The minimum area is 126.025 at [tex]$x=\frac{60 \pi}{(4+\pi)}[/tex] is 26.39, maximize the combined area of the circle and the square is 60.
Let the length of the wire (l)=60
Suppose the wire is cut into two parts as follows:
The length of the wire used for circle is x and the length of the wire used for square is (l-x)=(60-x).
So, length of the circle (x)= Perimeter of the of the circle
[tex]$$(x)=2 \pi r$$[/tex]
[tex]$r=\frac{x}{2 \pi} \quad[/tex] , r is the radius of the circular part.
Also, length of square (60-x)= Perimeter of the of the square
(60-x)=4 a
[tex]$a=\frac{60-x}{4} \quad$[/tex], a is the side of the square.
Hence, the total combined area of the circular region and the square region say A(x) is;
A(x)=area of circle + area of square
[tex]\\& A(x)=\pi r^2+a^2\end{aligned}$$[/tex]
Put the values of 'r' and 'a' and proceed as follows;
[tex]$$\begin{aligned}& A(x)=\pi\left(\frac{x}{2 \pi}\right)^2+\left(\frac{60-x}{4}\right)^2 \\& =\pi \frac{x^2}{4 \pi^2}+\left(15-\frac{x}{4}\right)^2\end{aligned}$$[/tex]
[tex]$=\frac{x^2}{4 \pi}+\left(15-\frac{x}{4}\right)^2$$[/tex]
So, [tex]$A(x)=\frac{x^2}{4 \pi}+\left(15-\frac{x}{4}\right)^2$[/tex].
For optimization, find the Critical points of [tex]$A^{\prime}(x)$[/tex];
Differentiate [tex]$A(x)=\frac{x^2}{4 \pi}+\left(15-\frac{x}{4}\right)^2$[/tex] with respect to x;
[tex]$$A^{\prime}(x)=\frac{2 x}{4 \pi}+2\left(15-\frac{x}{4}\right)\left(-\frac{1}{4}\right)$$[/tex]
Put [tex]$A^{\prime}(x)=0$[/tex] for critical points;
[tex]$$\begin{aligned}& A^{\prime}(x)=\frac{2 x}{4 \pi}+2\left(15-\frac{x}{4}\right)\left(-\frac{1}{4}\right)=0 \\& \frac{2 x}{4 \pi}=\left(15-\frac{x}{4}\right)\left(\frac{1}{2}\right) \\& \frac{4 x}{4 \pi}=\left(15-\frac{x}{4}\right) \\& \frac{4 x}{4 \pi}+\frac{x}{4}=15 \\& \frac{x(4+\pi)}{4 \pi}=15 \\& x=\frac{60 \pi}{(4+\pi)} \\& x=26.39\end{aligned}$$[/tex]
x=26.39 is the required critical point.
For maximum or minimum, find [tex]$A^{\prime \prime}(x)$[/tex] as follows:
[tex]$$\begin{aligned}& A^{\prime \prime}(x)=\frac{d}{d x}\left(\frac{2 x}{4 \pi}+\left(15-\frac{x}{4}\right)\left(-\frac{1}{2}\right)\right) \\& A^{\prime \prime}(x)=\frac{2}{4 \pi}+\frac{1}{8}\end{aligned}$$[/tex]
As [tex]$A^{\prime \prime}(x)=\frac{2}{4 \pi}+\frac{1}{8} > 0$[/tex] for every value of x. So, [tex]$A^{\prime \prime}(x) > 0$[/tex] for every critical point.
