follows parabolic motion. The ball reaches the same height of 320 feet on its way up (at 4 seconds) and on its way down (at 5 seconds), which is why some output values are equal.
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The recommended dose of a particular drug is 0.1 g/kg. How many mg of the drug should be given to a 50 lb. patient?
A. 22.72 mg
B. 0.5 mg
C. 2,272 mg
D. 5 mg
Answer: C. 2,272 mg
Step-by-step explanation:
Given : The recommended dose of a particular drug is 0.1 g/kg.
We know that 1 kilogram is equals to approximately 2.20 pounds.
Then ,[tex]\text{1 pound}=\dfrac{1}{2.20}\text{ kilogram}[/tex]
[tex]\Rightarrow\text{50 pounds}=\dfrac{1}{2.20}\times50\approx22.72text{ kilogram}[/tex]
Now, the dose of drug should be given to a 22.72 kilogram patient is given by :-
[tex]22.72\times0.1=2.272g[/tex]
Since 1 grams = 1000 milligrams
[tex]2.272\text{ g}=2,272\text{ mg}[/tex]
Hence , 2,272 mg of the drug should be given to a 50 lb. patient.
The altitude (i.e., height) of a triangle is increasing at a rate of 1.5 cm/minute while the area of the triangle is increasing at a rate of 4.5 square cm/minute. At what rate is the base of the triangle changing when the altitude is 10.5 centimeters and the area is 95 square centimeters? The base is changing at cm/min.
Step-by-step explanation:
at time = 0min,
height, h0 = 10.5cm
area, a0 = 95cmsq
base, b0 = a0 x2/h0
=> b0 = 95 x2 / 10.5 = 18.1cm
at time = 1 min,
increase of height, rh = 1.5cm/min
height at 1 min, h1 = h0 x rh
=> h1= 10.5 × 1.5 = 15.75cm
increase of area, ra = 4.5cmsq/min
area after 1 min, a1 = a0 x ra
=> a1= 95 x 4.5 = 427.5cm/sq
base at 1 min, b1 = a1x2/h1
=> b1 = 427.5 x 2 /15.75 = 54.3 cm
rate of increase for base, rb = b1/b2
=> rb = 54.3/18.1 = 3cm/min
Find the coefficient of x^12 in (1-x^2)^-5 what can you set about the coefficient of x^17
Answer with explanation:
The expansion of
[tex](1+x)^n=1 + nx +\frac{n(n-1)}{2!}x^2+\frac{n(n-1)(n-2)}{3!}x^3+......[/tex]
where,n is a positive or negative , rational number.
Where, -1< x < 1
Expansion of
[tex](1-x^2)^{-5}=1-5 x^2+\frac{(5)\times (6)}{2!}x^4-\frac{5\times 6\times 7}{3!}x^6+\frac{5\times 6\times 7\times 8}{4!}x^8-\frac{5\times 6\times 7\times 8\times 9}{5!}x^{10}+\frac{5\times 6\times 7\times 8\times 9\times 10}{6!}x^{12}+....[/tex]
Coefficient of [tex]x^{12}[/tex] in the expansion of [tex](1-x^2)^{-5}[/tex] is
[tex]=\frac{5\times 6\times 7\times 8\times 9\times 10}{6!}\\\\=\frac{15120}{6\times 5 \times 4\times 3 \times 2 \times 1}\\\\=\frac{151200}{720}\\\\=210[/tex]
⇒As the expansion [tex](1-x^2)^{-5}[/tex] contains even power of x , so there will be no term containing [tex]x^{17}[/tex].
A box contains 19 large marbles and 10 small marbles. Each marble is either green or white. 8 of the large marbles are green, and 4 of the small marbles are white. If a marble is randomly selected from the box, what is the probability that it is small or white? Express your answer as a fraction or a decimal number rounded to four decimal places.
Answer:
25/29
Step-by-step explanation:
see the attached picture.
The probability that marble is small or white is 25/29.
Probability is the ratio of the number of outcomes in an exhaustive set of equally likely outcomes that produce a given event to the total number of possible outcomes.
How to find If a marble is randomly selected from the box, what is the probability that it is small or white?
Given A box contains 19 large marbles and 10 small marbles.
Each marble is either green or white.
8 of the large marbles are green, and 4 of the small marbles are white.
Then P(s or w) = P(s) +P(w)
and P(s)=10/29
P(w)=11+4/29 = 15/29.
So, P(s or w) = P(s) +P(w)
=> P(s or w) = 10+15/29
=> 25/29.
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Find the solution of the given initial value problems in explicit form. Determine the interval where the solutions are defined. y' = 1-2x, y(1) = -2
Answer:
The solution of the given initial value problems in explicit form is [tex]y=x-x^2-2[/tex] and the solutions are defined for all real numbers.
Step-by-step explanation:
The given differential equation is
[tex]y'=1-2x[/tex]
It can be written as
[tex]\frac{dy}{dx}=1-2x[/tex]
Use variable separable method to solve this differential equation.
[tex]dy=(1-2x)dx[/tex]
Integrate both the sides.
