Julia is an intern at an architecture firm. She is given an assignment to create a bell-shaped structure that is symmetrical. She writes the function f(x)=3 squareroot -|x-2|+6 to model the structure. Find the piecewise function that matches this absolute value function. Then, graph the function using a graphing calculator and describe what you see.
Answers
Answer:
The given function is
f(x)= 3 [tex]\sqrt\left- |x \right-2 |+6[/tex] ⇒[Absolute value function]
F(x)= 1. 3[tex]\sqrt {-(x-2)+6}[/tex] =[tex]3\sqrt {-x+2+6}=3\sqrt{-x+8}[/tex] when (x-2≥0→x≥2) ⇒[ Piecewise function]
2. [tex]3\sqrt{-[-(x-2)]+6}=3\sqrt{x-2+6}=3\sqrt{x+4}[/tex] when [ x-2<0→x<2]⇒[ Piecewise function]
We see there is a shape which is is in the form of dome,point of intersection of two curves being (2,7.348), symmetrical on both side of line x=2.
Related Questions
You have an equally likely chance of choosing any number from 1 to 10. What is the probability that you choose a number greater than 6?
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(1/10)+(1/10)+(1/10)+(1/10)=4/10=2/5
Answer is 2/5
Answer:
2/5
Step-by-step explanation:
1 to 10, it is 10 numbers
and there are 4 numbers which are 7,8,9,10 are bigger than 6
so P =4/10=2/5
A gear rotates one degree each second. If it rotates for 2 minutes, how many degrees will it measure
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convert the following logarithmic equation to the equivalent exponential equation. Use the caret (^) to enter exponents. y=ln x
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f(x)=a^x
"a" is the base and "x" is the exponent.
2. It is important to know that the logarithmic functions and the exponential functionsare inverse. Then, you have:
y=ln x
e^y=e^(lnx)
e^y=x
3. Therefore, the answer is:
x=e^y
Answer:
To solve this problem, you need to remember that an exponential function has the following form:
f(x)=a^x
"a" is the base and "x" is the exponent.
2. It is important to know that the logarithmic functions and the exponential functionsare inverse. Then, you have:
y=ln x
e^y=e^(lnx)
e^y=x
3. Therefore, the answer is:
Step-by-step explanation:
what does this expression represent in words 12f+24
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line AB is parallel to line CD and line CD is perpendicular to line EF what can you conclude about AB and EF?
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Answer:
Line AB and line EF are perpendicular.
Hope this helps!
If line AB is parallel to line CD and line CD is perpendicular to line EF, we can conclude that line AB is also perpendicular to line EF.
Explanation:If line AB is parallel to line CD and line CD is perpendicular to line EF, we can conclude that line AB is also perpendicular to line EF.
The reason for this is that if two lines are parallel to the same line, they are parallel to each other. In this case, we have line AB and line CD both parallel to line EF. Then, because line CD is perpendicular to line EF, line AB must also be perpendicular to line EF.
pls help me on this math question.I am really confused
Answers
Answer:
640 m
Step-by-step explanation:
We can consider 4 seconds to be 1 time unit. Then 8 more seconds is 2 more time units, for a total of 3 time units.
The distance is proportional to the square of the number of time units. After 1 time unit, the distance is 1² × 80 m. Then after 3 time units, the distance will be 3² × 80 m = 720 m.
In the additional 2 time units (8 seconds), the ball dropped an additional
... (720 -80) m = 640 m
_____
Alternate solution
You can write the equation for the proportionality and find the constant that goes into it. If we use seconds (not 4-second intervals) as the time unit, then we can say ...
... d = kt²
Filling in the information related to the first 4 seconds, we have ...
... 80 = k(4)²
... 80/16 = k = 5
Then the distance equation becomes ...
... d = 5t²
After 12 seconds (the first 4 plus the next 8), the distance will be ...
... d = 5×12² = 5×144 = 720 . . . meters
That is, the ball dropped an additional 720 -80 = 640 meters in the 12 -4 = 8 seconds after the first data point.
It is recommended that one fire extinguisher be available for every 6,000 square feet in a building. Write and solve an equation to determine x, the number of fire extinguishers needed for a building that has 135,000 square feet.
Answers
Answer:
[tex]6000x=135000[/tex]
23 fire extinguisher.
