The form of the derivative of the absolute equation is
[tex]\begin{gathered} f(x)=\lvert x-a\rvert \\ \frac{dx}{dy}=\frac{\lvert x-a\rvert}{x-a} \\ \frac{dy}{dx}=\frac{x-a}{\lvert x-a\rvert} \end{gathered}[/tex]For the given function
[tex]f(x)=\lvert x-9\rvert[/tex]We will find the derivative from the left at x = 9
[tex]\frac{dy}{dx}=\frac{x-9}{\lvert x-9\rvert}[/tex]Substitute x by 9
[tex]\begin{gathered} \frac{dy}{dx}=\frac{9-9}{\lvert9-9\rvert} \\ \frac{dy}{dx}=\frac{0}{0} \end{gathered}[/tex]Then dy/dx does not exist (None)
The derivative from right at x = 9
[tex]\frac{dx}{dy}=\frac{\lvert x-9\rvert}{x-9}[/tex]Substitute x by 9
[tex]\begin{gathered} \frac{dx}{dy}=\frac{\lvert9-9\rvert}{9-9} \\ \frac{dx}{dy}=\frac{0}{0} \end{gathered}[/tex]Then dx/dy does not exist (None)
Melissa and brain are at the base of a mountain.melissa hikes to a location 27 meters above sea level.brain hikes to a location 21 meters below sea level. what us the diffrence if the hikers altitudes?
The point at the base of the mountain is described as point zero, that is they are still at sea level. If Melissa now moves to a position 27 metres above sea level, that is positive 5, assuming we are now measuring along a vertical number line. Brian however moves to a location which is 21 metres below sea level, or negative 21 along the vertical number line. Since we are measuring distances, the distance from point zero to both hikers' new position will be measured in absolute values only, that is both distances will be measured as positive.
Hence the distance between Melissa and Brian is;
Distance = 27 + 21
Distance = 48
The difference in altitude between both hikers is 48 metres
If y varies directly with x,and y is 14when x is 2,what is the value of x when y is 35
x=5
1) Since y varies directly with x, then we can write a table
x | y
2 14
35
2) Since y is 7 times the value of x, then we can state that
x | y
2 14
5 35
3) So this variation can be expressed as y=7x and x=5
Suppose y varies directly with x, and y=6 when x=-2. Find x when y=9
The fact that y varies directly with x means that they have a relation like:
[tex]y=kx[/tex]Where k can be any number. We know that y=6 when x=-2 which means that k is given by:
[tex]\begin{gathered} y=kx \\ 6=-2k \\ \frac{6}{-2}=k \\ k=-3 \end{gathered}[/tex]Then if y=9 we get:
[tex]\begin{gathered} y=-3x \\ 9=-3x \\ \frac{9}{-3}=x \\ x=-3 \end{gathered}[/tex]So x=-3 when y=9.
What is the value of x?x = ___ ydRound your answer to the nearest tenth
We need to use the cosine of the angle, in this case:
Cos(37°) = 30 yd / X
Thus, X = 30 / Cos(37) = 30 / 0.7986 = 37.56
Consider the expression 5c+2ad+10-3d*6k how many terms are there? How many factors are in second term? Identify them which term is a constant?
The expression 5c+2ad+10-3d*6k has 4 terms , second term has 3 factors and constant term is 10.
Given expression:
5c+2ad+10-3d*6k
5c+2ad+10-18dk
Here terms = 5c, 2ad, 10, 18dk = 4 terms.
factors in second term = 2,a,d = 3 factors.
constant term is 10.
Therefore he expression 5c+2ad+10-3d*6k has 4 terms , second term has 3 factors and constant term is 10.
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A projectile is launched upward with a velocity of 288 feet per second from the top of a 35-foot platform. What is the maximum height attained by the projectile?
The maximum height attained by the projectile is 403.79 meters = 1324.77 feet.
The height formula has the following formula:
h = h₀ + v₀t - gt²/2
In which h₀ is the initial height, v₀ is the initial speed and g = 9.8 m/s² is the gravity.