Hence, by Second Derivative test; A(x) is minimum at [tex]\\ $$x=\frac{60 \pi}{(4+\pi)}$.[/tex]
Therefore, minimum area is given as follows:
[tex]$$\begin{aligned}& A\left(\frac{60 \pi}{(4+\pi)}\right)=\left(\frac{60 \pi}{(4+\pi)}\right)^2 \frac{1}{4 \pi}+\left(15-\frac{60 \pi}{4(4+\pi)}\right)^2 \\\\& \text { As } x=\frac{60 \pi}{(4+\pi)}=26.39\end{aligned}$$[/tex]
So,
[tex]$$\begin{aligned}& A\left(\frac{60 \pi}{(4+\pi)}\right)=\left(\frac{60 \pi}{(4+\pi)}\right)^2 \frac{1}{4 \pi}+\left(15-\frac{60 \pi}{4(4+\pi)}\right)^2 \\& A(26.39)=(26.39)^2 \frac{1}{4 \pi}+\left(15-\frac{26.39}{4}\right)^2 \\& A(26.39)=55.42030+70.6020 \\& A(26.39)=126.025\end{aligned}$$[/tex]
Therefore, the minimum area is 126.025 at [tex]$x=\frac{60 \pi}{(4+\pi)}[/tex] is 26.39.
b. Note that, maximum area cannot be determined by the critical points as there is only one critical point that too corresponds to the minimum area.
For maximum area; consider the following;
Put x=0 and x=60 in A(x);
[tex]$$\begin{aligned}& A(0)=(0)^2 \frac{1}{4 \pi}+\left(15-\frac{0}{4}\right)^2 \\& A(0)=(15)^2 \\& A(0)=225 \\& A(60)=(60)^2 \frac{1}{4 \pi}+\left(15-\frac{60}{4}\right)^2 \\& A(60)=(60)^2 \frac{1}{4 \pi} \\& A(60)=286.47\end{aligned}$$[/tex]
So, at x=60, A(x) is maximum. That is in the case, when total length of the wire is used for circle.
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2. Brandon found that the number of snapper fish in an area can be represented by the formula
F = 1500(1.02)^t
where t represents the number of years since 2004. What is the y-intercept of this equation
and what does it represent?
The y-intercept represents the number of snapper fish in the area in 2004, which is the year that t = 0.
An area can be represented by the given formula:
[tex]F = 1500(1.02)^t[/tex]
where t represents the number of years since 2004.
The y-intercept of this equation is the point where the graph of the equation crosses the y-axis. In other words, it is the value of F when t is equal to 0.
Plugging in t = 0 into the equation above gives us:
F = 1500(1.02)⁰
= 1500(1)
= 1500
So the y-intercept of this equation is 1500.
Thus, the y-intercept reflects the amount of snapper fish caught in the area in 2004, the year t = 0.
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solar panels use the sun's energy to generate power, a town wants to install 28th solar panels in an array, which what are all the possible ways the panels could be installed
The possible arrangements would be:
1 by 282 by 144 by 7What is an Arrangement?Combination refers to the various ways that certain objects or numbers can be arranged. If the set already has an order, rearranging its elements, or placing its members in a linear or sequential order.
Given that solar panels use the sun's energy to generate power. A town wants to install 28 solar panels in an array.
The possible arrangements would be:-
( 1 row having 28 panels in series )( 2 rows having 14 panels in each series.)( 4 rows having 7 panels in each series.)Read more about arrangements here:
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Triangle ABC was dilated and translated to form similar triangle A'B'C'.
On a coordinate plane, 2 triangles are shown. Triangle A B C has points (0, 2), (2, 2), and (2, 0). Triangle A prime B prime C prime has points (negative 4, negative 1), (1, negative 1), and (1, negative 6).
What is the scale factor of the dilation?
One-fifth
Two-fifths
Five-halves
5/1
Answer; Five-halves
Step-by-step explanation:
Put triangles on graph
To find the answer you divide the A' to B' by the A to B distance.
So you divide 5/2
9) Consider the pattern of squares shown below: Which type of model, linear or exponential.
The given model is exponential.
Exponential function -The formula for an exponential function is f (x) = ax, where x is a variable and an is a constant that serves as the function's base and must be bigger than 0. The transcendental number e, or roughly 2.71828, is the most often used exponential function basis.
When b > 0 and b 1, an exponential function has the form f(x) = bx. The base is b, and the exponent is x, just like in any exponential expression. The development of bacteria is an illustration of an exponential function. Some germs multiply by two per hour.
As it is clear from the figure, 1 block has 2 boxes, then 4, 8.
So, it is in the form 2ⁿ.
∴ It is Exponential model.
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Complete question -
What is SAS congruence rule of triangle?
The SAS (Side-Angle-Side) Congruence Rule states that if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the two triangles are congruent.
The SAS (Side-Angle-Side) congruence rule is a method used to determine if two triangles are congruent. This rule states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.
To determine if two triangles are congruent using the SAS rule, we must first identify the two sides and the angle in one triangle that are congruent to the two sides and the angle in the other triangle. This means that they must have the same measures in terms of length and angle.
Once we have identified the corresponding sides and angles, we can use the SAS rule to determine if the two triangles are congruent. If the two sides and the angle of one triangle are congruent to the two sides and the angle of another triangle, then the two triangles are congruent.
However, if the two sides and the angle of one triangle are not congruent to the two sides and the angle of another triangle, then the two triangles are not congruent. The SAS rule is a simple and effective way to determine if two triangles are congruent.
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Can someone answer this question really quick
The Bobcats football coach logged the following yardage gains and losses over four plays of a game.
1. Gain 25x yards.
2. Gain 0.9y yards.
3. Lose 12y yards.
4. Lose 5.2x yards.
What is the net yardage for these four plays?
Enter your answer as an expression, like this: 42x+53y
The net yardage the Bobcats football coach logged over four plays of game is 19.8x - 11.1y
How to find the net yardageThe net yardage is calculated using the addition and subtraction operation as follows
For the gains the calculation is as follows
= Gain 25x yards + Gain 0.9y yards.
= 25x + 0.9y
For the lose the calculation is as follows
= Lose 12y yards + Lose 5.2x yards.
= 5.2x + 12y
the net yardage is solved as follows
net yardage = gains - loses
net yardage = (25x + 0.9y) - (5.2x + 12y)
removing the parenthesis and rearranging
net yardage = 25x + 0.9y - 5.2x - 12y
net yardage = 25x - 5.2x + 0.9y - 12y
net yardage = 19.8x - 11.1y
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It's the end of the school year, and your class of 250
students must decide the best place for the
yearbook signing party.
Everybody votes in a direct democracy!
☐ Select representatives for groups of students.
Why?
Plllls answer
Everybody votes in a direct democracy to select a place for the yearbook signing party.
What is direct democracy?Rules, institutions, and procedures known as "direct democracy" allow members of the public to vote directly on proposed constitutional amendments, laws, treaties, or policy decisions. Referendum and initiatives are the two most significant examples of direct democracy that are addressed in this primer.
Given:
It's the end of the school year, and your class of 250 students must decide the best place for the yearbook signing party.
To select the place:
They vote in direct democracy.
Because they have only 250 people.
And 250 people can have direct say.
Number of people is less.
So, selecting representatives would be not fair.
Therefore, everybody votes in a direct democracy.
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Help me please, math is hard
Answer:
see explanation
Step-by-step explanation:
using the identity
sin²Θ = 1 - cos²Θ
consider the left side
1 - [tex]\frac{sin^20}{1+cos0}[/tex]
= 1 - [tex]\frac{1-cos^20}{1+cos0}[/tex]
1 - cos²Θ is a difference of squares and factors as (1 - cosΘ)(1 + cosΘ)
= 1 - [tex]\frac{(1-cos0)(1+cos0)}{1+cos0}[/tex] ← cancel (1 + cosΘ) on numerator/ denominator
= 1 - (1 - cosΘ) ← distribute parenthesis by - 1
= 1 - 1 + cosΘ
= cosΘ
= right side
What is an equation of the line that passes through the points (8, 6) and
(-3,6)?
Answer:
y=6
Step-by-step explanation:
right!!
Which four statements describe the relationship represented by the model?