Step-by-step explanation:
Let x be the number of fire extinguishers.
We have been given that one fire extinguisher be available for every 6,000 square feet in a building. So the total area covered by x extinguisher will be 6,000x.
Since the building has an area of 135,000 square feet. This means that that total area covered by x extinguisher is 135,000 square feet. We can represent this information in an equation as:
[tex]6000x=135000[/tex]
Therefore, the equation [tex]6000x=135000[/tex] represents the number of fire extinguishers needed for a building that has 135,000 square feet.
Now we will solve for by dividing both sides of our equation by 6000.
[tex]\frac{6000x}{6000}=\frac{135000}{6000}[/tex]
[tex]x=\frac{135}{6}[/tex]
[tex]x=22.5[/tex]
Since we can not have 0.5 of a fire extinguisher, so we will round up our answer as:
[tex]x\approx 23[/tex]
Therefore, 23 fire extinguisher are needed for the building.
This figure consists of a rectangle and semicircle.
What is the perimeter of this figure?
Use 3.14 for pi.
24.00 ft
30.28 ft
34.28 ft
36.56 ft
HELP ASAP WILL GET BRAINLIEST AND 10 POINTS!! IF CORRECT
Answers
Answer:
30.28
Step-by-step explanation:
just did the k12 test so i thought i would help out:)
compare:
(a÷b)^2.........a^2/b^2
Answers
(a/b)^2 = (a/b)(a/b) = (a*a)/(b*b) = a^2/b^2
What is a sequence? a sequence is an unordered list of numbers. a sequence is an ordered list of numbers. a sequence is the sum of an?
Answers
Jayden has been reading a 275275275-page book for English class. On average, he has read about 101010 pages in 151515 minutes (\text{min})(min)left parenthesis, m, i, n, right parenthesis. With a deadline looming, Jayden realizes that he won't finish in time unless he skims through the rest of the book. If he skims the book, he can cover 151515 pages in 10\,\text{min}10min10, space, m, i, n. If it takes Jayden 555 hours and 50\,\text{min}50min50, space, m, i, n to finish the book (reading and skimming combined), how many pages did Jayden have to skim through?
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r: reading
s: skimming
We write the system of equations that adapts to the problem:
r + s = 275
(15/10) r + (10/15) s = 355
We solve the system of equations:
r = 206
s = 69
Note: see attached image for system solution.
Answer:
Jayden had to skim 69 pages through
Leo painted for 2/3 of an hour each day on Monday, Tuesday, Thursday and Friday. How long did Leo paint this week
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Leo painted for 2/3 of an hour each day on Monday, Tuesday, Thurday, and friday. Thus, the total time of the week for which he painted is 1.58% of the total time.
What is a fraction?A fraction represents a part of a whole number or, more commonly, any number of equal parts. A fraction can be defined as the parts of a certain size, for example, one-half, eight-fifths, three-quarters, etc.
The time for which Leo painted in this week can be calculated as:
Leo painted 2/3 of an hour each day for monday, tuesday, thursday, and friday
Therefore, the time of painting in an hour = 40 minutes
2/3 × 60 = 120/ 3 = 40 minutes = 40 × 4 = 160 minutes
The number of minutes in a day,
1 day = 60 × 24 = 1,440 minutes
The number of minutes in seven days,
7 days = 1440 × 7 = 10,080 minutes
Therefore, the total time for which Leo painted = time for which he painted in minute divided by total time in minutes
Total time for which Leo painted = 160/ 1440 = 0.0158 or 1.58%
Therefore, total time for which he painted in this week is 0.0158 of total time
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javier has 4 juice boxes three are grape flavored write two equivalent fractions thatdescribe the part of the juice boxes that is grape
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= (3x)/(4x) for any non-zero number x
can someone help me with this pls
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try using demos it helps a lot with number lines and graphs (:
Please help:
Solve using logarithms. Round the solution to the nearest hundredth. Show your work.