Given, We are working with the gravity in meters, so we must convert the feet measures to meters.
Each feet has 0.3048 meters.
So 288 feet per second = 87.78 meters per second.
35 foot = 10.66 meters.
This means that:
h₀ = 10.66, v = 87.78
so,
h(t) = 10.66 + 87.78 - 4.9t²
The maximum height is attained at the moment of time in which the velocity is 0. The velocity is the derivative of the height. So:
v(t) = h'(t) = -9.8t+87.78
v(t) = 0
9.8t = 87.78
t = 8.96
The maximum height is attained at 8.96s. This height is
h(t) = 10.66 + 87.78t - 4.9t²
h(8.96) = 10.66 + (87.78×8.96) - (4.9×(8.96)²)
= 10.66 + 786.50 - 393.37
= 403.79 m
The maximum height attained by the projectile is 403.79 meters = 1324.77 feet.
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Risky drivers: An automobile insurance company divides customers into three categories: good risks, medium risks, and poor risks. Assume that of a total of 11,210 customers, 7761 are good risks, 2499 are medium risks, and 950 are poor risks. As part of an audit, one customer is chosen at random. Round youranswers to four decimal places if necessary.Part 1 of 2(a) What is the probability that the customer is a good risk?The probability that the customer is a good risk isXŚPart: 1/2Part 2 of 2(b) What is the probability that the customer is not a poor risk?The probability that the customer is not a poor risk isXŚ
Given data:
Total: 11210
7761 good risk
2499 medium risk
950 poor risk
a) Probability of choose a customer with good risk (gr):
[tex]P(gr)=\frac{#customer\text{ }gr}{#total}=\frac{7761}{11210}=0.6923[/tex]Th eprobability that the customer is a good risk is 0.6923b) Probability that a customer is not a poor risk (pr):
[tex]P(no\text{ }pr)=1-P(pr)=1-\frac{#custormer\text{ }pr}{#total}=1-\frac{950}{11210}=\frac{11210-950}{11210}=\frac{10260}{11210}=0.9153[/tex]The probability that the customer is not a poor risk is 0.9153Sam is paid $50 per room that he paint and he paint room in exactly two hours on sunday sam hopes to make at least $150 painting rooms and can work for exactly 10 hours which of the following sets represents the range of hours H that Sam can work without violating his monetary or restriction
Since Sam can paint 5 rooms in 10 hours, since:
[tex]\frac{10\text{hours}}{2\text{hours}}=5\text{ rooms}[/tex]then Sam would have to paint at least 3 rooms to make $150.
The range of hours would be from 6 to 10 hours, since 3 rooms takes 6 hours to paint.
combining functionsConsider the following functions. f(-2) = -10 and g(-2) = -11Find (f +g)(-2). (f + g) (-2) =
f(-2) = -10 and g(-2) = -11
Find (f +g)(-2).
we have that
(f +g)(-2)=f(-2)+g(-2)
substitute the given values
(f +g)(-2)=-10+(-11)=-21
need help with geometry problem number 12 ( ignore my writing )
Given
Height of man = 6ft
Shadow of man = 9ft
Shadsw of building = 322.5ft
Find
Height of building
Explanation
At a particular time in the day the proportion of the shadow of man will be equal to that of the building
So triangle ABC will be similar to PQR
Hence the ratio of their corresponding sides will be equal
[tex]\begin{gathered} \frac{AB}{BC}=\frac{PQ}{QR} \\ \frac{6}{9}=\frac{PQ}{322.5} \\ PQ=322.5\times\frac{2}{3} \end{gathered}[/tex]Therefore,
PQ = 215ft
which is the required length
Final Answer
The height of the building is 215ft
Select the correct answer from each drop-down menu.Given: and Prove:
Answer:
Given that,
[tex]GH\cong JH[/tex]GH and JH are congruent.
[tex]IG\cong IJ[/tex]IG and IJ are congruent.