The four statemen that describes the relationship of the model
there are many possible masses of the cubethe mass of the cube is twice the mass of the cylinderWhen a cylinder has a mass of one the mass of the cube is twoIf the mass of the cylinder is s and the mass of the cube is c, 2s = cWhat is a model?A model refers to images, objects variables meant to depict a relation
In the model provided the relationship existing between the models is that the mass are equal hence there will be an equality sign to balance the relationship in the left and right hand side
examining the model shows that
3 cylinders = 1 cylinder + 1 cube
3 cylinders - 1 cylinder = 1 cube
2 cylinders = 1 cube
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need help assap please.
option C again correct if I'm wrong
An arithmetic sequence has a constant difference between each consecutive pair of terms. This is similar to the linear functions that have the form y=mx+b. A geometric sequence has a constant ratio between each pair of consecutive terms. This would create the effect of a constant multiplier.
what is the simplified form of square root of 140
What is 1.4m + 3.7n +1 subtracted from 2.3m−1.2n ?
Responses
−0.9m+4.9n+1
negative 0.9 m plus 4.9 n plus 1
0.9m−2.5n−1
0.9 m minus 2.5 n minus 1
0.9m−4.9n−1
The value of the mathematical statement is (d) 0.9m - 4.9n - 1
How to evaluate the mathematical statement?From the question, we have the following parameters that can be used in our computation:
A mathematical statement
The mathematical statement is give as
1.4m + 3.7n +1 subtracted from 2.3m−1.2n
When this is represented as an expression
We have the following representation
2.3m−1.2n - (1.4m + 3.7n +1)
Express properly
2.3m − 1.2n - (1.4m + 3.7n + 1)
Open the brackets
So, we have the following representation
2.3m − 1.2n - (1.4m + 3.7n + 1) = 2.3m − 1.2n - 1.4m - 3.7n - 1
Collect the like terms
So, we have the following representation
2.3m − 1.2n - (1.4m + 3.7n + 1) = 2.3m - 1.4m − 1.2n - 3.7n - 1
Evaluate the like terms
So, we have the following representation
2.3m − 1.2n - (1.4m + 3.7n + 1) = 0.9m - 4.9n - 1
Hence, the expression is (d) 0.9m - 4.9n - 1
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What is the acute angle of ABC?
The acute angle of a triangle is an angle that measures less than 90°. Hence, in the triangle ABC, an acute angle is any angle that is smaller than 90°.
An angle is the figure created by two rays whose termini coincide. The measure of an angle is the amount of rotation required to bring one ray onto the other. A degree or angle that is less than 90° is said to be acute. This means that the two rays forming the angle can be brought together without making a full rotation.
A right angle is an angle having a measurement of precisely 90°. A right angle is formed when the two rays are perpendicular to each other, meaning they form a 90-degree angle. A right angle is often symbolized with a small square at the angle's vertex.
If an angle measures more than 90°, it is called an obtuse angle. An obtuse angle is formed when the two rays forming the angle can be brought together by making more than one full rotation. Obtuse angles are larger than right angles, but smaller than straight angles (which measure 180°).
It's important to note that the terms "acute," "right," and "obtuse" are not just limited to angles. They can also be used to describe other geometric shapes, such as triangles and circles. In these cases, the definitions may vary slightly, but the basic concept remains the same.
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Find the value of x
Answer:
x = 33
Step-by-step explanation:
67 = 2x + 1
66 = 2x
x = 33
2(33) + 1 = 67
Use substitution to solve the following system of equation
X =
y =
4x + y = 28
y = 3x
Answer:
x = 4
y = 12
Step-by-step explanation:
Replace all occurrences of y with 3x in each equation.
7x = 28
y = 3x
Divide each term in 7x = 28 by 7 and simplify.
x = 4
y = 3x
Replace all occurrences of x with 4 in each equation.
y = 12
x = 4
The solution to the system is the complete set of ordered pairs that are valid solutions.
(4,12)
The result can be shown in multiple forms.