(you will need to use a calculator at some point)
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So, the first thing we are going to do is take natural logarithm, [tex]ln[/tex], to both sides of our equation:
[tex]ln(5^{x+2} )=ln(9^{2x+3} )[/tex]
Now, by the power rule of logarithms:
[tex](x+2)ln(5)=(2x+3)ln(9)[/tex]
[tex] \frac{(x+2)ln(5)}{2x+3} =ln(9)[/tex]
[tex] \frac{x+2}{2x+3}= \frac{ln(9)}{ln(5)} [/tex]
[tex] \frac{x+2}{2x+3} =1.365[/tex]
[tex]x+2=1.365(2x+3)[/tex]
[tex]x+2=2.73x+4.095[/tex]
[tex]-1.73x=2.095[/tex]
[tex]x=- \frac{2.095}{1.73} [/tex]
[tex]x=-1.211[/tex]
Using our calculator we can conclude that [tex]x[/tex] is approximately -1.211
Use the chain rule to find ∂z/∂s and ∂z/∂t. z = x5y7, x = s cos(t), y = s sin(t)
Answers
Using the Chain Rule of differentiation, we can solve for ∂z/∂s and ∂z/∂t given the functions for z, x, and y. The Chain Rule is fundamental for calculus by finding the derivative of composite functions.
Explanation:To solve for ∂z/∂s and ∂z/∂t with z = x5y7, x = s cos(t), and y = s sin(t), we'll employ the use of the Chain Rule of differentiation. First, we consider ∂z/∂s. As z is x raised to the power of 5 then multiplied by y to the power of 7, we can express ∂z/∂x and ∂z/∂y. Proceeding using the chain rule, which in simpler terms could be stated as 'the derivative of the outside times the derivative of the inside', we get the values of ∂z/∂s and ∂z/∂t. The chain rule in calculus is an important tool when differentiating composite functions.
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Ethan has a small coin collection. Nine of his coins he collected from traveling. Six of his coins he received from friends. What is the ratio of coins he collected from friends to the total coin in Ethan's collection?
2:3
2:5
9:6
1:5
Answers
Answer:
2:5!
Step-by-step explanation:
Have a great day!
consider the quadratic equation x^2=4x-5.How many solutions does the equation have?
Answers
Please help me with question 22
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Erase line A. For the pair of lines B and C, the marked angles are same side exterior angles. They are supplementary
(2x+18) + (6x+26) = 180
8x + 44 = 180
8x = 180-44
8x = 136
x = 136/8
x = 17
Since x = 17,
smaller angle = 2x+18
smaller angle = 2*17+18
smaller angle = 34+18
smaller angle = 52 degrees
What is the slope of the line which passes through (4, 5) and (0, 1)?
Undefined
1
0
4
Answers
slope = 1
Explanation:
slope is calculated using the following rule:
slope = (y2 - y1) / (x2 - x1)
The given points are (4,5) which represent (x1,y1) and (0,1) which represent (x2 , y2)
Substitute with these points in the above equation to get the slope as follows:
slope = (1-5) / (0-4) = 1
Hope this helps :)
Answer:
1
Step-by-step explanation:
Find r(t) if r'(t) = 3t2i + 4t3j + t k and r(1) = i + j.
Answers
Final answer:
To find the position vector r(t) given its derivative and an initial condition, we integrate each component of the derivative and apply the initial condition to solve for constants of integration. The final position function is r(t) = t³i + t⁴j + (0.5t² - 0.5)k.
Explanation:
To find r(t) given that r'(t) = 3t²i + 4t³j + tk and r(1) = i + j, we integrate each component of r'(t) with respect to t. The integral of a derivative returns the original function plus a constant of integration, which we can solve using the initial condition provided.
For the i component: ∑ 3t² dt = t³+ C1
For the j component: ∑ 4t³ dt = t⁴ + C2
For the k component: ∑ t dt = 0.5t² + C3
Applying the initial condition r(1) = i + j, we substitute t = 1 into r(t) to solve for the constants of integration:
(1) + C1 = 1, so C1 = 0
(1) + C2 = 1, so C2 = 0
0.5(1) + C3 = 0, so C3 = -0.5
Therefore, the position function r(t) is given by t³i + t⁴j + (0.5t² - 0.5)k.
r(t) = t^3i + t^4j + ((1/2)t^2 - 1/2)k.
To find r(t) given r'(t) = 3t^2i + 4t^3j + tk and r(1) = i + j, we need to integrate r'(t) with respect to t.