The third side of the triangle GHI is HI and also for the triangle JHI is HI
HI is common for both the triangle GHI and triangle JHI.
we get,
[tex]HI\cong HI[/tex]By using SSS congrurnce criteria: SSS (Side – Side – Side) Congruence. If the three sides of one triangle are equal to the three sides of another triangle, then the two triangles are congruent.
we conclude that,
[tex]\Delta GHI\cong\Delta JHI[/tex]Hence proved.
Twin brothers, Andy and Brian, can mow their grandparent's lawn together in 60 minutes. Brian could mow the lawn by himself in 22 minutes more than it would take Andy. How long would ittake each person mow the lawn alone?lespleesIt would take Andy minutes to mow the lawn by himself(Simplify your answer.)It would take Brian minutes to mow the lawn by himself(Simplify your answer.)
STEP - BY - STEP - EXPLANATION
What to find?
The time taken for each person to mow the lawn alone.
Given:
Time take for the two to mow the lawn together = 60 minutes.
Brian could mow himself 22 minutes more than it would take andy.
Let x be the time taken for Andy to mow the lawn.
Let x + 22 be the time taken for Brian to mow the lawn.
Step 1
Form the equation.
[tex](\frac{1}{x}+\frac{1}{x+22})\times60=1[/tex]Step 2
Divide both-side of the equation by 60.
[tex]\frac{1}{x}+\frac{1}{x+22}=\frac{1}{60}[/tex][tex]\frac{x+22+x}{x(x+22)}=\frac{1}{60}[/tex][tex]\frac{2x+22}{x^2+22x}=\frac{1}{60}[/tex]Step 3
Cross-multiply.
[tex]x^2+22x=60(2x+22)[/tex]Step 4
Open the parenthesis.
[tex]x^2+22x=120x+1320[/tex][tex]x^2+22x-120x-1320=0[/tex][tex]x^2-98x-1320=0[/tex]Step 5
Solve the above using factorization method.
[tex]\begin{gathered} x^2-110x+12x-1320=0 \\ \\ x(x-110)+12(x-110)=0 \\ \\ (x-110)(x+12)=0 \end{gathered}[/tex]Either (x-110) = 0 or x+12 =0
x =110 or x =-12
Since there is no negative timing, we will consider only the positive value.
Hence, x=110
Therefore,
The time taken Andy to mow = 110 minutes.
The time taken for Brian to mow = x+ 22 = 110+22 = 132
ANSWER
It takes Brian 132 minutes to mow the lawn himself.
It takes Andy 110 minutes to mow the lawn himself.
These tables represent an exponential function. Find the average rate ofchange for the interval from x = 8 to x = 9.хyInterval0110 to 13291 to 2Average rateof change2]x36]x318]*3543x3162]*34863272 to 34813 to 44 to 5524367295 to 6O A. 13,122O B. 3O C. 19,683D. 6561
Average rate can be calculated like the slope
[tex]\frac{y2-y1}{x2-x1}[/tex]where (x2,y2) is a right point yo (x1,y1)
But we need the values of x=8 and x=9
we realize the change between values of y is the last value by 3 then if
[tex]\begin{gathered} 6\longrightarrow729 \\ 7\longrightarrow729\times3=2187 \\ 8\longrightarrow2187\times3=6561 \\ 9\longrightarrow6561\times3=19683 \end{gathered}[/tex]we have the corresponding values for x=8 and x=9
now replace on the formula of the slope using the points (8 , 6561) and (9 , 19683)
where (9 , 19683) is (x2,y2) and (8 , 6561) is (x1,y1)
[tex]\begin{gathered} \frac{19683-6561}{9-8} \\ \\ \frac{13122}{1}=13122 \end{gathered}[/tex]the avreage rate of change for the interval 8 to 9 is 13122
We realize the values of the average are multiplied by 3 too, then we can fi
Find the measure in degrees of the smallest angle in the triangle.
