Answer:
sin(x°) = 11/s
Step-by-step explanation:
The tangent is the ratio of sine to cosine, so ...
tan(x°) = sin(x°)/cos(x°)
Multiplying by cos(x°) gives ...
sin(x°) = cos(x°)·tan(x°) = (r/s)·(11/r)
sin(x°) = 11/s
Answer:
sin x° = 11 divided by s
Step-by-step explanation:
Given,
tan x° = 11 divided by r
[tex]\implies tan x^{\circ}=\frac{11}{r}[/tex]
Also, cos x° = r divided by s
[tex]\implies cos x^{\circ}=\frac{r}{s}[/tex]
We know that,
[tex]\frac{sinx^{\circ}}{cos x^{\circ}}=tan x^{\circ}[/tex]
[tex]\implies sinx^{\circ}= tan x^{\circ}\times cos x^{\circ}[/tex] ( by cross multiplication )
By substituting the values,
[tex]sin x^{\circ}=\frac{11}{r}\times \frac{r}{s}=\frac{11r}{rs}=\frac{11}{s}[/tex]
⇒ sin x° = 11 divided by s
The graphs of f(x)=2x+2 and g(x)=2(2)^x are shown.
What are the solutions to the equation 2x+2=2(2) ^2 ?
Select each correct answer.
0
1
2
4
Answer:
{0, 1}
Step-by-step explanation:
The x-coordinates of the points of intersection are 0 and 1. These are the solutions to f(x)=g(x).
The solution to the given system of equations is [0, 1].
What is the solution of two equations?
A solution of a system (two equations) in two variables is an ordered pair that makes both the equations true, that solution corresponds to the intersection point of the two equations.
According to the given question.
We have a graph for the two equations [tex]2x+2[/tex] and [tex](2)2^{x}[/tex].
From the graph we can se that both the equations or a system of equations are coinciding from 2 to 4. And for 2 to 4 the x-coordinate is 0 to 1.
Therefore, the solution to the given system of equations is [0, 1].
Find out more information about solution to the system of equations:
https://brainly.com/question/9351049
#SPJ2
he amount of carbon-14 present in animal bones t years after the animal's death is given by P(t)equals=Upper P 0 e Superscript negative 0.00012097 tP0e−0.00012097t. How old is an ivory tusk that has lost 26% of its carbon-14?
Answer:
t = 2489 years
Step-by-step explanation:
The equation you need for this is
[tex]N=N_{0}e^{kt}[/tex]
where N is the amount AFTER the decomposition, N-sub-0 is the initial amount, k is the decomposition constant and t is time in years.
If we are told that the tusk LOST 26% of its carbon-14, that means 74% of it remains from the initial 100% it had.
Filling in:
[tex]74=100e^{-.00012097t}[/tex]
Begin by dividing both sides by 100 to get a decimal of .74:
[tex].74=e^{-.00012097t}[/tex]
The goal is to get that t out of the exponential position in which it is currently sitting. Do this by "undoing" the e. Do THAT by taking the natural log of both sides because a natural log "undoes" an e. This is due to the fact that the base of a natural log is e.
[tex]ln(.74)=ln(e^{-.00012097t})[/tex]
The ln and the e disappear on the right side, leaving
ln(.74) = -.00012097t
Plug ln(.74) into your calculator to get
-.3011050928 = -.00012097t
t = 2489
triangle CRV has side lengths that measure 10 centimeters, 12 centimeters, and 15 centimeters. Which of the following best describes this type of triangle?
A) scalene triangle
b) equilateral triangle
c) isosceles triangle
d) obtuse triangle
Answer:
A. Scalene, which just means three unequal sides.
Step-by-step explanation:
An equilateral triangle has three equal sides and an isosceles triangle two, so it's not those.
Obtuse means there's at least one obtuse angle, always opposite the longest side. Here we have 10^2+12^2=244 which is greater than 15^2=225, indicating the angle opposite 15 is acute, so not this one either.
Answer:
A) Scalene triangle.
Step-by-step explanation:
We have been given that triangle CRV has side lengths that measure 10 centimeters, 12 centimeters, and 15 centimeters. We are asked to determine the type of triangle CRV.
We can see that the given measure of all sides are different. We know that a triangle is known a scalene triangle, when all its sides has different measures.
Since all the sides of triangle CRV has different measure, therefore, triangle CRV is a scalene triangle and option A is the correct choice.