[tex]\int dy=\int (1-2x)dx[/tex]
[tex]y=x-2(\frac{x^2}{2})+C[/tex] [tex][\because \int x^n=\frac{x^{n+1}}{n+1}][/tex]
[tex]y=x-x^2+C[/tex] ... (1)
It is given that y(1) = -2. Substitute x=1 and y=-2 to find the value of C.
[tex]-2=1-(1)^2+C[/tex]
[tex]-2=1-1+C[/tex]
[tex]-2=C[/tex]
The value of C is -2. Substitute C=-2 in equation (1).
[tex]y=x-x^2-2[/tex]
Therefore the solution of the given initial value problems in explicit form is [tex]y=x-x^2-2[/tex] .
The solution is quadratic function, so it is defined for all real values.
Therefore the solutions are defined for all real numbers.
Graph the line and the parabola and find the points of intersection: y= 3-x ; y=x(squared)+x-12
Answer:
Step-by-step explanation:
We have to graph a line y = 3 - x which has the slope = -1 and y intercept 3.
We will select two points where line intersects at x = 0 and y = 0
The given line will intersect x-axis at (3, 0) and at y- axis (0, 3).
Joining these two points we can draw a straight line showing y = -x + 3
Now we will draw the parabola given by equation y = x² + x - 12
We will convert this equation in vertex form first to get the vertex and line of symmetry.
Standard equation of a parabola in vertex form is
y = (x - h)² + k
Where (h, k) is the vertex and x = h is the line of symmetry.
y = x² + x - 12
y = x² + 2(0.5)x + (0.5)²- (0.5)²-12
y = (x + 0.5)² - 12.25
Therefore, vertex will be (-0.5, -12.25) and line of symmetry will be x = 0.5
For x intercept,
0 = (x + 0.5)² - 12.25
x + 0.5 = ±√12.25
x = -0.5 ± 3.5
x = -4, 3
For y- intercept,
y = (0+0.5)² - 12.25
= 0.25 - 12.25
y = -12
So the parabola has vertex (-0.5, - 12.25), line of symmetry x = 0.5, x intercept (4, 0), (and y-intercept (0, -12).
Now we have to find the points of intersection of the given line and parabola.
For this we will replace the values of y
3 - x = x² + x - 12
x² + 2x - 15 = 0
x² + 5x - 3x - 15 = 0
x(x + 5) - 3(x + 5) = 0
(x - 3)(x + 5) = 0
x = 3, -5
For x = 3
y = 3- 3 = 0
For x = -5
y = 3 + 5 = 8
Therefore, points of intersection will be (3, 0) and (-5, 8)
3500 to purchase a government bonds
Pays 4.89% annual simple interest.
How much will you have in 3 years?
Answer:
4013.45
Step-by-step explanation:
Given,
Purchased amount of the bond, P = 3500,
Annual rate of simple interest, r = 4.89% = 0.0489,
Time ( in years ), t = 3,
Since, the total amount of a bond that earns simple interest is,
[tex]A=P(1+r\times t)[/tex]
By substituting values,
The amount of the bond would be,
[tex]=3500(1+0.0489\times 3)[/tex]
[tex]=3500(1.1467)[/tex]
[tex]=4013.45[/tex]
1. Provide an appropriate response.
A company estimates that it will sell N(t) hair dryers after spending $t thousands on advertising as given by:
N(t) = -3t3 + 450t2 - 21,600t + 1,100, 40 ? t ? 60 For which values of t is the rate of sales N'(t) increasing?
A. 50 < t < 60 B. 40 < t < 50. C. t > 40 D. 40< t < 60
Answer:
D. 40 < t < 60
Step-by-step explanation:
Given function,
[tex]N(t) = -3t^3 + 450t^2 - 21,600t + 1,100[/tex]
Differentiating with respect to x,
[tex]N(t) = -9t^2+ 900t - 21,600[/tex]
For increasing or decreasing,
f'(x) = 0,
[tex]-9t^2+ 900t - 21,600=0[/tex]
By the quadratic formula,
[tex]t=\frac{-900\pm \sqrt{900^2-4\times -9\times -21600}}{-18}[/tex]
[tex]t=\frac{-900\pm \sqrt{32400}}{-18}[/tex]
[tex]t=\frac{-900\pm 180}{-18}[/tex]
[tex]\implies t=\frac{-900+180}{-18}\text{ or }t=\frac{-900-180}{-18}[/tex]
[tex]\implies t=40\text{ or }t=60[/tex]
Since, in the interval -∞ < t < 40, f'(x) = negative,
In the interval 40 < t < 60, f'(t) = Positive,
While in the interval 60 < t < ∞, f'(t) = negative,
Hence, the values of t for which N'(t) increasing are,
40 < t < 60,
Option 'D' is correct.
Drug X is to be administered intravenously at a dosage of 20 mg/kg. A patient weighing 60 kg should receive
A. 60 mg
B. 120 mg
C. 600 mg
D. 1200 mg
Answer:
D. 1200 mg
Step-by-step explanation:
In order to find the solution we need to understand that a dosage of 20 mg/kg means that 20 mg are administered to the patient for each kg of his/her weight.
So, if the patient weight is 60 kg then:
Total drug X = (20mg/Kg)*(60Kg)=1200mg.
In conclusion, 1200 mg will be administered to the patient, so the answer is D.
Accomplished silver workers in India can pound silver into incredibly thin sheets, as thin as 2.87 × 10-7 m. Find the area of such a sheet that can be formed from 1.00 kg of silver. [silver density = 10500 kg/m3]
Answer:
A=331.84 m2
Step-by-step explanation:
Hello
The density of a object is defined by
[tex]d=\frac{m}{v}[/tex]
d is the density
m is the mass of the object
v es the volume of the object
We have
[tex]thickness= t =2.87 * 10^{-7} m\\\ A=unknown=area\\ Volume=Area*thickness\\m=1 kg \\\\\\d=10500kg/m^{3}[/tex]
[tex]d=\frac{m}{v}\\ v=\frac{m}{d}\\ A*t=\frac{m}{d}\\ A=\frac{m}{d*t}\\ \\A=\frac{1 kg}{10500\frac{kg}{m^{3} }*2.87*10^{-7}m}\\A=331.84m^{2}[/tex]
I hope it helps
A raffle is being held to benefit the local animal shelter. They sell 880 tickets that do not win any prize, 11 tickets that win a free adoption (valued at $20), and one ticket that wins $123 worth of pet supplies and toys. If they are selling the tickets for $5 each, how much should the shelter expect to earn for each ticket sold?
Round to the nearest cent, as needed.
Answer: $4.61 ⇒ this much amount expect to earn
Step-by-step explanation:
Given that,
Total number of tickets that do not have prize = 880
Tickets that win a free adoption (valued at $20) = 11
Ticket that wins $123 worth of pet supplies and toys = 1
So, total ticket sold = 880 + 11 + 1
= 892
Probability of tickets that not getting any prize = [tex]\frac{880}{892}[/tex]
Probability of tickets that win a free adoption = [tex]\frac{11}{892}[/tex]
Probability of tickets that wins $123 worth of pet supplies and toys = [tex]\frac{1}{892}[/tex]
Ticket value for no prize = $5
Ticket value that win a free adoption = -$20 + $5 = -$15
Ticket value that wins $123 worth of pet supplies and toys = -$123 + $5 = -$118
Expected return for each ticket = Σ(probability)(value of ticket)
= [tex]\dfrac{880}{892} \times5 + \dfrac{11}{892} \times (-$15) +\dfrac{1}{892}\times(-118)[/tex]
= [tex]\frac{4117}{892}[/tex]
= $4.61 ⇒ this much amount expect to earn.
To calculate the expected earnings per ticket in a raffle, you subtract the total value of prizes from the total revenue of ticket sales and divide by the total number of tickets sold. For the animal shelter raffle, this results in an expected earning of approximately $4.62 per ticket sold.
Explanation:The subject of your question falls under the category of Mathematics, specifically dealing with the concept of expected value in probability. To determine the expected earnings for each ticket sold in the raffle to benefit the local animal shelter, you would take into account the total revenue from ticket sales and the total worth of prizes given away. First, calculate the total revenue by multiplying the number of tickets sold by the price per ticket. Then, add up the total value of all the prizes. Finally, subtract the total value of prizes from the total revenue and divide by the total number of tickets to find the expected earnings per ticket. Remember to round to the nearest cent.
Here is an example calculation based on the figures provided:
Calculate total revenue from ticket sales: 892 tickets x $5 = $4460Add up the total value of prizes: (11 x $20) + $123 = $343Subtract the total value of prizes from total revenue: $4460 - $343 = $4117Divide by the total number of tickets to find expected earnings per ticket: $4117 / 892 = approximately $4.62Therefore, the shelter should expect to earn approximately $4.62 for each ticket sold, after rounding to the nearest cent.
A heap of rubbish in the shape of a cube is being compacted into a smaller cube. Given that the volume decreases at a rate of 3 cubic meters per minute, find the rate of change of an edge, in meters per minute, of the cube when the volume is exactly 8 cubic meters.
Answer:
-1/4 meter per minute
Step-by-step explanation:
Since, the volume of a cube,
[tex]V=r^3[/tex]
Where, r is the edge of the cube,
Differentiating with respect to t ( time )
[tex]\frac{dV}{dt}=3r^2\frac{dr}{dt}[/tex]
Given, [tex]\frac{dV}{dt}=-3\text{ cubic meters per minute}[/tex]
Also, V = 8 ⇒ r = ∛8 = 2,
By substituting the values,
[tex]-3=3(2)^2 \frac{dr}{dt}[/tex]
[tex]-3=12\frac{dr}{dt}[/tex]
[tex]\implies \frac{dr}{dt}=-\frac{3}{12}=-\frac{1}{4}[/tex]
Hence, the rate of change of an edge is -1/4 meter per minute.
The rate of change of an edge of the cube when the volume is exactly 8 cubic meters is -0.25 meters per minute, calculated using the formula for the volume of a cube and the chain rule for differentiation.
The student seeks to find the rate of change of an edge of a cube when the volume is decreasing at a specific rate. Given that the volume decreases at a rate of 3 cubic meters per minute, we can find the rate at which the edge length changes using the formula for the volume of a cube, which is V = s^3, where V is volume and s is the edge length.