Answer:
Explanation:
The sum of the angles in a triangle is 180 degrees. The angles in the given triangle are 2x, 6x + 4 and 2x + 6
Thus,
2x + 6x + 4 + 2x + 6 = 180
By collecting like terms, we have
2x + 6x + 2x + 4 + 6 = 180
10x + 10 = 180
10x = 180 - 10 = 170
x = 170/10
x = 17
The smallest angle in the triangle is 2x. Thus,
Smallest angle = 2 * 17
Smallest angle = 34 degrees
Find dy/dx by implicit differention.1) x^4 + x^2y^2 + y^3 = 5(2) Sin (x+y) = cosx + casy
The given equation is:
[tex]x^4+x^2y^2+y^3=5[/tex]The differential is given as:
[tex]4x^3+2xy^2+2yx^2\frac{dy}{dx}+3y^2\frac{dy}{dx}=0[/tex]Make dy/dx the subject of the formula:
[tex]\begin{gathered} 2yx^2\frac{dy}{dx}+3y^2\frac{dy}{dx}=-(4x^3+2xy^2) \\ \\ \frac{dy}{dx}(2x^2y+3y^2)=-(4x^3+2xy^2) \\ \\ \frac{dy}{dx}=\frac{-(4x^3+2xy^2)}{2x^2y+3y^2} \end{gathered}[/tex]and your while you have the following paper money 7 singles 352 tens + 620 is what is the probability of a fraction you would draw a 5 and then a 20
You have 7 ones
3 fives
2 tens
6 twenties
This is a total of 7+3+2+6 = 18 bills
First we want to know the probabilitity that you will draw a five
P(five dollar bill) = number of fives/ total = 3 / 18 = 1/6
Now we have 7 ones
2 fives
2 tens
6 twenties
This is a total of 7+2+2+6 = 17 bills left ( assuming you get to keep the bill)
Now we want to know the probability of getting a twenty dollar bill
P( twenty dollar bill) = number of twenties / total = 6/ 17
Multiply the probabilities together
1/6 * 6/17 = 1/17
The probability of getting a five and then a twenty is 1/17
The Lofoten Islands in Norway (one of Mr. Maier's favorite places) has a latitude of 68.4711 ° north of the equator. What is the linear speed as the earth rotates at that latitude? Use 3961.3 miles for the radius of the earth.
The equations are the linear velocity and angular moment respectively.
Since we have that the rotation of the Earth takes 24 hours, we transform it into seconds, that is:
[tex]24\cdot60\cdot60=86400[/tex]So, it has a period of 86400 seconds.
We now, transform the radius to the IS (from miles to meters), that is:
[tex]3961.3\text{miles}=6375.1\operatorname{km}[/tex]And, since the latitude is 68.4711° we solve in the function given at the start, that is:
[tex]w=\frac{2\pi}{86400}\Rightarrow w=7.272205217\cdot10^5[/tex]Then we divide this value by the time it takes to do a revolution of the Earth, the previously calculated 86400 seconds, that is:
[tex]v=wr\Rightarrow w=(7.272205217\cdot10^{-5})(6375.1)[/tex][tex]\Rightarrow v\approx0.464[/tex]So, the linear velocity at that latitude is approximately 0.464 Km/s.
Tim bought a new computer for his office for $1200. He read that thecomputer depreciates (loses value) at a rate of $200 per year. What will bethe value of the computer after 3 years? *
If each year the computer loses $200 of it's value, in 3 years it'll have lost 3 times that amount i.e. $600.
So, after 3 years the computer will have $600 less the value than when Tim bought it:
[tex]1200-600=600[/tex]After 3 years, the value of the computer will be $600
The loss of the computer's value can be model as:
[tex]y=-200x+1200[/tex]Where 1200 is the initial value, 200 is how much it's devalued per year. 'x' represents the years since Tim bought the computer and 'y' represents it's value after 'x' years.
(a) Ivanna is driving on the freeway at a constant speed. She then speeds up to pass a truck. After passing the truck, she exits the freeway and slowsdown.SpeedSpeedSpeedSpeedTimeTimeTimeTimeOO
Ok, so
Ivanna is driving on the freeway at a constant speed. She then speeds up to pass a truck. After passing the truck, she exits the freeway and slows
down.
