Answer:
Step-by-step explanation:
a. The equation for the cost of AT&T would be $100 + $15n, where n is the number of phones on the account. The equation for the cost of Sprint would be $200.
We can graph these equations on a coordinate plane with the x-axis representing the number of phones on the account and the y-axis representing the monthly cost. The graph for the cost of AT&T would be a straight line starting at the point (0,100) and going through the point (1,115), (2,130), etc. The graph for the cost of Sprint would be a horizontal line at the y-coordinate of 200.
b. In order to determine which company would be cheaper for Carly's family, we can simply plug in the number of phones they have (5) into each of the equations we derived in part a. This would give us a cost of $200 for AT&T and $200 for Sprint. In this case, both companies would be equally cheap for Carly's family.
Alternatively, we could graph the equations and see which company's line is lower on the y-axis for the given number of phones (5). This would also show that both companies would be equally cheap for Carly's family in this case.
2. Sandy Sellers works 30 hours per week
with an hourly rate of pay of $15.00 per
hour. She has deductions from each
paycheck of 12% for federal income tax,
2.84% for North Carolina income tax, 6.2%
for Social Security, and 1.45% for Medicare.
What is Sandy's net take home pay each
week?
Answer:
$348.79
Step-by-step explanation:
You want to know Sandy Sellers' net weekly pay if she earns $15 per hour for 30 hours' work and has deductions of ...
12% federal tax2.84% NC tax6.2% SS1.45% Medicare.Gross PaySellers' net pay will be her gross pay, less the taxes. Those taxes are all percentages of her gross pay.
gross pay = (hourly rate) × (hours worked)
gross pay = ($15.00/hour) × (30 hours) = $450
DeductionsThe deductions from her pay total ...
12% +2.84% +6.2% +1.45% = 22.49% . . . . of her gross pay
The dollar amount of the deductions is ...
0.2249 × $450 = $101.205 ≈ $101.21
Net PayThen Sellers' net pay is ...
net = gross - deductions
net = $450 -101.21 = $348.79
Sandy Sellers' take-home pay each week is $348.79.
PLS HURRY 30 PTS Write the math sentence as an equation: Negative six times the sum of a number and 18.3 is 27.9
−6(r + 18.3) = 27.9
−6(r − 18.3) = 27.9
−6r + 18.3 = 27.9
−6r − 18.3 = 27.9
Answer: A
Step-by-step explanation:
From the statement.
Times = Multiplication
The sum of a number and 18.3 = (r + 18.3)
Equals = 27.9
Therefore,
-6 x (r + 18.3) = 27.9
To multiply a decimal number by 10,
O move the decimal point one place to the right.
move the decimal point two places to the right.
O move the decimal point one place to the left.
O move the decimal point two places to the left.
Answer:
[tex]\huge{\boxed{\text{A) move the decimal point one place to the right}}}}[/tex]
Step-by-step explanation:
Think about the following scenarios
4.00 * 10
529.00 * 10
4 * 10 = 40 (4.00 * 10 = 40.0)
529 * 10 = 5290 (5290.00 * 10 = 5290.0)
As you can see, the decimal point is moving one space to the right when multiplying by ten.
In the decimal system, moving the decimal point one place to the right multiplies by ten, while moving the decimal point one place to the left multiplies by 1/10.
Hope it helps :) and let me know if you want me to elaborate.
Please help, picture of question is attached
Bearing is a topic that deals with angles and distance. Thus, the required answers to the given question are:
i. Distance of 1034.2 miles
ii. Bearing of [tex]158^{o}[/tex]
The speed of an object is the rate of movement of the object with time.