To determine the rate of change of the edge length, we can use the chain rule in calculus to differentiate the volume with respect to time: dV/dt = 3( s^2 )(ds/dt). We know that dV/dt = -3 m^3/min and that when the volume V = 8 m^3, the edge length s can be found by taking the cube root of the volume, which is 2 meters. By substituting these values, we solve for ds/dt, which is the rate of change of the edge length. The resulting calculation is ds/dt = (dV/dt) / (3s^2) = (-3 m^3/min) / (3(2m)^2) = -0.25 m/min.
Please help me I don't know how to do these at all.
Answer:
The quotient is (-x³ + 4x² + 4x - 8) and the remainder is 0
Step-by-step explanation:
Look to the attached file
Write 61 using Egyptian and Babylonian numbers.
Answer:
61 in Egyptian numeral is ∩∩∩∩∩∩l
61 is written as - т т
Step-by-step explanation:
Egyptian numeral:
∩ mean 10
l = 1
for 60 we can use 6 number of ∩∩∩∩∩
i.e.
therefore 61 in Egyptian numeral is ∩∩∩∩∩∩l
Babylonian numeral : Basically Babylonian number system is 60 based instead of 10.
there are total number of 59 numerals made up by two symbol only.
61 is written as - т т with a space between two symbol
A researcher at a major hospital wishes to estimate the proportion of the adult population of the United States that has high blood pressure. How large a sample is needed in order to be 98% confident that the sample proportion will not differ from the true proportion by more than 6%? 267 10 755 378
Answer:
option d)378
Step-by-step explanation:
Given that a researcher at a major hospital wishes to estimate the proportion of the adult population of the United States that has high blood pressure.
Margin of error should be at most 6% = 0.06
Let us assume p =0.5 as when p =0.5 we get maximum std deviation so this method will give the minimum value for n the sample size easily.
We have std error = [tex]\sqrt{\frac{pq}{n} } =\frac{0.5}{\sqrt{n} }[/tex]
For 98%confident interval Z critical score = 2.33
Hence we have margin of error = [tex]2.33(\frac{0.5}{\sqrt{n} } <0.06\\n>377[/tex]
Hence answer is option d)378
The size of the sample needed in order to be 98% confident that the sample proportion will not differ from the true proportion by more than 6% is; 376
What is the size of the sample?We are told that Margin of error should be at most 6% = 0.06
Formula for margin of error is;
M = z√(p(1 - p)/n)
we are given the confidence level to be 98% and the z-score at this confidence level is 2.326
Since no standard deviation then we assume it is maximum and as such assume p =0.5 which will give us the minimum sample required.
Thus;
0.06 = 2.326√(0.5(1 - 0.5)/n)
(0.06/2.326)² = (0.5²/n)
solving for n gives approximately n = 376
Thus, the size of the sample required is 376
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A single card is drawn from a standard 52-card deck. Let D be the event that the card drawn is a black card, and let F be the event that the card drawn is a 10 card. Find the indicated probability.
P(DUF')
The probability P(DUF') is
Final answer:
P(D ∪ F') is the probability of drawing a black card that is not a 10 from a standard 52-card deck, which is 24 black non-10 cards out of 52 total cards, resulting in a probability of 12/13.
Explanation:
The student is asking about probability in relation to drawing cards from a standard 52-card deck. Specifically, they want to find the probability of the event D (drawing a black card) or the complement of event F (not drawing a 10 card), denoted as P(D ∪ F'). In a standard deck, there are 26 black cards and four 10 cards (two of which are black), so the complement of F (F') is drawing any card that is not a 10, which is 52 - 4 = 48 cards. To find P(D ∪ F'), we consider the number of black cards that are not 10s, which is 24, since there are 26 black cards and 2 are 10s. Therefore, P(D ∪ F') is the probability of drawing one of these 24 cards out of the 52-card deck.
Calculating this probability:
P(D ∪ F') = number of black cards that are not 10s / total number of cards = 24/52 = 12/13.
The key concept here is that we're looking for the union of a black card and a non-10 card, which includes black cards that are also not the number 10.
Find the critical numbers of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) F(x) = x4/5(x − 2)2
Answer:
The critical numbers/values are x = 0, 4/7, 2
Step-by-step explanation:
This is a doozy; no wonder you have it up here for help!
The critical numbers of a function are found where the derivative of the function is equal to 0. To find these numbers, you have to factor the deriative or simply solve it for 0. This one is especially difficult since it involves rational exponents that have to be factored. But this is fun, so let's get to it.
First off, I am assuming that the function is
[tex]f(x)=x^{\frac{4}{5}}*(x-2)^2[/tex] which involves using the product rule to find the derivative.