Notice that if the speed is constant, that means that the speed won't increase or decrease its value.
Then, the proper graph could be:
And the correct answer for (a) is c.
Now, the sight-seeing ship is stopped in the water for an hour. This means that it didn't change its distance. Then, the captain heads the ship back to the shore at a constant rate, which means that the distance to the shore is 0. The ship stays there for a while and then returns to the open sea.
Now, the correct graph will be:
The correct option is a.
Rationalize the denominator of the fraction below. What is the newdenominator?
To rationalize the denominator we have to multiply it by the conjugate, as follows:
[tex](3+\sqrt{6})(3-\sqrt{6})=3^2-(\sqrt{6}\rparen^2[/tex]The latter considering the property (a-b)(a+b)=a^2-b^2.
Finally we have:
[tex]3^2-6=9-6=3[/tex]The the answer is D. 3
John takes out a loan for $ 12 , 000 at a simple interest rate of 5% to be paid back in 36 monthly installments. What is the amount of the monthly payments?CORRECTION: The interest rate is 7.9%.
From the statement, we know that:
• John takes a loan for A₀ = $12,000,
,• at a simple interest rate of r = 7.9% = 0.079per year,
,• to be paid back in 36 monthly instalments.
1) The total amount to be paid is given by:
[tex]A=A_0\cdot(1+r\cdot n_{years})=\text{\$12,000}\cdot(1+0.079\cdot3)=\text{\$14,844.}[/tex]2) The monthly payments are given by:
[tex]m=\frac{A}{n_{months}}=\frac{\text{\$14,844}}{36}\cong\text{\$412.33.}[/tex]AnswerThe monthly payments are $412.33.
Let Z be a standard normal random variable. Calculate the following probabilities using the ALEKS calculator. Round your responses to at least three decimal places.
(a)
P (Z > -1.62 ) = 0.94738
(b)
P (Z ≤ 1.72) = 0.95728
(c)
P (-058 < Z < 1.91 ) = 0.69097
Z<1.91 = 0.97193
Z<-0.58 = 0.28096
0.97193 - 0.28096 = 0.69097
14) Which of the following is NOT a rational number?a. product of 15 and .25b. sum of 2/5 and 1/2c. the sum of 2+√√4 and 15-√4d. product of 20 and √6
If we multiply 20 by squareroot 6, the answer would be 20(squareroot6). So it is a rational numbersquare root.
But in a, if we multiply 0.25 that is 1/4, by 15 it will be 15/4, so. itis a rational number
In b:
[tex]\frac{2}{5}+\frac{1}{2}=\frac{4+5}{10}=\frac{9}{10}[/tex]It is rational.
However, let's do the operation.
[tex](2+\sqrt{4})+(15\text{ - }\sqrt{4})=(2+2)+(15\text{ - }2)=4+13=17[/tex]Therefore, it is a natural number and not a rational. The answer is c
2x + 9 + 3x + x = __x + __Fill in the empty spaces to make this equation have one solution
Answer:
2x + 9 + 3x + x = 7x + 5
Explanation:
The expression on the left side is equal to
2x + 9 + 3x + x
Adding the like terms, we get
(2x + 3x + x) + 9
6x + 9
Then, the given equation is
2x + 9 + 3x + x = __x + __
To make this equation have one solution, the coefficient of x on the right side has to be different from 6, which is the coefficient of 6x + 9.
Therefore, we can fill the empty spaces as
2x + 9 + 3x + x = 7x + 5
Solving this equation, we get:
(2x + 3x + x) + 9 = 7x + 5
6x + 9 = 7x + 5
6x + 9 - 5 = 7x + 5 - 5
6x + 4 = 7x
6x + 4 - 6x = 7x - 6x
4 = x
Therefore, the only solution is x = 4.
Graph the function for the given domain.-2x - 10y = 10 , D: (-5, 0, 5, 10)
In order to graph the function, we need first to find the coordinates of the points. To do so, we need to apply the values of x given in the domain and calculate the corresponding values of y. So we have that:
[tex]\begin{gathered} x=-5\colon \\ -2\cdot(-5)-10y=10 \\ 10-10y=10 \\ -10y=0 \\ y=0 \\ \\ x=0\colon \\ -2\cdot(0)-10y=10 \\ -10y=10 \\ y=-1 \\ \\ x=5\colon \\ -2\cdot(5)-10y=10 \\ -10-10y=10 \\ -10y=20 \\ y=-2 \\ \\ x=10\colon \\ -2\cdot(10)-10y=10 \\ -20-10y=10 \\ -10y=30 \\ y=-3 \end{gathered}[/tex]So the points are (-5, 0), (0, -1), (5, -2) and (10, -3).
Graphing these points, we have that:
Which undefined geometric term is described as an infinite set of points that has length but not width?distancelineplanesphere
A line (option B)
Explanation:
A line contains an infinite number of points. It has no width. It is one dimentional.
A plane has is two dimentional. Hence it has awidth and length.
The correct answer is a line
You spin the spinner twice.5243What is the probability of landing on a number less than 3 and then landing on a 5?Simplify your answer and write it as a fraction or whole number.Submit
To get the probability of an event, we need two things:
1. The total number of possibilities
2. The total number of possibilities considered favorable
For the first event A which is landing on a number less than 3, we only have one favorable possible which is landing on 2.
For the second event B which is landing on a 5, we also only have one favorable possible which is landing a 5 itself.
Now, for both events, the total number of possibilities is 4.
So, the probability of landing on a number less than 3 is 1/4 while the probability of landing on a 5 is also 1/4.
So, the probability of landing on a number lesser than 3 AND landing on a 5 is:
[tex]P(A\text{ }and\text{ }B)=P(A)\times P(B)[/tex][tex]P(A\text{ }and\text{ }B)=\frac{1}{4}\times\frac{1}{4}[/tex][tex]P(A\text{ }and\text{ }B)=\frac{1}{16}[/tex]Answer:
The probability of landing on a number lesser than 3 AND landing on a 5 on the next spin is 1/16.
the following table represents the probability distribution of the number of vacations X taken last year for a randomly chosen family. compute the standard deviation
We have to use the formula for standard deviation of a probability distribution:
[tex]\sigma=\sqrt[]{\sum^{}_{i\mathop=0}(x_i-\mu)^2\cdot P(x_i)}[/tex]x P(x) x*P(x) (xi - μ)^2*P(x)
0 0.11 0 0.180
1 0.64 0.64 0.050
2 0.13 0.26 0.067
3 0.1 0.3 0.296
4 0.02 0.08 0.148
The expected value μ would be the sum of the values of the third column of the table.
Therefore μ = 1.28
The sum of the values of the fourth column would be: 0.7416
Taking the square root of the last value, we have: 0.861
The answer is option D
Check for Understanding 1eteofLook at the function table below and select the correct equation rule.tly510714918Select one:Of(0) =Of(0) = 21 +1Of(1) = 20Of(1) = 21 - 1Check
C) y= 2x
1) Examining that table, we can notice that the value of y is precisely twice the value of x
x | y
5 10
7 14
9 18
2) Hence, we can state since y is twice x that the function can be written as:
y=2x
3) So the answer is C) y= 2x
Mona bought 3 3/8 pounds of cheese. She used 2 3/4 pounds to make sandwiches. Write and solve an equation to find how much cheese is left.
In this case, we'll have to carry out several steps to find the solution.
Step 01:
Data:
cheese:
purchased amount = 3 3/8 lb
used amount = 2 3/4
Step 02:
equation:
remaining amount = purchased amount - used amount
[tex]\begin{gathered} \text{remaining amount = 3 3/8 - 2 3/4 }=\text{ (3 + }\frac{3}{8})\text{ - (2 + }\frac{3}{4}) \\ \text{ } \end{gathered}[/tex][tex]\text{remaining amount = }\frac{27}{8}\text{ - }\frac{11}{4}=\frac{5}{8}[/tex]The answer is:
5/8 lb = 0.625 lb