William's yard has a perimeter of 2 2/6. The length is 3/6. What is the width?
Evaluate the step function for the given input values. g(x) = g(2) = g(–2) = g(5) =
Answer:
Step-by-step explanation:
g(2) = g(–2) = g(5) is never true for the step function. One " = " symbol per pair of g values, please.
The step function, which you're calling "g(x)," is 0 from -infinity up to but not including 0. It's 1 from x just greater than 0 through infinity.
Thus:
g(2) = 1 because x is greater than 0.
g(-2) = 0 because x is less than 0.
g(5) = 1 because x is greater than 0.
Answer:
g(2)= 3
g(-2)= -4
g(5)= 5
1. If the domain of a function f(x) is the set {10, 20, 30}. What does the information tell you about f-1(x)?
2. If the graph of a function f(x) includes the point (3, 0), what point must the graph of f-1(x) include? Explain.
3.The first term in an arithmetic sequence is 2. The twelfth term is 211. Find the value of n so that an = 135.
PART A (1 in diagram)
Each mapping diagram represents a function because none of the elements in the inputs maps on to two different elements in the outputs.
1. The domain of the given function f(x) is the set {10, 20, 30}.
All we can say about
[tex] {f}^{ - 1} (x)[/tex]
is that, it has a range of {10, 20, 30}.
This is because the domain of a function becomes the range of its inverse function.
2. If the graph of a function f(x) includes the point (3, 0), then the graph of the inverse function ,
[tex] {f}^{ - 1} (x) [/tex]
will contain the point (0,3).
Reason:The point on the graph of the inverse function is obtained by reflecting the corresponding point on the graph of the function in the line y=x. We just have to swap the coordinates.
In simple terms the domain of the function becomes the range of the inverse function and vice versa.
3. If the first term in an arithmetic sequence is 2.
Then we have a=2.
If the twelfth term is 211, then
[tex]a + 11d = 211...(1)[/tex]
Put a=2 into this equation.
[tex]2 + 11d = 211[/tex]
Solve for d.
[tex]11d = 211 - 2[/tex]
[tex]11d = 209[/tex]
[tex]d = \frac{209}{11} = 19[/tex]
If
[tex]a_n = 135[/tex]
then
[tex]a + (n - 1)d = 135[/tex]
Substitute a=2 and d=19
[tex]2 + (n - 1)19 = 135[/tex]
[tex]19(n - 1) = 133[/tex]
[tex](n - 1) = \frac{133}{19} [/tex]
[tex]n - 1 = 7[/tex]
[tex]n = 7 + 1 = 8[/tex]
Therefore n=8
he temperature in Tampa, Florida is 15 degrees warmer than twice the temperature in Chicago, Illinois. The temperature in Tampa is 75 degrees. Write an equation to determine the temperature in Chicago.
2x + 15 = 75
2x + 75 = 15
2x − 15 = 75
2x − 75 = 15
30 = 2x + 10
Answer:
2x+15=75
Step-by-step explanation:
Let the temperature of chicago be x
75-15=2x
Reversing the equation
2x+15=75
An equation to determine the temperature in Chicago will be 2x + 15 = 75. so option A is correct.
What is a system of equations?A system of equations is two or more equations that can be solved to get a unique solution. the power of the equation must be in one degree.
Let the temperature of Chicago be represented x.
75 - 15 = 2x
Reversing the equation;
2x + 15 = 75
An equation to determine the temperature in Chicago will be 2x + 15 = 75. so option A is correct.
Learn more about equations here;
https://brainly.com/question/10413253
#SPJ2
Evaluate................
Answer:
h(8q²-2q) = 56q² -10q
k(2q²+3q) = 16q² +31q
Step-by-step explanation:
1. Replace x in the function definition with the function's argument, then simplify.
h(x) = 7x +4q
h(8q² -2q) = 7(8q² -2q) +4q = 56q² -14q +4q = 56q² -10q
__
2. Same as the first problem.
k(x) = 8x +7q
k(2q² +3q) = 8(2q² +3q) +7q = 16q² +24q +7q = 16q² +31q
_____
Comment on the problem
In each case, the function definition says the function is not a function of q; it is only a function of x. It is h(x), not h(x, q). Thus the "q" in the function definition should be considered to be a literal not to be affected by any value x may have. It could be considered another way to write z, for example. In that case, the function would evaluate to ...
h(8q² -2q) = 56q² -14q +4z
and replacing q with some value (say, 2) would give 196+4z, a value that still has z as a separate entity.