speed = [tex]\frac{distance}{time}[/tex]
So that:
distance = speed x time
i. In the given question; let the initial distance covered be represented by d, so that:
d = 250 x 2
= 500 miles
ii. Let the second distance covered by the pilot be represented by D, so that:
D = 300 x 2
= 600 miles
iii. Thus let the distance to the starting point be represented by x. From Cosine rule, we have:
[tex]c^{2}[/tex] = [tex]a^{2}[/tex] + [tex]b^{2}[/tex] - 2abCos C
[tex]x^{2}[/tex] = [tex]500^{2}[/tex] + [tex]600^{2}[/tex] - 2(500 x 600)Cos [tex]140^{o}[/tex]
= 250000 + 360000 - 600000(-0,7660)
= 610000 + 459600
= 1069600
x = [tex]\sqrt{1069600}[/tex]
= 1034.22
x = 1034.2 miles
Using the Sine rule to determine the required internal angle, we have:
[tex]\frac{a}{Sin A}[/tex] = [tex]\frac{c}{Sin C}[/tex]
[tex]\frac{500}{Sin A}[/tex] = [tex]\frac{1034.22}{Sin 140}[/tex]
Sin A = [tex]\frac{500*0.6428}{1034.22}[/tex]
= 0.3108
A = [tex]Sin^{-1}[/tex] 0.3108
= 18.11
A = [tex]18^{o}[/tex]
To determine angle B, we have:
140 + 18 + B = 180 (sum of angles in a triangle)
158 + B = 180
B = 180 - 158
= 22
B = [tex]22^{o}[/tex]
Thus, the required bearing is 180 - 22 = 158
Therefore the distance and bearing required to fly to starting point are 1034.2 miles and [tex]158^{o}[/tex] respectively.
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Please help work due at 11:59( look at photo for answer choices)
The vertex labels of Δ QRT are : ∠2 label is Q, ∠8 label is T and the remaining vertex label is R. The vertex labels for ΔSTR are: ∠5 label is S, ∠9 label is R and the remaining label is T.
What is a triangle?The polygon having three sides and three vertices are known as triangle.
If we separate the two triangles as shown in the figure. We can find the vertex labels with the help of the angles marked in the figure.
For Δ QRT, the vertex labels are:
∠2 is Q, ∠1 is R and ∠8 is T
Similarly. for ΔSTR, the vertex labels are:
∠5 is S, ∠6 is T, and ∠9 is R
Hence, For Δ QRT, we can label as ∠2 is Q, ∠1 is R and ∠8 is T and for ΔSTR we can label as ∠5 is S, ∠6 is T, and ∠9 is R.
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Solve the system: x + 2y = 30 -2x - 2y = -38 ordered pair
Answer:
(8, 11)
x=8
y=11
Step-by-step explanation:
I solved using elimination.
I set the equations over one another getting x+2y=30 over -2x-2y=-38.
I chose elimination because the y values were already the opposite so it can easily cancel each other out. After it get's canceled out I get x=30 over -2x=-38. When using elimination we always add so you'd add x+ -2x and 30+ -38. When you add x + -2x you'd get -1x. When you add 30 + -30 you'd get -8. Now you have 1x=-8 divide both sides by -1 and you get x=8. Now you plug in 8 to one of the equations. I chose to plug it into the 1st equation which is x+ 2y=30. But because I plugged in the 8 the equation is now 8+2y=30. I subtracted 8 on both sides canceling the 8 out. So now I have 2y=22. divide both sides by 2 and x=11. So your order pair is (8, 11)
(Hopefully you were able to understand it..)
A football player throws a ball down the side line. The ball's height above the
ground is modeled by the equation h=-8t^2+24t+4, where h is the height
in feet and t is the time in seconds.
At what two times will the ball be at a height of 16 feet?
Answer: t₁=1.5-0.5√3 c ≈0.634 c t₂=1.5+0.5√3 c ≈2.366 c
Step-by-step explanation:
h=-8t²+24t+4 h=16 feet t₂=?
[tex]16=-8t^2+24t+4\\\\16-16=-8t^2+24t+4-16\\\\0=-8t^2+24t-12\\[/tex]
Divide both parts of the equation by -4:
[tex]0=2t^2-6t+3[/tex]
Thus,
[tex]2t^2-6t+3=0[/tex]
a=2 b=-6 c=3
D=b²-4ac
D=(-6)²-4(2)(3)
D=36-24
D=12
√D=√12
√D=√(4*3)
√D=2√3
[tex]t=\frac{-bб\sqrt{D} }{2a} \\\\t=\frac{-(-6)б2\sqrt{3} }{2(2)} \\\\t=\frac{6б2\sqrt{3} }{4} \\\\t_1=1.5-0.5\sqrt{3}\ c \\\\t_2=1.5+0.5\sqrt{3}\ c[/tex]
I Need help on this problem! please!