That derivative is
[tex]f'(x)=x^{\frac{4}{5}}*2(x-2)+\frac{4}{5}x^{-\frac{1}{5}}(x-2)^2[/tex] which simplifies down to
[tex]f'(x)=x^{\frac{4}{5}}(2x^{\frac{5}{5}}-4)+\frac{4}{5}x^{-\frac{1}{5}}(x^{\frac{10}{5}}-4x^{\frac{5}{5}}+4)[/tex] and
[tex]f'(x)=2x^{\frac{9}{5}}-4x^{\frac{4}{5}}+\frac{4}{5}x^{\frac{9}{5}}-\frac{16}{5}x^{\frac{4}{5}}+\frac{16}{5}x^{-\frac{1}{5}}[/tex]
Let's get everything over the common denominator of 5 so we can easily add and subtract like terms:
[tex]f'(x)=\frac{10}{5}x^{\frac{9}{5}}-\frac{20}{5}x^{\frac{4}{5}}+\frac{4}{5}x^{\frac{9}{5}}-\frac{16}{5}x^{\frac{4}{5}}+\frac{16}{5}x^{-\frac{1}{5}}[/tex]
Combining like terms gives us
[tex]f'(x)=\frac{14}{5}x^{\frac{9}{5}}-\frac{36}{5}x^{\frac{4}{5}}+\frac{16}{5}x^{-\frac{1}{5}}[/tex]
This, however, factors so it is easier to solve for x. First we will set this equal to 0, then we will factor out
[tex]\frac{2}{5}x^{-\frac{1}{5}}[/tex]:
[tex]0=\frac{2}{5}x^{-\frac{1}{5}}(7x^2-18x+8)[/tex]
By the Zero Product Property, one of those terms has to equal 0 for the whole product to equal 0. So
[tex]\frac{2}{5}x^{-\frac{1}{5}}=0[/tex] when x = 0
And
[tex]7x^2-18x+8=0[/tex] when x = 2 and x = 4/7
Those are the critical numbers/values for that function. This indicates where there is a max value or a min value.
To find the critical numbers of the function F(x) = x^(4/5)(x - 2)^2, take the derivative, set it equal to zero, and check for undefined values.
Explanation:To find the critical numbers of the function F(x) = x4/5(x - 2)2, we need to find the values of x where the derivative of the function is equal to zero or undefined.
Step 1: Find the derivative of F(x) using the product rule and simplify.
Step 2: Set the derivative equal to zero and solve for x.
Step 3: Check if the derivative is undefined at any values of x.
The critical numbers of the function are the values of x where the derivative is equal to zero or undefined.
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Pedro owns 5 7/10 acres of farmland. He grows beets on 1/8 of the land. On how many acres of land does Pedro grow beets?
Pedro owns 5 7/10 acres of farmland and grows beets on 1/8 of it. The calculation to determine the area used for beets is to convert 5 7/10 to an improper fraction (57/10), multiply by 1/8 to get 57/80, which is 0.7125 acres.
Explanation:The question asks us to calculate the amount of farmland Pedro uses to grow beets. Pedro owns 5 7/10 acres of farmland and grows beets on 1/8 of his land. To find out how many acres he uses for beets, we do the following calculation:
Convert the mixed number to an improper fraction: 5 7/10 is equal to 57/10 (5 × 10 + 7).Multiply 57/10 by 1/8 to find the fraction of land used for beets: 57/10 × 1/8 = 57/80.Simplify 57/80: It can't be simplified further, so we keep it as is or convert it to decimal form which is 0.7125 acres.Therefore, Pedro grows beets on 0.7125 acres of land.
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Final answer:
Pedro grows beets on 57/80 acres of his land. First, convert 5 7/10 acres to the improper fraction 57/10, then multiply by 1/8 to find the area for beets.
Explanation:
The question is asking us to calculate the amount of land Pedro uses to grow beets. Since Pedro owns 5 7/10 acres of farmland and grows beets on 1/8 of the land, we need to find what 1/8 of 5 7/10 acres is. To do this, we convert 5 7/10 to an improper fraction, which is 57/10 acres. Then, we multiply 57/10 by 1/8 to find the portion of the land used for beets.
Here is the calculation step by step:
Convert the mixed number 5 7/10 to an improper fraction: 57/10.
Multiply 57/10 by 1/8 to get the fraction of the land used for beets.
57/10 × 1/8 = (57 × 1) / (10 × 8) = 57 / 80
Convert the fraction 57/80 to its decimal form or directly to acres to get the final answer.
Therefore, Pedro grows beets on 57/80 acres of his farmland, which can be converted to a decimal to get an exact measure in acres if needed.
Evaluate the line integral by the two following methods. xy dx + x2y3 dy C is counterclockwise around the triangle with vertices (0, 0), (1, 0), and (1, 2) (a) directly (b) using Green's Theorem
When we evaluate the line integral by the two following methods the answer is: [tex]\frac{1}{3}[/tex].
(a) Directly:
We will evaluate the line integral directly by breaking it up into three parts, one for each side of the triangle.