In short, I believe the offered answers are misleading with respect to how you would treat function definitions in the real world.
A surveyor, Toby, measures the distance between two landmarks and the point where he stands. He also measured the angles between the landmarks in degrees.
the triangle has
two sides(65,55)
angles (40,30)
What is the distance, x, between the two landmarks? Round the answer to the nearest tenth.
32.5 m
42.1 m
85.1 m
98.5 m
The Set Up:
x² = (Side1)² + (Side2)² - 2[(Side1)(Side2)]
Solution:
cos(Toby's Angle) • x² = 55² + 65² - 2[(55)(65)] cos(110°)
x² = 3025 + 4225 -7150[cos(110°)]
x² = 7250 - 2445.44x =
√4804.56x = 69.31m
The distance, x, between two landmarks is 69.31m.
Note: The answer choices given are incorrect.
Answer:
98.5 m
Step-by-step explanation:
Refer the attached figure
AB = 55
AD = 65
∠ABC=40°
∠ADC = 30°
We are supposed to find the distance between the two landmarks i.e. BD = BC+CD
In ΔABC
[tex]Cos \theta = \frac{Base}{Hypotenuse}[/tex]
[tex]Cos 40^{\circ} = \frac{BC}{AB}[/tex]
[tex]0.76604444= \frac{BC}{55}[/tex]
[tex]0.76604444 \times 55 =BC[/tex]
[tex]42.132442 =BC[/tex]
In ΔADC
[tex]Cos \theta = \frac{Base}{Hypotenuse}[/tex]
[tex]Cos 30^{\circ} = \frac{CD}{AD}[/tex]
[tex]0.8660254= \frac{CD}{65}[/tex]
[tex]0.8660254 \times 65 =CD[/tex]
[tex]56.291651 =CD[/tex]
So, BD = BC+CD=42.132442+56.291651=98.424≈ 98.5
Hence the distance between the two landmarks is 98.5 m.
Which of the following constants can be added to x 2 - x to form a perfect square trinomial?
A) 1/4
B) 1/2
C) 1
Using the given points, determine Δy.
(-3, -5) and (0, 10)
A. Δy = 3
B. Δy = 5
C. Δy = 13
D. Δy = 15
[tex]\bf (\stackrel{x_1}{-3}~,~\stackrel{y_1}{-5})\qquad (\stackrel{x_2}{0}~,~\stackrel{y_2}{10}) \\\\\\ slope = m\implies \cfrac{\stackrel{\Delta y}{ y_2- y_1}}{\stackrel{\Delta x}{ x_2- x_1}}\implies \cfrac{10-(-5)}{0-(-3)}\implies \cfrac{10+5}{0+3}\implies \cfrac{\stackrel{\stackrel{\Delta y}{\downarrow }}{\boxed{15}}}{3}[/tex]
Graph the following piecewise function.
Shown below
Step-by-step explanation:A piecewise function is a function defined by two or more equations. In this problem, we have a function defined by two equations.
First, a quadratic equation:
[tex]x^2+2[/tex]
This is a parabola that opens upward and starts at the point [tex](-5,27)[/tex] and whose vertex is the point [tex](0,2)[/tex]. Keep in mind that [tex]x=-5[/tex] is included in the domain of the function, and we know this by the symbol ≤ that includes the equality.
Second, a linear equation:
[tex]x-4[/tex]
The graph of this is a linear function with slope [tex]m=1[/tex] and starts at the point [tex](3,-1)[/tex]. Keep in mind that [tex]x=3[/tex] is included in the domain of the function by the same symbol ≤
Moreover, at [tex]x=3[/tex] there is a jump discontinuity.
If f(x) = 3х2 + 5x, find f(-2).
-22
-9
2. 22
Answer:
2
Step-by-step explanation:
f(x) = 3х^2 + 5x
Let x = -2
f(-2) = 3(-2)^2 + 5(-2)
= 3*4 -10
= 12-10
=2
PEOPLE THAT KNOW GEOMETRY HELP A BRO OUT
Answer: Second option.