The pairs of triangles that are congruent are;
A. Δ1 and Δ5
B. Δ1 and Δ2
C. Δ2 and Δ3
D. Δ3 and Δ5
E. Δ1 and Δ3
F. Δ3 and Δ4
G. Δ1 and Δ4
H. Δ4 and Δ5
What are congruent triangles?Congruent triangles are triangles that have all three sides and all three angles on each triangle having the same length and the same measure.
The congruent angles are indicated by the number of markings
The angles in the figures are;
Angle with one arc
Angles with two arcs marks
Angles with three arc marks
The congruent sides are indicated by one dash, and two dashes
Therefore;
Δ2 s congruent to Δ1 is congruent to Δ3 by Angle-Side-Angle rule of congruency
Similarly, we have;
Δ3 is congruent to Δ4 by Angle-Side-Angle congruency postulate
Therefore;
Triangle Δ2 is congruent to Δ3 by Angle-Side-Angle congruency postulate
From the diagram, the one arc, two arcs, and three arc angles form a linear pair, therefore;
Therefore, Δ4 is congruent to Δ5
Δ1 is congruent to Δ4
Δ2 is congruent to Δ5, therefore; Δ3 is congruent to Δ5
Similarly;
Δ1 is congruent to Δ3
Δ1 is congruent to Δ2, therefore , therefore, Δ1 is congruent to Δ5
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David is running a concession stand at a soccer game. He sells nachos and sodas. Nachos
cost $1.50 each and sodas cost $0.50 each. At the end of the game, David made a total of
$78,50 and sold a total of 87 nachos and sodas combined. Which system of equations
represents this situation?
Answer:
x + y = 87
1.5x + 0.5y = 78.50
Step-by-step explanation:
Let x be the number of nachos sold and y be the number of sodas sold
x nachos will cost 1.50x dollars
y sodas will cost 0.50y dollars
So total sales = 1.5x + 0.5y and we know he made $78.50
so one equation is
1.5x + 0.5y = 78.50
Total sold = x + y and we know that this number is 87
So the second equation is
x + y = 87
Find the quotient of 6x^3-+18x^2-18x+12 , x-2
In linear equation, 96 is the quotient.
What in mathematics is a linear equation?
A linear equation is a first-order (linear) term plus a constant in the algebraic form y=mx+b, where m is the slope and b is the y-intercept.
Sometimes, the aforementioned is referred to as a "linear equation of two variables," where x and y are the variables.
f(x) = 6x^3-+18x^2-18x+12
x - 2 = 0
x = 2
f( x) = 6 * 8 + 18 * 4 - 18 * 2 + 12
= 48 + 72 - 36 + 12
= 132 - 36
= 96
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Fill in the blanks to complete the equation that describes the diagram.
Answer:
-9
Step-by-step explanation:
-3+-6=-9
The graph displaying the prices of laptops at two different stores is shown. You are helping your friend Marcus decide how much money to spend on a laptop. Which of the following statements are true? Pls help
b,d and e option are correct
What is statement?
A statement, like "Pizza is tasty," is a declaration that something is true. In the fields of law, finance, and government, there are more types of statements. Every statement makes a claim or a point.
These statements are true:
The least Marcus would pay for a laptop from electro's is $500
Marcus can accept to pay about $700 for a laptop from electro's and $1500 for a laptop from bits and bytes.
The middle 50% of laptops from electro's are priced between $500 and $900.
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Please help with this math qustion
Answer:155m
Step-by-step explanation:GK is congruent to HI
Tami earned 71 out of 80 points on a test. What percentage did Tami earn on the test?
Answer:
88.75% is the percentage Tami earned.
Need help to see if this is has a euler path?
If a graph G has an Euler path, then it must include precisely two odd vertices. Or, to put it another way, G cannot have an Euler path if the number of odd vertices is less than 2.