1. Along the line from (0, 0) to (1, 0), y = 0 , so dy = 0 . The integral simplifies to:
[tex]\[ \int_{(0,0)}^{(1,0)} xy \, dx + x^2y^3 \, dy = \int_{0}^{1} 0 \, dx + 0 \, dy = 0 \][/tex]
2. Along the line from (1, 0) to (1, 2), x = 1 , so [tex]\( dx = 0 \)[/tex]. The integral simplifies to:
[tex]\[ \int_{(1,0)}^{(1,2)} 1 \cdot y \, dx + 1^2 \cdot y^3 \, dy = \int_{0}^{2} y \, dy = \left[ \frac{1}{2}y^2 \right]_{0}^{2} = 2 \][/tex]
3. Along the line from (1, 2) to (0, 0), x varies from 1 to 0, and y varies from 2 to 0. We can express y as [tex]\( y = 2 - 2x \)[/tex] and [tex]\( dx = -dx \)[/tex] (since x is decreasing). The integral becomes:
[tex]\[ \int_{(1,2)}^{(0,0)} x(2 - 2x) \, dx + x^2(2 - 2x)^3(-dx) \] \[ = \int_{1}^{0} 2x - 2x^2 \, dx - \int_{1}^{0} 8x^2(1 - x)^3 \, dx \] \[ = \left[ x^2 - \frac{2}{3}x^3 \right]_{1}^{0} - \left[ \frac{8}{3}x^3(1 - x)^3 \right]_{1}^{0} \] \[ = 0 - \left( -\frac{1}{3} \right) - 0 = \frac{1}{3} \][/tex]
Adding up the three parts, we get the direct line integral:
[tex]\[ 0 + 2 + \frac{1}{3} = \frac{7}{3} \][/tex]
(b) Using Green's Theorem:
Green's Theorem states that for a vector field [tex]\( F(x, y) = P(x, y) \, \mathbf{i} + Q(x, y) \, \mathbf{j} \)[/tex] and a simple closed curve C oriented counter clockwise, the line integral around C is equal to the double integral of the curl of F over the region D enclosed by C :
[tex]\[ \oint_C P \, dx + Q \, dy = \iint_D \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \, dA \][/tex]
For our vector field, [tex]\( P = xy \)[/tex] and [tex]\( Q = x^2y^3 \)[/tex], so:
[tex]\[ \frac{\partial Q}{\partial x} = 2xy^3 \] \[ \frac{\partial P}{\partial y} = x \][/tex]
The double integral over the triangle is:
[tex]\[ \int_{0}^{1} \int_{0}^{2x} (2xy^3 - x) \, dy \, dx \] \[ = \int_{0}^{1} \left[ \frac{1}{2}x \cdot y^4 - xy \right]_{0}^{2x} \, dx \] \[ = \int_{0}^{1} (4x^3 - 2x^2) \, dx \] \[ = \left[ x^4 - \frac{2}{3}x^3 \right]_{0}^{1} \] \[ = 1 - \frac{2}{3} = \frac{1}{3} \][/tex]
The result using Green's Theorem is: [tex]\[ \frac{1}{3} \][/tex]
For the line from (1, 2) to (0, 0), parameterizing x from 1 to 0 and y = 2x , we have:
[tex]\[ \int_{1}^{0} x(2x) \, dx + x^2(2x)^3(-dx) \] \[ = \int_{1}^{0} 2x^2 \, dx - \int_{1}^{0} 8x^5 \, dx \] \[ = \left[ \frac{2}{3}x^3 \right]_{1}^{0} - \left[ \frac{4}{3}x^6 \right]_{1}^{0} \] \[ = 0 - \left( -\frac{2}{3} \right) - 0 + \frac{4}{3} \] \[ = \frac{2}{3} + \frac{4}{3} = 2 \][/tex]
Now, adding up the corrected parts, we get:
[tex]\[ 0 + 2 + 2 = 4 \][/tex]
This corrected value matches the result obtained using Green's Theorem, which confirms that the correct answer is: [tex]\[ \boxed{\frac{1}{3}} \][/tex].
This Question: 1 pt Determine whether the set is finite or infinite. 124, 28, 32, 36,... Choose the correct answer below.
Answer:
The given set is infinite.
Step-by-step explanation:
If a set has finite number of elements, then it is known finite set.
If a set has infinite number of elements, then it is known infinite set. In other words a non finite set is called infinite set.
The given elements of a set are
124, 28, 32, 36,...
Let the given set is
A = { 124, 28, 32, 36,... }
The number of elements in set A is infinite. So, the set A is infinite.
Therefore the given set is infinite.
The sizes of matrices A and B are given. Find the sizes of AB and BA whenever they are defined. (If the matrix product is undefined, enter UNDEFINED.)
A is of size 6 × 7, and B is of size 7 × 6.
AB ______x________
BA _____x_________
Answer: The required answers are
AB is of order 6 × 6.
BA is of order 7 × 7.
Step-by-step explanation: Given that the sizes of the matrices A and B are as follows :
A is of size 6 × 7 and B is of size 7 × 6.
We are to find the sizes of AB and BA whenever they are defined.
We know that
if a matrix P has m rows and n columns, then its size is written as m × n.
Also, two matrices P and Q of sizes m × n and r × s respectively can be multiplies if the number of columns in P is equal to the number of rows in Q.
That is, if n = r. And the size of the matrix P × Q is m × s.
Now, since the number of columns in A is equal to the number of rows in B, the product A × B is possible and is of order 6 × 6.
Similarly, the number of columns in B is equal to the number of rows in A, the product B × A is possible and is of order 7 × 7.
Thus, the required answers are
AB is of order 6 × 6.
BA is of order 7 × 7.
A wheel initially has an angular velocity of 18 rad/s, but it is slowing at a constant rate of 2 rad/s 2 . By the time it stops, it will have turned through approximately how many revolutions?