Step-by-step explanation:
Given the transformation [tex]T:(x,y)[/tex]→[tex](x+3,y+1)[/tex] for the ordered pair (4,3), you can find the preimage point through this procedure:
1) Find the x-coordinate of the preimage point. You know that:
[tex]x+3=4[/tex]
So you must solve for "x":
[tex]x=4-3\\x=1[/tex]
2) Find the y-coordinate of the preimage point. You know that:
[tex]y+1=3[/tex]
So you must solve for "y":
[tex]y=3-1\\y=2[/tex]
Therefore, the preimage point is: (1,2)
What is the circumference of circle P?
Express your answer in terms of .
AB = 14 in.
Answer:
[tex]C=14\pi\ in[/tex]
Step-by-step explanation:
we know that the circumference is equal to
[tex]C=\pi D[/tex]
we have
[tex]D=14\ in[/tex]
substitute
[tex]C=\pi (14)[/tex]
[tex]C=14\pi\ in[/tex]
Answer:
Circumference of given circle = 14π in
Step-by-step explanation:
Points to remember
Circumference of circle = 2πr
Where 'r' is the radius of circle
To find the circumference of circle
Here diameter AB = 14 in
radius, r = 14/2 = 7 in
Circumference = 2πr
= 2 * * 7
= 14π in
Therefore the correct answer is,
Circumference = 14π in
What is the following sum 4(5square x^2y)+3(5 square x^2y
Answer:
The answer would be [tex]7(\sqrt[5]{x^{2}y } )[/tex]
Step-by-step explanation: I got it right on Edge 2020
Need help with a math question
Answer:
[tex]d =\sqrt{(b-0)^2 +(c-a)^2}[/tex]
Step-by-step explanation:
We know that the distance between two points is calculated using the following formula
[tex]d =\sqrt{(x_2-x_1)^2 +(y_2-y_1)^2}[/tex]
In this case we look for the RS distance
Then
The starting point is: (0, a)
The final point is (b, c)
So
[tex]x_2 = b\\x_1 = 0\\y_2 = c\\y_1=a[/tex]
The distance is:
[tex]d =\sqrt{(b-0)^2 +(c-a)^2}[/tex]
Rectangular windows are being made into a wall of windows for an office building. Each wall is 14 feet tall and 12 feet wide. The wall will be divided into x columns and into x + 7 rows. Find the area of one of these windows.
Answer:
168/(x² +7x)
Step-by-step explanation:
The height of each window is 14/(x+7), and the width of each window is 12/x. The area of each window is the product of its height ans width:
area = (14/(x+7))(12/x) = 168/(x(x +7))
area = 168/(x² +7x)
_____
Comment on the problem
There is not enough information given to determine suitable values for x. If x is 42, each window is a square 3 3/7 inches on a side.
Can anyone help me?!?!?!?!
Answer:
B
Step-by-step explanation:
A number is called a "decreasing number" if each digit in the number is less than the digit to its left. For example, 87420 is a decreasing number. How many five-digit decreasing numbers are there?
Answer:
252
Step-by-step explanation:
The question is fully equivalent to asking how many subsets of length 5 there are of 10 objects (digits 0–9). That number is 10C5, where nCk is the number of ways to choose k objects from a list of 10. The value of that is ...
nCk = n!/(k!(n-k)!)
There are 252 ways to choose 5 numbers from the digits 0-9:
10C5 = 10!/(5!(10-5)!) = 10·9·8·7·6/(5·4·3·2·1) = 9·4·7 = 252
_____
The order of the selection doesn't matter, because the selected digits are always arranged in decreasing order to form a decreasing number.
Answer:
Could you please post a video or give more simple explanations to someone who does not have prior knowledge of such problems?
Step-by-step explanation:
If a denotes some event, what does upper a overbar denote? if p(a)equals0.995, what is the value of p(upper a overbar)? if p(a)equals0.995, is upper a overbar unlikely?
[tex]P(A)=0.995\\P(A')=0.005[/tex]
So [tex]P(A')[/tex] is pretty much unlikely.
95. A 20-gallon alcohol-water solution contains 15% pure alcohol. How much alcohol should
be added to make a new solution that is 20% alcohol ?
Answer:
1.25 gallons of alcohol
Step-by-step explanation:
Let x represent the amount of alcohol to add to the mix. Then the total amount of alcohol in the mix is ...
0.15×20 + x = 0.20×(20 +x)
3 +x = 4 + 0.2x . . . . simplify
0.8x = 1 . . . . . . . . . . add -3-0.2x
x = 1/0.8 = 1.25 . . . . divide by 0.8
1.25 gallons of alcohol should be added to make 21.25 gallons of 20% alcohol.