What is the Euler Theorem?A graph lacks Euler pathways if it contains more than two odd-degree vertices. 2. A graph contains at least one Euler path if it is connected and has exactly two vertices of odd degree or fewer. 3. If a network is connected and has 0 vertices of odd degree, it must contain at least one Euler circuit.If a graph G has an Euler path, then it must include precisely two odd vertices. Or, to put it another way, G cannot have an Euler path if the number of odd vertices is less than 2. In other words, v must have an even vertex.To learn more about Euler path refer to:
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Which inequality does the graph represent?
A two-dimensional graph with an x-axis and a y-axis is given. A dotted line is passing through the co-ordinates (1,0), and (-1,2). The region below the dotted line is shaded in grey.
A. y < –x – 1
B. y < –x + 1
C. y < x – 1
D. y < x + 1
The inequality that the graph represent is y < -x + 1. The correct option is B. y < –x + 1
Determining the inequality that represents a graphFrom the question, we are to determine the inequality that represents the given graph.
From the given information,
The line is passing through the coordinates (1, 0) and (-1, 2).
Using the formula for the equation of a line,
Using the formula,
(y - y₁)/(x - x₁) = (y₂ - y₁)/(x₂ - x₁)
Then,
(y - 0)/(x - 1) = (2 - 0)/(-1 - 1)
y/(x-1) = 2/-2
y/(x -1) = -1
y = -1(x -1)
y = -x + 1
Since the line is dotted and the shaded region is below the line, the inequality becomes
y < -x + 1
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how do i get an equation perpendicular to 5/4x + 2 that passes through the point (-6,2)
Answer: -4/5x - 22/5
Step-by-step explanation:
To find an equation of a line that is perpendicular to a given line and passes through a given point, we need to first find the slope of the given line. The slope of a line is the ratio of the change in the y-coordinate to the change in the x-coordinate of any two points on the line. For example, the slope of the line 5/4x + 2 is 5/4.
To find the equation of a line that is perpendicular to a line with slope m, we need to use the equation:
y - y1 = -1/m(x - x1)
where (x1, y1) is the given point that the line passes through. In this case, the given line has slope 5/4 and the given point is (-6,2), so the equation of the line perpendicular to 5/4x + 2 that passes through (-6,2) is:
y - 2 = -1/((5/4))(x + 6)
y - 2 = -4/5(x + 6)
y - 2 = -4/5x - 24/5
y = -4/5x - 22/5
Therefore, the equation of the line perpendicular to 5/4x + 2 that passes through (-6,2) is -4/5x - 22/5 (option "-4/5x - 22/5").
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A plane slices a double-napped cone parallel to the base of the cone. Which conic section is formed?
Answer:
Parabola
Step-by-step explanation:
A conic section is a curve obtained by the intersection of a plane with the surface of a (double-napped) cone, When the plane is parallel to the edge of one cone.
Thus, The intersection is a parabola.
Answer: degenerate hyperbola
Step-by-step explanation:
degenerate hyperbola because its the thickest of the plane so it can be formed.
If log102=0.3010, then log210 is equal to:
If log₁₀2=0.3010, the value of log₂10 will be 3.3×10⁻⁴.Logarithm identity is used to find the value of the log₂10.
What is a logarithm?It is another way to represent the power of numbers, and we say that 'b' is the logarithm of 'c' with base 'a' if and only if 'a' to the power 'b' equals 'c'.
[tex]\rm a^b = c\\log_ac =b[/tex]
It is given that, log₁₀2=0.3010
We have to find the log₂10
The following logarithm identity is used as,
logₐn=logₙa
log₂10 =1/ log₁₀2
log₂10 =1/ 0.3010
log₂10 =3.3×10⁻⁴
Thus, if log₁₀2=0.3010, the value of log₂10 will be 3.3×10⁻⁴. Logarithm identity is used to find the value of the log₂10.
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The formula V equals 1.225 √a is used to estimate how far a person can see on a clear day,where V is the visibility in miles and A is the altitude in feet. What is Ty's visibility if out the upper window of a building with a altitude of 1,602 feet? Round your answer to the nearest whole number.