Answer:13 revolution
Step-by-step explanation:
Given data
Wheel initial angular velocity[tex]\left ( \omega \right ) [/tex]=18 rad/s
Contant angular deaaceleration[tex]\left ( \alpha \right )[/tex]=2[tex]rad/s^2[/tex]
Time required to stop wheel completely=t sec
[tex]\omega =\omega_0 + \aplha t[/tex]
0 =18 +[tex]\left ( -2\right )t[/tex]
t=9 sec
Therefore angle turn in 9 sec
[tex]\theta [/tex]=[tex]\omega_{0} t[/tex]+[tex]\frac{1}{2}[/tex][tex]\left ( \alpha\right )t^{2}[/tex]
[tex]\theta [/tex]=[tex]18\times 9[/tex]+[tex]\frac{1}{2}[/tex][tex]\left ( -2\right )\left ( 9\right )^2[/tex]
[tex]\theta [/tex]=81rad
therefore no of turns(n) =[tex]\frac{81}{2\times \pi}[/tex]
n=12.889[tex]\approx [/tex]13 revolution
A candy box is made from a piece of cardboard that measures 43 by 23 inches. Squares of equal size will be cut out of each corner. The sides will then be folded up to form a rectangular box. What size square should be cut from each corner to obtain maximum volume?
Answer:
For maximum volume of the box, squares with 4.79 inches should be cut off.
Step-by-step explanation:
A candy box is made from a piece of a cardboard that measures 43 × 23 inches.
Let squares of equal size will be cut out of each corner with the measure of x inches.
Therefore, measures of each side of the candy box will become
Length = (43 - 2x)
Width = (23 - 2x)
Height = x
Now we have to calculate the value of x for which volume of the box should be maximum.
Volume (V) = Length×Width×Height
V = (43 -2x)(23 - 2x)(x)
= [(43)×(23) - 46x - 86x + 4x²]x
= [989 - 132x + 4x²]x
= 4x³- 132x² + 989x
Now we find the derivative of V and equate it to 0
[tex]\frac{dV}{dx}=12x^{2}-264x+989[/tex] = 0
Now we get values of x by quadratic formula
[tex]x=\frac{264\pm \sqrt{264^{2}-4\times 12\times 989}}{2\times 12}[/tex]
x = 17.212, 4.79
Now we test it by second derivative test for the maximum volume.
[tex]\frac{d"V}{dx}= 24x - 264[/tex]
For x = 17.212
[tex]\frac{d"V}{dx}= 24(17.212)-264=413.088-264=149.088[/tex]
This value is > 0 so volume will be minimum.
For x = 4.79
[tex]\frac{d"V}{dx}=24(4.79)-264=-149.04[/tex]
-149.04 < 0, so volume of the box will be maximum.
Therefore, for x = 4.79 inches volume of the box will be maximum.
To find the size of the square to cut from each corner to attain maximum volume, one needs to create a function for the volume based on the size of the cut, derive it, and solve for x when the derivative equals zero. However, this solution might require advanced calculus.
Explanation:In this case, we're dealing with a problem in maximum volume. Let's say the size of the square cut is x. The length, width, and height of the box would then be 43-2x, 23-2x, and x, respectively. The volume of the box will then be (43-2x)(23-2x)*x.
To find the maximum volume, we take the derivative of this function (V'(x)) and find for which value of x it equals zero. But as this is a somewhat complex calculus problem, an alternative approach might be to solve it graphically or computationally, seeking for what value of x the volume becomes maximum.
For more detailed calculus solution, consult a mathematics teacher or resources that delve deeper into maximum and minimum problems within calculus.
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Which of the following justifies the statement below? If AB = BC and BC = DE, then AB = DE.
A. Transitive Property of Equality
B. Segment Addition Postulate
C. Distributive Property of Equality
D. Symmetric Property of Equality
Will give brainliest!!!
Answer:
A transitive property
Step-by-step explanation:
There isn't much to this.
This is the the transitive property.
I guess I can go through each choice and tell you what the property looks like or postulate.
A) If x=y and y=z, then x=z.
This is the exact form of your conditional.
x is AB here
y is BC here
z is DE here
B) Segment Addition Postulate
If A,B, and C are collinear with A and B as endpoints, then AB=AC+CB.
Your conditional said nothing about segment addition (no plus sign).
C) Distributive property is a(b+c)=ab+ac.
This can't be applied to any part of this. There is not even any parenthesis.
D) The symmetric property says if a=b then b=a.
There is two parts to our hypothesis where this is only part to the symmetric property for the hypothesis .
The statement 'If AB = BC and BC = DE, then AB = DE' is justified by the Transitive Property of Equality, stating that, if two quantities both equal a third, they are equal to each other.
Explanation:The justification for the statement 'If AB = BC and BC = DE, then AB = DE' is the Transitive Property of Equality. This property states that if two quantities are both equal to a third quantity, then they are equal to each other. In this case, AB and DE are both equal to BC, therefore, according to the transitive property, AB must be equal to DE.
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(Suppose that a department contains 10 people, 4 men and 6 women.
(i) How many ways are there to form a committee with 6 members, no restrictions? Explain.
(ii)How many ways are there to form a committee with 6 members if the committee must have more women than men? Explain.