For the following sequence determine the common difference (if it is an arithmetic sequence) or the common ratio (if it is a geometric sequence).
Answer:
the common ratio is 1/√7
Step-by-step explanation:
The differences are not constant, but the ratio is:
7/(7√7) = 1/√7
√7/7 = 1/√7
The common ratio is 1/√7.
The given sequence, 7√7, 7, √7, ..., is a geometric sequence with a common ratio of √7.
A geometric sequence is a sequence of numbers in which each term is equal to the previous term times a constant value, called the common ratio. In the given sequence, we can see that each term is equal to the previous term times √7. For example, the second term, 7, is equal to the first term, 7√7, times √7. The third term, √7, is equal to the second term, 7, times √7. And so on.
To find the common ratio of a geometric sequence, we can divide any term of the sequence by the previous term. For example, we can divide the second term, 7, by the first term, 7√7, to get:
7 / 7√7 = √7
We can also divide the third term, √7, by the second term, 7, to get:
√7 / 7 = √7
In both cases, we get the same answer, √7. This tells us that the common ratio of the geometric sequence is √7.
Therefore, the answer is common ratio = √7.
To learn more about geometric sequence here:
https://brainly.com/question/33243139
#SPJ3
need help with a math question
Answer:
Z = 27°
Step-by-step explanation:
The sum of the internal angles of a triangle is always equal to 180 °. Note that the triangle shown whose angles are z and 63 ° is a right triangle. Therefore it has an angle of 90 °. Then we can write the following equation:
[tex]z + 63\° +90\°= 180\°\\\\z = 180\° - 63\°-90\°\\\\z = 27\°[/tex]
Finally z = 27°
ANSWER
[tex]z = 27 \degree[/tex]
EXPLANATION
The diagonals of a rhombus bisect each other at right angles.
Hence each of the four angles at the center by are 90° each.
This means that:
[tex]z + 63 + 90 = 180[/tex]
Sum of interior angles of a triangle.
[tex]z + 153= 180[/tex]
[tex]z = 180 - 153[/tex]
This simplifies to.
[tex]z = 27 \degree[/tex]
The unit cost, in dollars, to produce bins of cat food is $3 and the fixed cost is $6972. The price-demand function, in dollars per bin, is
p
(
x
)
=
253
−
2
x
Find the cost function.
C
(
x
)
=
Find the revenue function.
R
(
x
)
=
Find the profit function.
P
(
x
)
=
At what quantity is the smallest break-even point?
Answer:
Revenue , Cost and Profit Function
Step-by-step explanation:
Here we are given the Price/Demand Function as
P(x) = 253-2x
which means when the demand of Cat food is x units , the price will be fixed as 253-2x per unit.
Now let us revenue generated from this demand i.e. x units
Revenue = Demand * Price per unit
R(x) = x * (253-2x)
= [tex]253x-2x^2[/tex]
Now let us Evaluate the Cost Function
Cost = Variable cost + Fixed Cost
Variable cost = cost per unit * number of units
= 3*x
= 3x
Fixed Cost = 6972 as given in the problem.
Hence
Cost Function C(x) = 3x+6972
Let us now find the Profit Function
Profit = Revenue - Cost
P(x) = R(x) - C(x)
= [tex]253x-2x^2 - (3x + 6972)[/tex]
= [tex]253x-3x-2x^2-6972\\= 250x-2x^2-6972\\=-2x^2+250x-6972\\[/tex]
Now we have to find the quantity at which we attain break even point.
We know that at break even point
Profit = 0
Hence P(x) = 0
[tex]-2x^2+250x-6972=0\\[/tex]
now we have to solve the above equation for x
Dividing both sides by -2 we get
[tex]x^2-125x+3486=0[/tex]
Now we have to find the factors of 3486 whose sum is 125. Which comes out to be 42 and 83
Hence we now solve the above quadratic equation using splitting the middle term method .
Hence
[tex]x^2-42x-83x+3486=0\\x(x-42)-83(x-42)=0\\(x-42)(x-83)=0\\[/tex]
Either (x-42) = 0 or (x-83) = 0 therefore
if x-42= 0 ; x=42
if x-83=0 ; x=83
Smallest of which is 42. Hence the number of units at which it attains the break even point is 42.
The cost function is C(x) = 3x + 6972, the revenue function is R(x) = 253x - 2x², and the profit function is P(x) = 250x - 2x²- 6972. The smallest break-even point occurs at 16 bins.
To solve the given problem, let's break it down step-by-step:
Cost FunctionThe cost function includes both the fixed cost and the variable cost. The fixed cost is $6972, and the unit cost to produce one bin of cat food is $3. Therefore, the cost function C(x) where x is the number of bins is:
C(x) = 3x + 6972
Revenue FunctionThe price-demand function is given by p(x) = 253 - 2x. Revenue R(x) is the product of the price per bin and the number of bins sold, so:
R(x) = x * (253 - 2x) = 253x - 2x²
Profit FunctionThe profit function is the revenue function minus the cost function. So, the profit function P(x) is:
P(x) = R(x) - C(x) = (253x - 2x²) - (3x + 6972) = 250x - 2x² - 6972
Break-Even PointTo find the smallest break-even point, we need to solve the equation where profit equals zero:
0 = 250x - 2x² - 6972
Rewriting the equation, we get:
2x² - 250x + 6972 = 0
Solving this quadratic equation using the quadratic formula x = [-b ± sqrt(b² - 4ac)] / 2a, where a = 2, b = -250, and c = 6972:
x = [250 ± sqrt(62500 - 4*2*6972)] / 4
x = [250 ± sqrt(62500 - 27888)] / 4
x = [250 ± sqrt(34612)] / 4
x = [250 ± 186.01] / 4
So, the two solutions for x are approximately:
x = (250 + 186.01) / 4 = 109
x = (250 - 186.01) / 4 = 15.997 ≈ 16
The smallest break-even point is at a quantity of 16 bins.
The cost function is C(x) = 3x + 6972, the revenue function is R(x) = 253x - 2x², and the profit function is P(x) = 250x - 2x² - 6972. The smallest break-even point occurs at 16 bins.
Can someone help me do part two please? It’s very important send a picture or something. I don’t even care if you tell me the steps in word form. Please help
1. "Create your own circle on a complex plane."
The equation of a circle in the complex plane can be written a number of ways. For center c (a complex number) and radius r (a positive real number), one formula is ...
|z-c| = r
If we let c = 2+i and r = 5, the equation becomes ...
|z -(2+i)| = 5
For z = x + yi and |z| = √(x² +y²), this equation is equivalent to the Cartesian coordinate equation ...
(x -2)² +(y -1)² = 5²
__
2. "Choose two end points of a diameter to prove the diameter and radius of the circle."
We don't know what "prove the diameter and radius" means. We can show that the chosen end points z₁ and z₂ are 10 units apart, and their midpoint is the center of the circle c.
For the end points of a diameter, we choose ...
z₁ = 5 +5iz₂ = -1 -3iThe distance between these is ...
|z₂ -z₁| = |(-1-5) +(-3-5)i| = |-6 -8i|
= √((-6)² +(-8)²) = √100
|z₂ -z₁| = 10 . . . . . . the diameter of a circle of radius 5
The midpoint of these two point should be the center of the circle.
(z₁ +z₂)/2 = ((5 -1) +(5 -3)i)/2 = (4 +2i)/2 = 2 +i
(z₁ +z₂)/2 = c . . . . . the center of the circle is the midpoint of the diameter
__₁₂₃₄
3. "Show how to determine the center of the circle."
As with any circle, the center is the midpoint of any diameter (demonstrated in question 2). It is also the point of intersection of the perpendicular bisectors of any chords, and it is equidistant from any points on the circle.
Any of these relations can be used to find the circle center, depending on the information you start with.
As an example. we can choose another point we know to be on the circle:
z₄ = 6-2i
Using this point and the z₁ and z₂ above, we can write three equations in the "unknown" circle center (a +bi):
|z₁ - (a+bi)| = r|z₂ - (a+bi)| = r|z₄ - (a+bi)| = rUsing the formula for the square of the magnitude of a complex number, this becomes ...
(5-a)² +(5-b)² = r² = 25 -10a +a² +25 -10b +b²
(-1-a)² +(-3-b)² = r² = 1 +2a +a² +9 +6b +b²
(6-a)² +(-2-b)² = r² = 36 -12a +a² +4 +4b +b²
Subtracting the first two equations from the third gives two linear equations in a and b:
11 -2a -21 +14b = 0
35 -14a -5 -2b = 0
Rearranging these to standard form, we get
a -7b = -5
7a +b = 15
Solving these by your favorite method gives ...
a +bi = 2 +i = c . . . . the center of the circle
__
4. "Choose two points, one on the circle and the other not on the circle. Show, mathematically, how to determine whether or not the point is on the circle."