Ty's visibility if out the upper window of a building with an altitude of 1,602 feet is 49.
What is altitude?
The height of an object in relation to a planet's or a natural satellite's surface, like sea level or land. the degree to which a celestial body is elevated above the horizon.
Here, we have
Given
The formula V = 1.225 √a is used to estimate how far a person can see on a clear day, where V is the visibility in miles and A is the altitude in feet.
We have to find Ty's visibility out the upper window of a building with an altitude of 1,602 feet.
We know that
V = 1.225 √a
a = 1,602 feet.
V = 1.225 √1602
V = 49
Hence, Ty's visibility out the upper window of a building with an altitude of 1,602 feet is 49.
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Match the graph to the slope of that function
Choices:
-3/2
-1/2
2
-1/5
Also pls answer my other questions
For the first graph, slope = 1/5.
For the second graph, slope = -1/2.
For the third graph, slope = 2.
For the fourth graph, slope = 1.42.
What is the slope of the line?
The slope of a line is a measure of its steepness. Mathematically, the slope is calculated as "rise over run" (change in y divided by change in x).
[tex]m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}[/tex]
For the first graph, the coordinates will be (-15, 0) and (0, 3)
Slope = 3 - 0/0-(-15)
= 3/15
Slope = 1/5
For the second graph, the coordinates will be (0, 5) and (10, 0)
Slope = 0 - 5/10- 0
= -5/10
Slope = -1/2
For the third graph, the coordinates will be (2.5, 0) and (0, -5)
Slope = -5-0/0 - 2.5
= -5/-2.5
Slope = 2
For the fourth graph, the coordinates will be (-2.8, 0) and (0, 4)
Slope = 4 - 0/0 - (-2.8)
= 4/2.8
Slope = 1.42
Hence,
For the first graph, slope = 1/5.
For the second graph, slope = -1/2.
For the third graph, slope = 2.
For the fourth graph, slope = 1.42.
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What is the function rule? Select the correct choice and fill in any answer boxes in your choice below.
A. y=_____
(Simplify your answer. Use integers or fractions for any numbers in the expression.)
B. There is no function rule.
B. There is no function rule
pls help i will give brainliest
The table represents a quadratic function C(t).
t C(t)
2 4
3 1
4 0
5 1
6 4
What is the equation of C(t)?
C(t) = −(x − 4)2
C(t) = (x − 4)2
C(t) = −x2 + 4
C(t) = x2 + 4
The quadratic equation of the function C(t) is; C(t)= (x - 4)²
How to find the quadratic function?
We are given the coordinates as;
When t = 2, C(t) = 4
When t = 3, C(t) = 1
When t = 4, C(t) = 0
When t = 5, C(t) = 1
When t = 6, C(t) = 4
The general formula formula for quadratic equation is;
y = ax² + bx + c
where c is y-intercept
We are given that x = 4, when y = 0. Thus, x = 4 is a root of the equation.
Put x = 2and y = 4 into the equation to get;
4a + 2b + c = 4 -----(1)
Put x = 3 and y = 1 into the equation to get;
9a + 3b + c = 1 -----(2)
Put x = 4 and y = 0 into the equation to get;
16a + 4b + c = 0 -----(3)
Using simultaneous equation solver gives;
a = 1, b = -8 and c = 16. Thus, our equation is;
y = x² - 8x + 16
By factorization;
y = x² - 4x - 4x + 16
y = x(x - 4) - 4(x - 4)
y = (x - 4)²
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A loan used for buying a home is called a mortgage. The Fortunato family is buying a $430,000 home. They are taking out a 30-year
mortgage at a rate of 8%.
a. Compute the monthly payment.
b. Find the total of all of the monthly payments for the 30 years.
c. What is the finance charge?
d. Which is greater, the interest or the original cost of the home?
Answer:
a. $3155.19
b. $1,135,868.40
c. $705,868.40
d. interest
Step-by-step explanation:
You want to find the monthly payment, total repaid, and finance charge on a 30-year mortgage for $430,000 at the rate of 8%.
a. PaymentYou can use a spreadsheet, financial calculator, app, or the amortization formula to find the payment amount.