Point Form:
(4,12)
Equation Form:
x = 4,y = 12
Hope I helped <3
Find the radius of convergence R of the series. n Ś (x - a)", b>0 bn n = 1 R= b Find the interval of convergence of the series. (Enter your answer using interval notation.) X
Answer: (-b + a, b + a).
Step-by-step explanation:
To find the radius of convergence of the series, we use the formula:
R = 1/limsup|bn|^(1/n)
To find the interval of convergence, we can use the following:
If R > 0, the series converges for all |x-a| < R and diverges for |x-a| > R.
If R = 0, the series converges only at x = a and diverges for all other values of x.
If R = ∞, the series converges for all values of x.
Therefore, the interval of convergence of the series is (-R + a, R + a).
In this case, the radius of convergence R is given as b, and the interval of convergence is (-b + a, b + a).
Question #3:
4x + 5y ≥ 10
Line: Solid or Dashed?
Shade: Above or Below?
m=
b=
Are these points solution: (0,2)
(3,4)
(-2,-2)
Answer:
See Below
Step-by-step explanation:
Answer:
Line: Solid
Shade: Above
m = -4/5
b = 2
(0,2) - True -Yes
(3,4) - True - Yes
(-2,-2) - False - No
Step-by-step explanation:
Given:
4x + 5y ≥ 10
Questions:
Line: Solid or Dashed?
Shade: Above or Below?
m= [?]
b= [?]
Are these points solution:
(0,2)(3,4)(-2,-2)Solve:
Inequalities that use '< or >' symbols are plotted with a dashed line. Due to that dashed line means the solution doesn't include the line itself.
< = Less than
> = Greater than
Inequalities that use '≤ or ≥' symbols are plotted with a solid line. Due to that solid line means the solution does includes the line itself
≤ = Less than or equal to
≥ = Greater than or equal to
Therefore, based on the given, the line is solid. Which also means the solution does includes the line itself.
We also know that the given info with the inequalities of "≥".
"≥" Means greater than or equal to which means shade above. If it was "≤ " which means less than or equal to you have to shade below.
To find the m and b we first have to know that;
y = mx + b
m = Slope
b = y-intercept
To find the Slope(m) and b(y-intercept) we first have to put 4x + 5y ≥ 10 in slope-intercept form.
4x + 5y ≥ 10
4x-4x + 5y ≥ 10-4x {Subtract 4x from both sides}
5y ≥ -4x + 10
5y ≥ -4x + 10 {Divided all by 5}
5y/5 ≥ -4x/5 + 10/5
y =-4/5x+2
Hence, now we know that;
m = -4/5
b = 2
To see if the following points {(0,2) (3,4) (-2,-2)} are solutions for 4x + 5y ≥ 10
All we have to do is to substitute it in.
Starting with (0,2)
x = 0
y = 2
4(0) + 5(2) ≥ 10
0 + 10 ≥ 10
10≥ 10
True. 10 isn't greater than 10 but- It's equal to 10. ≥ Means greater than or equal to Hence, (0,2) is a solution.
Next we have (3,4)
x = 3
y = 4
4(3) + 5(4) ≥ 10
12 + 20 ≥ 10
32 ≥ 10
True. 32 is greater than 10 - Hence, (3,4) is a solution.
Finally we have (-2,-2)
x = -2
y = -2
4(-2) + 5(-2) ≥ 10
-8 + -10 ≥ 10
-18 ≥ 10
False. Although you may probably think 18 is greater than 10 it is not correct. -18 isn't greater than 10 due to that it is negative.
RevyBreeze
Use properties to rewrite the given equation. Which equations have the same solution as 2.3p – 10.1 = 6.5p – 4 – 0.01p? Select two options. 2.3p – 10.1 = 6.4p – 4 2.3p – 10.1 = 6.49p – 4 230p – 1010 = 650p – 400 – p 23p – 101 = 65p – 40 – p 2.3p – 14.1 = 6.4p – 4
The equation with the same solutions are; 2.3·p - 10.1 = 6.5·p - 4 - 0.01·p
How to find properties of equality?The properties of equality can be used to rewrite the equations in
simplified form.
Two equations that have the same solution as the given equation are;
2.3p-10.1=6.49.p-4
230.p-1010=650.p-400-p
Reasons:
The given equation is; 2.3·p - 10.1 = 6.5·p - 4 - 0.01·p
Simplifying the above equation by collecting like terms gives;
2.3·p - 10.1 = 6.5·p - 0.01·p - 4 = 6.49·p - 4
2.3·p - 10.1 = 6.49·p - 4
Multiplication property of equality:
Both sides of an equation will remain equal, when both sides are
multiplied by the same amount.
Multiplying both sides of the original equation by 100 gives the same
solution as follows;
100 × (2.3·p - 10.1) = 100 × (6.5·p - 4 - 0.01·p)
230·p - 1010 = 650·p - 400 - p
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Answer:
B-2.3p – 10.1 = 6.49p – 4 and C- 230p – 1010 = 650p – 400 – p
Step-by-step explanation:
Rectangle abcd is dilated to form rectangle a′b′c′d′. what is the dilation factor? what is the center of dilation? select all that apply
Answer:
The answer is C
Step-by-step explanation:
4. a driver is entering a round-about at 25 mph and it is icy. is the driver safer to be in the inside lane or the outer lane? explain your reasoning and how it relates to the equation for fc
The driver is safer in the inside lane. The equation for fc=mv^2 states that the force of centripetal acceleration increases with velocity, so the faster the driver goes, the greater the force of centripetal acceleration.
This means that in an icy roundabout, it is safer to drive in the inside lane since the inside lane has a lower speed limit and thus a lower force of centripetal acceleration.
The driver is safer in the inside lane of a roundabout when it is icy, as the equation for fc=mv^2 states that the force of centripetal acceleration increases with velocity, so the lower speed inside lane will have a lower force of centripetal acceleration.
The driver is safer in the inside lane. The equation for fc=mv^2 states that the force of centripetal acceleration increases with velocity, so the faster the driver goes, the greater the force of centripetal acceleration.
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How do you find the domain and range of a sequence?
A sequence is a function whose domain is the set of natural numbers or a subset of the natural numbers.
The domain basically can be defined a set of all possible inputs for the function.
The range is basically the difference between the highest and the lowest numbers of a sequence.
To find the range, first put all the numbers in order. Then subtract the lowest number from the highest. The difference gives you the range of the list.
The domain of a sequence consists of the natural (counting) numbers 1, 2, 3, 4, ...Whereas the range of a sequence consists of the terms of the sequence.
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A car can travel 75 miles on 4 gallons of gas at this rate,how many gallons of gas are needed to travel 300 miles?
Answer:
16 gallons of gas are needed to travel 300 miles
Step-by-step explanation:
75x4=300
4x4=16
Identify the quadratic form of x^4-10x^2+9=0. And now Factor and solve for X.
The Quadratic form is (x² - 1)(x² - 9) = 0 and the value of x is {1, -1, 3, -3}.
What is Quadratic form ?
In mathematics, a polynomial with all terms of degree two is called a quadratic form.
Given, x^4-10x^2+9=0
here, a + b = -10
and, ab = 9
So, a = -1 and b = -9
So, It can be written in Quadratic form as :
(x² - 1)(x² - 9) = 0
Now, we can solve as follows:
x² - 1 = 0 and, x² - 9 = 0
(x+1)(x-1) = 0 (x+3)(x-3)=0
x = ±1 and, x = ±3
∴ Value of x = {1, -1, 3, -3}
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Can you tell me what the answer is
The simplified form of the expression is ∛5/3x
What is an expression?Expressions in maths are mathematical statements that have a minimum of two terms containing numbers or variables, or both, connected by an operator in between.
Given is an expression, [tex]\sqrt[3]{\frac{10x^{5} }{54x^{8} } }[/tex]
Simplifying, we get,
[tex]\sqrt[3]{\frac{10x^{5} }{54x^{8} } }\\= \sqrt[3]{\frac{10x^{3+2} }{2*27x^{3+3+2} } }\\= \frac{x}{3x^{2} } \sqrt[3]{\frac{10x^{2} }{2x^{2} } }\\= \frac{\sqrt[3]{5} }{3x}[/tex]
Hence, the expression is ∛5/3x
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Select all the correct answers. Which expressions are equal to [tex]6^3 * 2^6/2^3[/tex] ?
Step-by-step explanation:
With bodmas rule
first should be solve divide
so 2^6/2^3=2^3
6^3*2^3=1728
Find the derivative of the function. h(x) = 1 - x2 sin-1(x) -x2-x h'(x) = 1+x 1-x sin-1 x +1 -x2 +1
The derivative of the function h(x) = 1 - x^2 sin^-1(x) -x^2-x is h'(x) = - x^2/(sqrt(1-x^2)) - 2x - 1.
What is derivative ?
The derivative of a function is a measure of how much the output of the function changes with respect to the change in its input. It is represented by the symbol "d/dx" or "∂/∂x" and it is a fundamental concept in calculus.
First, we will apply the chain rule to the term -x^2 sin^-1(x)
Let u = sin^-1(x)
du/dx = 1/(sqrt(1-x^2))
so, -x^2sin^-1(x) = -x^2u
then dh/du = -x^2
then dh/dx = -x^2 * (du/dx) = -x^2/(sqrt(1-x^2))
Next, we will differentiate the rest of the function ,
h(x) = 1 - x^2sin^-1(x) - x^2 - x
h'(x) = - x^2/(sqrt(1-x^2)) - 2x - 1
So, the derivative of the function h(x) = 1 - x^2 sin^-1(x) -x^2-x is h'(x) = - x^2/(sqrt(1-x^2)) - 2x - 1.
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Drag each tile to the correct box. Not all tiles will be used. The current population of a city is about 2,130,000 people, and it’s increasing at a rate of 10 percent per year. Arrange the steps in the correct order to show how to determine the population of the city in five years.
Answer:
y = 2/5x + 10 (1st box)
y = -1/2x + 1/2 (2nd box)
y = 3/2x - 11/2 (3rd box)
y = 4/3 x - 7/3 (4th box)
y = 3/4x - 10 (5th box)
Step-by-step explanation:
find and sketch the domain of the function. f(x, y) = ln(25 − x2 − 25y2)
The domain is the area enclosed by the circle, which is all the points inside the unit circle.
The domain of a function is the set of all possible input values (x, y) for which the function is defined and produces a valid output.
For f(x, y) = ln(25 − x2 − 25y2), the function is defined as long as 25 − x2 − 25y2 > 0. This is because the natural logarithm (ln) is only defined for positive numbers. To find the domain, we can set 25 − x2 − 25y2 > 0 and solve for x and y.
25 − x2 − 25y2 > 0
x2 + 25y2 < 25
x2 / 25 + y2 < 1
This is the equation of a unit circle centered at (0,0) with a radius of 1.
So the domain is the area enclosed by the circle, which is all the points inside the unit circle.
You can graph this on the x-y plane.
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Express tan() in terms of sec() for in Quadrant II.
It can express as tan (θ) = -√(sec² (θ) - 1).
What is the quadrant?A quadrant is defined as an area contained by the x and y axes, which there are four quadrants in a graph.
We have to express tan(θ) in terms of sec(θ) for θ in Quadrant II.
As per the trigonometry identity, we know that
sec² (θ) - tan² (θ) = 1
tan² (θ) = sec² (θ) - 1
Apply the square root property to both sides, and we get
tan (θ) = ± √(sec² (θ) - 1)
Since θ in Quadrant II, so we take a negative value
tan (θ) = -√(sec² (θ) - 1)
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