The points we choose are ...
z₃ = 3 -2iz₄ = 6 -2iWe can show whether or not these are on the circle by seeing if they satisfy the equation of the circle.
|z -c| = 5
For z₃: |(3 -2i) -(2 +i)| = √((3-2)² +(-2-i)²) = √(1+9) = √10 ≠ 5 . . . NOT on circle
For z₄: |(6 -2i) -(2 +i)| = √((6 -2)² +(2 -i)²) = √(16 +9) = √25 = 5 . . . IS on circle
What is required to derive the equations of a parabola, ellipse, and a hyperbola?
What application does Cavalieri’s principle have with solid figures?
Answer:
You need to have some idea where you want to start if you're going to derive equations for these. You can start with a definition based on focus and directrix, or you can start with a definition based on the geometry of planes and cones. (The second focus is replaced by a directrix in the parabola.) In general, these "conics" represent the intersection between a plane and a cone. Perpendicular to the axis of symmetry, you have a circle. At an angle to the axis of symmetry, but less than parallel to the side of the cone, you have an ellipse. Parallel to the side of the cone, you have a parabola. At an angle between the side of the cone and the axis of the cone, you have a hyperbola. (See source link.)
You can also start with the general form of the quadratic equation.
.. ±((x-h)/a)^2 ± ((y-k)/b)^2 = 1
By selecting signs and values of "a" and "b", you can get any of the equations. (For the parabola, you probably need to take the limit as both k and b approach infinity.)
The distance from one corner of a rectangular garden to the other is 13 ft. The length of the garden is 7 ft longer than the width. Write a quadratic equation to find the dimension of the garden. Solve the equation and find the area of the garden in square feet.
Answer:
dimensions: 12 ft by 5 ftarea: 60 ft²Step-by-step explanation:
Let x represent the shorter dimension in feet. Then the longer one is x+7 and the Pythagorean theorem tells us the relation of these to the diagonal is ...
x² + (x+7)² = 13²
2x² +14x + 49 = 169 . . . . eliminate parentheses
x² +7x -60 = 0 . . . . . subtract 169 and divide by 2
(x +12)(x -5) = 0 . . . . factor the equation
x = -12 or +5 . . . . . . . only the positive value of x is useful here.
The short dimension is 5 ft, so the long dimension is 12 ft. The area is their product, 60 ft².
_____
Comment on finding the area
The quadratic equation above can be rearranged and factored as ...
x(x +7) = 60
Since the dimensions of the garden are x and (x+7), this product is the garden's area. This equation tells us the area is 60. We don't actually have to find the dimensions.
HELP meeeeeeeeeee!!!!!!!!!!
Answer: B
Step-by-step explanation:
How can you tell whether an exponential equation models growth or decay? Use the general form of an exponential expression to explain your answer
Answer:
Step-by-step explanation:
The general form of an exponential equation for growth is
[tex]y=(1+r)^x[/tex]
and for decay is
[tex]y=(1-r)^x[/tex]
In general, if the number inside the parenthesis (the growth or decay rate) is greater than 1, it's a growth problem. If the number inside the parenthesis is greater than 0 but less than 1 (in other words a positive fraction), it's a decay problem.
Final answer:
To determine if an exponential equation represents growth or decay, examine the base of the expression: a growth model has a base greater than 1, while a decay model has a base between 0 and 1. Exponential growth is illustrated by a J-shaped curve, whereas logistic growth follows an S-shaped curve.
Explanation:
The general form of an exponential function is f(t) = a*b^t, where a is the initial amount, b is the base, and t is the time.
Growth is modeled when the base b is greater than 1. This signifies that the quantity is increasing over time. For example, with a base of 2, the sequence would be 2, 4, 8, 16, and so forth, representing that the population doubles at each time interval.
In contrast, decay is modeled when the base b is between 0 and 1. This indicates that the quantity is decreasing over time, such as in the case of radioactive decay or depreciation of assets.
Exponential growth is often represented by a 'J-shaped' curve, which depicts how a population may grow faster as the population becomes larger. On the other hand, logistic growth, which is more realistic in natural populations due to factors like limited resources, follows an 'S-shaped' curve where growth levels off at carrying capacity.