Step 1: Integrate each component of r'(t) separately:
∫3t^2 dt = t^3 + C1 (integration with respect to t)
∫4t^3 dt = t^4 + C2 (integration with respect to t)
∫tk dt = (1/2)t^2k + C3 (integration with respect to t)
Step 2: Combine the results to get r(t):
r(t) = (t^3 + C1)i + (t^4 + C2)j + ((1/2)t^2k + C3)
Step 3: Use the given initial condition r(1) = i + j to find the values of the constants C1, C2, and C3:
r(1) = (1^3 + C1)i + (1^4 + C2)j + ((1/2)(1)^2k + C3)
i + j = i + j + (1/2)k + C3
Comparing the coefficients of k, we get:
(1/2) + C3 = 0
C3 = -1/2
Therefore, r(t) = (t^3 + C1)i + (t^4 + C2)j + ((1/2)t^2 - 1/2)k
What will the range for the function f(g)=3g-5 for the domain {-1.5, 2, 4}? Please show your work!
Suppose a van gets 22 mi/gal. The distance traveled D(g) is a function of the gallons used. The rule D(g) = 22g should be used for all parts.
Part A: How far did the van travel if it used 10.5 gallons of gas? PLEASE SHOW YOUR WORK!
Part B: How far did the van travel if it used 20 gallons of gas? PLEASE SHOW YOUR WORK!
Answers
use the babylonian method to approximate square root of 24 to the nearest hundredth.
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Divide 24 by 12; 24/12=2. Average this answer with our guess: (12+2)/2=7. This is our new guess.
24/7=3.428571429. Average this with our guess of 7: (3.428571429+7)/2=5.214285715. This is our new guess.
24/5.214285715=4.602739726. Averaging with our guess: (4.602739726+5.214285715)/2=4.90851272. New guess!
24/4.90851272=4.889464766. Averaging with our guess: (4.889464766+4.90851272)/2=4.898988743. New guess! You can see as we go through our guesses are closer and closer to the same number...)
24/4.898988743=4.898970228. Averaging: (4.898970228+4.898988743)/2=4.898979486. At this point our answer is the same every time down to the hundred-thousandth. Our estimate to the nearest hundredth would be 4.90.
The circle is centered at the point (-3,4) and has a radius of length 3. What is its equation?
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Find the longer diagonal of a parallelogram having sides of 10 and 15 and an angle measure of 120° between them. 13.2 18.0 21.8 23.1
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to get the length of the longest diagonal we use the cosine rule which states that----------> c²=a²+b²-2abCos(C)
where a,b and c are the sides and C is the angle.
therefore
a=15,b=10,C=120,c=?
to solve for the length C we shall substitute the values
c²=15²+10²-2*15*10*cos 120
c²=225+100-300*cos120
c²=325-(300*(-0.5))
c²=325-(-150)c²=475
c=√475
c=21.79-------------> c=21.8 units
since this is the opposite side to the largest angle, we therefore conclude that the longer diagonal of the parallelogram is 21.8 units
the answer is 21.8 units
Answer:
21.8
Step-by-step explanation:
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Quadrilateral ABCD is inscribed in this circle.
What is the measure of angle C?
Enter your answer in the box.
Answers
But first, we should solve for x
Opposite angles add up to 180 degrees.
So we'll use angles B and D
x + 10 + x + 24 = 180
combine like terms
2x + 34 = 180
subtract 34 from both sides
2x = 146
divide both sides by 2
x = 73
Now let's solve for the angles
Angle D: x + 24 = 97
Angle B: x + 10 = 83
Angle A: x + 15 = 88
Opposite angles add up to 180
we know that angle A is 88
C + 88 = 180
subtract 88 from both sides
C = 92
Hope this helps!
Answer:
92 is correct
Step-by-step explanation:
paul is in charge of the egg toss at the world egg day fair. there are 48 people participating in teams of 8 people. each team needs 1 egg. paul is buying eggs in cartons that each have 6 eggs. how many cartons does paul need?
Answers
Final answer:
Paul needs to buy 1 carton of eggs, as there are 6 teams needing 1 egg each, and one carton contains 6 eggs.
Explanation:
Paul needs to calculate how many cartons of eggs to buy for the egg toss at the world egg day fair.
With 48 people participating in teams of 8, we have a total of 48/8 = 6 teams, and each team needs 1 egg.
Since each carton contains 6 eggs, Paul needs to buy 6/6 = 1 carton to provide one egg for each team.