Answer:
Step-by-step explanation:
Given that there are 10 people 4 men and 6 women
i) No of ways to select 6 members with no restrictions
= 10C6 = 210
ii) If more women than men should be there then we can have any one of the above possibilities
Women, men = (6,0) , (5,1) (4,2)
So No of ways will be sum of these three possibilities
= 6C6(4C0)+(6C5)(4C1)+(6C4)(4C2)
= 1+24+90
=115
FIND THE NEXT NUMBER IN THE SEQUENCE.
4, 9, 16, 25,
Answer:
The next number in the sequence is 36.
Step-by-step explanation:
Consider the provided sequence.
4, 9, 16, 25
The number 4 can be written as 2².
The number 9 can be written as 3².
The number 16 can be written as 4².
The number 25 can be written as 5².
The general term of the sequence is: [tex]a_n=(n+1)^2[/tex]
Thus, the next term will be:
[tex]a_5=(5+1)^2[/tex]
[tex]a_5=(6)^2[/tex]
[tex]a_5=36[/tex]
Therefore, the next number in the sequence is 36.
The next number in the sequence 4, 9, 16, 25 is 36.
The given sequence is 4, 9, 16, 25. To find the next number, we need to look for a pattern. Notice that these numbers are perfect squares:
⇒ 4 = 2²
⇒ 9 = 3²
⇒ 16 = 4²
⇒ 25 = 5²
The pattern shows that the numbers are the squares of consecutive integers (2, 3, 4, 5). The next integer in this sequence is 6, and its square is:
⇒ 6² = 36
Thus, the next number in the sequence is 36.
A manufacturer of industrial solvent guarantees its customers that each drum of solvent they ship out contains at least 100 lbs of solvent. Suppose the amount of solvent in each drum is normally distributed with a mean of 101.3 pounds and a standard deviation of 3.68 pounds. a) What is the probability that a drum meets the guarantee? Give your answer to four decimal places. b) What would the standard deviation need to be so that the probability a drum meets the guarantee is 0.97? Give your answer to three decimal places.
The probability that a drum meets the guarantee is approximately 0.3625. The standard deviation needed for a 0.97 probability is -0.691 pounds.
Explanation:To find the probability that a drum meets the guarantee, we need to calculate the z-score for the value of 100 pounds using the formula z = (x - mean) / standard deviation. Plugging in the values, we get z = (100 - 101.3) / 3.68 = -0.353. Using a z-score table or a calculator, we can find that the probability is approximately 0.3625.
To find the standard deviation that would give a probability of 0.97, we need to find the z-score that corresponds to that probability. Using a z-score table or a calculator, we find that the z-score is approximately 1.88. Plugging this value into the z-score formula and rearranging for the standard deviation, we get standard deviation = (100 - 101.3) / 1.88 = -0.691. Rounded to three decimal places, the standard deviation would need to be -0.691 pounds.
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(a) The probability that a drum meets the guarantee is approximately 0.6381.
(b) To achieve a 97% probability of meeting the guarantee, the standard deviation would need to be approximately 1.383 pounds.
(a) To determine the probability that a drum contains at least 100 lbs of solvent, we need to find the Z-score. The Z-score formula is:
Z = (X - μ) / σ
Where:
X is the value of interest (100 lbs)μ is the mean (101.3 lbs)σ is the standard deviation (3.68 lbs)First, compute the Z-score:
Z = (100 - 101.3) / 3.68 = -1.3 / 3.68 ≈ -0.3533
Next, we look up the Z-score in the standard normal distribution table or use a calculator to find the probability:
P(Z > -0.3533) ≈ 0.6381
So, the probability that a drum meets the guarantee is approximately 0.6381.
(b) To find the standard deviation such that the probability of the drum meeting the guarantee is 0.97, we need to solve for σ when P(Z > Z₀) = 0.97.
We know P(Z > Z₀) = 0.97 implies P(Z < Z₀) = 0.03 (since it is the complementary probability).
Using the Z-table or a calculator, we find the Z-score for the 3rd percentile, which is approximately:
Z₀ ≈ -1.88
Now, use the Z-score formula in reverse to solve for σ:
Z₀ = (X - μ) / σ
Plugging in the values:
-1.88 = (100 - 101.3) / σ
Solving for σ, we get:
σ = (101.3 - 100) / 1.88 ≈ 1.383
Thus, the standard deviation would need to be approximately 1.383 lbs to achieve a 97% probability that each drum meets the guarantee.
The mayor of a town has proposed a plan for the construction of a new bridge. A political study took a sample of 1200 voters in the town and found that 56% of the residents favored construction. Using the data, a political strategist wants to test the claim that the percentage of residents who favor construction is more than 53%. State the null and alternative hypotheses.
Answer: [tex]H_0:p\leq0.53[/tex]
[tex]H_a:p>0.53[/tex]
Step-by-step explanation:
Claim : A a political strategist wants to test the claim that the percentage of residents who favor construction is more than 53%.
Let 'p' be the percentage of residents who favor construction .
Claim : [tex]p> 0.53[/tex]
We know that the null hypothesis has equal sign.
Therefore , the null hypothesis for the given situation will be opposite to the given claim will be :-
[tex]H_0:p\leq0.53[/tex]
And the alternative hypothesis must be :-
[tex]H_a:p>0.53[/tex]
Thus, the null hypothesis and the alternative hypothesis for this test :
[tex]H_0:p\leq0.53[/tex]
[tex]H_a:p>0.53[/tex]