A = P(r/12)/(1 -(1 +r/12)^(-12t))
where P is the principal amount of the loan, r is the annual interest rate, and t is the number of years.
A = 430,000(0.08/12)/(1 -(1 +0.08/12)^(-12·30)) ≈ 3155.19
The monthly payment is $3,155.19.
b. Total repaidThe total of the 12·30 = 360 payments is ...
360 · $3155.19 = $1,135,868.40
It will take about $1,135,868.40 to pay off the loan.
c. Finance chargeThe finance charge is the difference between the amount repaid and the original loan amount:
$1135868.40 -430000 = $705,868.40
The finance charge is $705,868.40.
d. ComparisonThe finance charge at $706k is considerably greater than the original cost of the home, $430k.
The finance charge is greater.
Kelly recorded the following data while observing the temperature, in degrees Fahrenheit, of a chemical liquid cooling in her science class.
What is the rate of change for the temperature of this liquid?
The rate of change of temp w.r.t time is -1.875°F
What is rate of change?
The rate of change (ROC) measures how quickly a variable alters over a predetermined amount of time. When discussing momentum, the term "rate of change" (ROC) is frequently used. It is typically defined as the ratio of a change in one variable to a matching change in another; visually, the rate of change is represented by the slope of a line. The Greek letter delta () is frequently used to represent the ROC.
The term "rate of change" (ROC) describes the rate at which something changes over time.
Thus, it is not the amount of individual changes themselves but rather the acceleration or slowdown of changes (i.e., the pace).
Time Temperature Difference
4 89.5 -
10 78.25 78.25 - 89.5 = -11.25
16 67.00 67 - 78.25 = -11.25
There is a common difference of -11.25 in each successive term.
Therefore, cooling of a chemical represents a linear function.
Rate of change of temp from 4 min to 10 min
=78.25 - 89.5 = -11.25
= -11.25/6
=-1.875
So, the rate of change of temp w.r.t time is -1.875°F
which means temperature is decreasing continuously when the time is increasing
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leo has 2 less than twice the toys gen, z has. if Leo has 6 toys how many does zen have?
Answer:10
Step-by-step explanation:10 times two is twelve and minus 2 is ten.
I’m a computer graphics design program, Mario created a segment with endpoints at ( 1, 5) and ( 7, 3). He then reflected the segment about a vertical line through the segment’s own midpoint. WHAT ARE THE COORDINATES OF THE ENDPOINTS OF THE NEW SEGMENTS?
A. (5, 1) and (3, 7)
B (7, 5) and (1, 3)
C. (7, 3) and (1, 5)
D. (5, 1) and (7, 3)
The coordinates of the endpoints after the transformation is (c) (7, 3) and (1, 5)
How to determine the coordinates of the endpointsFrom the question, we have the following parameters that can be used in our computation:
Segment with endpoints at (1, 5) and (7, 3)
Also from the question, we have:
The points are reflected the segment about a vertical line through the segment’s own midpoint
When this is done, this means that the points would swap positions
So, we have the following representations
(1, 5) and (7, 3) ⇒ (7, 3) and (1, 5)
Hence, the coordinates of the endpoints is (c) (7, 3) and (1, 5)
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For the following, find the length of AB:
(Use \large \pi=3.14 when necessary and round your final answer to the hundredths.)
Arc AB =
A circle is a two-dimensional figure with a radius and circumference of
2πr.
The arc length of AB is 10.05.
What is a circle?
A circle is a two-dimensional figure with a radius and circumference of
2πr.
We have,
The radius of the circle = 12
The circumference of the given circle.
= 2πr
This means,
The whole circle is 360°.
= 360/360 x 2πr
= 2πr
Now,
We need to find 48°
So,
The arc length of AB.
= 48/360 x 2πr
= 48/360 x 2 x 3.14 x 12
= 48/30 x 2 x 3.14
= 10.048
Rounding to the nearest hundredths.
= 10.05
Thus,
The arc length of AB is 10.05.
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Answer: 10.05
Step-by-step explanation:
The arc length of AB.
= 48/360 x 2πr
= 48/360 x 2 x 3.14 x 12
= 48/30 x 2 x 3.14
= 10.048
Rounding to the nearest hundredths.
= 10.05
The arc length of AB is 10.05.
Use cylindrical coordinates. Evaluate E(x+y+z)dV where E is the solid in the first octant that lies under the paraboloid z=4x2y2.
Evaluation of E(x+y+z)dV where E is the solid in the first octant that lies under the paraboloid z = 4 - x² - y² is [tex]\frac{128}{15} + \frac{8\pi}{3}[/tex]
Cylindrical coordinates are a set of three coordinates used to locate a point in the cylindrical coordinate system.
When the polar coordinates are extended to a three-dimensional plane, an additional z coordinate is added.
These three measurements combine to form cylindrical coordinates. The coordinates describe two distances and one angle.
According to the question,
We have to find the area under paraboloid
Given : E is a region that lies under paraboloid z = 4 - x² - y² and in the first octant.
The intersection of this paraboloid with the xy plane will give z = 0,
z = 4 - x² - y² , z = 0, that is a circle x² +y² = 4 with radius 2
Therefore , 0 ≤ z ≤ 4 - r²
Therefore, the region E in cylindrical coordinates is
[tex]E = ( r , \theta , z ) | 0 \leq \theta \geq \frac{\pi}{2} , 0 \leq 4 \geq , 0 \leq z \geq 4 - r^2[/tex]
f(x , y ,z ) = x + y + z
=> f(x , y ,z ) = rcosθ + rsinθ + z
=> f(x , y ,z ) = r(cosθ + sinθ) + z
Integration ,
[tex]{\int \int \int }\limits_E {z} \,dv[/tex]
=> [tex]\int\limits^{\frac{\pi}{2}}_0 \int\limits^{2}_0 \int\limits^{4 - r^2}_0 {[r ( cos \theta + sin \theta) + z] r \, dzdrd\theta[/tex]
simlifying,
[tex]= \int\limits^{\frac{\pi}{2}}_0 \int\limits^{2}_0 \int\limits^{4 - r^2}_0 {[r^2 ( cos \theta + sin \theta) + rz] \, dzdrd\theta\\= \int\limits^{\frac{\pi}{2}}_0 \int\limits^{2}_0 [r^2 ( cos \theta + sin \theta)z + r\frac{z^2}{2}] \, drd\theta[/tex]
=> [tex]\int\limits^{\frac{\pi}{2}}_0 \int\limits^{2}_0 [(4r^{2} - r^{4})( cos \theta + sin \theta) + \frac{r(4 - r^2)^2}{2}] \, drd\theta[/tex]
=> [tex]\int\limits^{\frac{\pi}{2}}_0 [(\frac{4}{3} r^{3} - \frac{r^{5}}{5})( cos \theta + sin \theta) + \frac{(4 - r^2)^3}{12}]^{2}_{0} \, d\theta[/tex]
Putting the limit and solving,
=> [tex]\int\limits^{\frac{\pi}{2}}_0 [(cos \theta + sin\theta )\frac{64}{15} + \frac{16}{3} \theta ]\, d\theta[/tex]
=> [tex][(sin\theta - cos\theta )\frac{64}{15} + \frac{16}{3} \theta ]^\frac{\pi}{2} _0[/tex]
Evaluating the limits,
[tex]\int\int\int\limits_E {x + y +z } \,dV = \frac{128}{15} + \frac{8\pi}{3}[/tex]
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the midpoint of AB is at (0,4). If A=(3,0), find B.
If the midpoint of AB is at (0,4). and A = (3,0) then B is at point (-3, 8)
How to find point BThe formula to determine the center point of a straight line using the coordinates of its endpoints is known as the midpoint formula in coordinate geometry.
The formula is of the form
X= (x₂ - x₁)/2
Y = (y₂ - y₁)/2
solving for B (x₂, y₂)
0 = (x₂ + 3)/2
0 = x₂ + 3
x₂ = -3
4 = (y₂ + 0)/2
8 = y₂ + 0
y₂ = 8
point B = (-3, 8)
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