Consider the diagram and the paragraph proof below. Given: Right △ABC as shown where CD is an altitude of the triangle Prove: a2 + b2 = c2 Because △ABC and △CBD both have a right angle, and the same angle B is in both triangles, the triangles must be similar by AA. Likewise, △ABC and △ACD both have a right angle, and the same angle A is in both triangles, so they also must be similar by AA. The proportions and are true because they are ratios of corresponding parts of similar triangles. The two proportions can be rewritten as a2 = cf and b2 = ce. Adding b2 to both sides of first equation, a2 = cf, results in the equation a2 + b2 = cf + b2. Because b2 and ce are equal, ce can be substituted into the right side of the equation for b2, resulting in the equation a2 + b2 = cf + ce. Applying the converse of the distributive property results in the equation a2 + b2 = c(f + e).
Answers
This is because since f + e = c,
then a² + b² = c(f + e)
a² + b² = c(c)
a² + b² = c²
This is the correct answer
Answer:
Because f+e=c
Therefore ,[tex]a^2+b^2=c^2[/tex]
Step-by-step explanation:
Given A right triangle ABC as shown in figure where CD is an altitude of the triangle.
To prove that [tex]a^2+b^2=c^2[/tex]
Proofe: Given [tex]\triangle ABC\; and\; \triangle CBD[/tex] both are right triangle and both triangles have common angle B si same.
Therefore , two angles of two triangles are equal .
Hence, [tex]\triangle ABC \sim\triangle CBD[/tex] by using AA similarity.
Similarity property: when two triangles are similar then their corresponding angles are equal and their corresponding side are in equal proportion.[tex]\frac{a}{f} =\frac{c}{a}[/tex]Similarly , [tex]\triangle ABC \sim \triangle ACD[/tex] by AA similarity property . Because both triangles are right triangles therefore, one angle of both triangles is equal to 90 degree and both triangles have one common angle A is same .
[tex]\therefore[/tex] [tex]\frac{b}{c} =\frac{e}{b}[/tex]
The corresponding parts of two similar triangles are in equal proportion therefore , two proportion can be rewrite as
[tex]a^2=cf[/tex] (I equation)
and [tex]b^2=ce[/tex] (II equation)
Adding [tex]b^2[/tex] to both sides of firs equation
[tex]a^2+b^2=cf+b^2[/tex]
Because [tex]b^2=ce[/tex] and ce can be substituted into the right side of equation wevcan write as
[tex]a^2+b^2=ce+cf[/tex]
Applying the converse of distributive property we can write
[tex]a^2+b^2=c(f+e)[/tex]
Distributive property: a.(b+c)= a.c+a.b
[tex]a^2+b^2=c^2[/tex]
Because f+e=[tex]c^2[/tex]
Hence proved.
Which equation represents a nonlinear function?
x(y – 5) = 2
y – 2(x + 9) = 0
3y + 6(2 – x) = 5
2(y + x) = 0
Answers
_____________________________________________________
Consider:
_____________________________________________________
Choice: [A]: " x(y–5) = 2 " ;
Divide each side by "x" ;
" [x(y – 5)] / x = 2/x " ;
→ y – 5 = 2/x ;
Add "5" to each side of the equation:
y – 5 + 5 = 2/x + 5 ;
→ y = 2/x + 5 ; not a line; since one cannot divide by "zero" ; there would be no "point" on the graph at "x = 0". So, this answer choice: [A] is correct.
_______________________________________________________
Choice [B]:
y -2x -18 = 0
y - 2x = 18
y = 18 + 2x ; y = 2x + 18 ; is a line.
_________________________________________________
Choice C) 3y + 12 - 6x = 5 ;
3y = 5 - 12 + 6x ;
3y = -7 + 6x ; 3y = 6x - 7 ; y = 6x/3 - 7/3 ; y = 2x - 7/3 ; is a line.
_________________________________________________
Choice: [D]:
2(y+x) = 0 ;
[2*(y+x)] / 2 = 0/2 ; y + x = 0 ; y = -x ; is a line.
_________________________________________________
Answer:
The correct answer is: A: x(y – 5) = 2
Step-by-step explanation:
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A roadway repair crew has fixed 42 mi of the roadway. That number of miles is 30% of the total number of miles that need to be repaired
Answers
Answer:
The answer is 140 miles 100x42 is 4200 divide that by 30= 140Step-by-step explanation: