Solution
[tex]2\leq y[/tex]Assume that x and y are both differentiable functions of t and find the required values of dy/dt and dx/dt.
[tex]x^2+y^2=400[/tex]
a) find dy/dt given x=16, y=12 and dy/dt=7
b) find dx/dt given x=16, y=12, and dy/dt =-3
For the given equation: x² + y² = 400, the required values of dy/dt and dx/dt are [(-28)/3] and 4 respectively.
What are differentiable functions?
If the derivative f '(a) exists at each point in its domain, then f(x) is said to be differentiable at the point x = a. Given two functions g and h, where y = g(u) and u = h(x). A function is referred to as a composite function if its definition is y = g [h (x)] or goh(x). Therefore, fog is also differentiable and (fog)'(x) = f'(g(x) if g (x) and h (x) are two differentiable functions. g’(x).
Given, the equation for x and y is: x² + y² = 400
Differentiating the equation above with respect to t using chain rule, we have: (2x)(dx/dt) + (2y)(dy/dt) = 0 -(i)
Rearranging (i) for dy/dt, we have: dy/dt = (-x/y)(dx/dt) - (ii)
Again, rearranging (i) for dx/dt, we have: dx/dt = (-y/x)(dy/dt) - (iii)
For (a), x = 16, y = 12 and dx/dt = 7, thus dy/dt using (ii) can be written as:
dy/dt = (-x/y)(dx/dt) = (-16/12)*7 = (-4/3)*7 = (-28)/3
For (b), x = 16, y = 12 and dy/dt = -3, thus dx/dt using (iii) can be written as:
dx/dt = (-y/x)(dy/dt) = (-12/16)*(-3) = (4/(-3))*(-3) = 4
Therefore, for the given equation: x² + y² = 400, the required values of dy/dt and dx/dt are [(-28)/3] and 4 respectively.
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The data can be modeled by the following system of linear equations.
-3x+10y = 160
x+2y=164
Equation 1
Equation 2
Equation 1 is modeled for the percentage of never-married American adults, y, x years after 1970 and Equation 2 is modeled for the percentage of married
American adults, y, x years after 1970. Use these models to complete parts a and b.
a. Determine the year, rounded to the nearest year, when the percentage of never-married adults will be the same as the percentage of married adults. For
that year, approximately what percentage of Americans, rounded to the nearest percent, will belong to each group?
In year
the percentage of never-married adults will be the same as the percentage of married adults. For that year, approximately % percentage of
Americans will belong to each group.
After 4 years the percentage of never-married adults will be the same as the percentage of married adults.
The data can be modeled by the following system of linear equations.
-3x+10y = 160
x+2y=164
Multiply the second equation with 3
-3x + 10y = 160 .....equation 1
3x + 6y = 492........equation 2
adding equation 1 and 2
16y = 652
y = 40.75
x + 2y = 164
x = 164 - 2 (40.75)
x = 82.5
Let the number of years be t
-3x+10y x t = x+2y
t = 4x - 8y
t = 330 - 326
t = 4 years
Therefore, after 4 years the percentage of never-married adults will be the same as the percentage of married adults.
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Model Real Life You have 3 toy bears. Yohave more yo-yos than toy bears. How mamore yo-yos do you have?
Solution
Step 1
Let the number of yo-yos than toy bears = x
In how many ways can Joe, Mary, Steve, Tina and Brenda be seated around a round table?241220
The number of people to be seated around the table, n=5.
Now, n=5 people can be seated in a circle in (n-1)! ways.
[tex](n-1)!=(5-1)=4!\text{ =4}\times3\times2\times1=24[/tex]Therefore, Joe, Mary, Steve, Tina and Brenda can be seated around the round table in 24 ways.
solve the problem by defining a variable and writing an equation
Randy and Wade started riding a bike at noon. Noon is 12 pm. Both of them are heading towards each other and 60km.
let
speed of wade = x
speed of Randy = 4 + x
They met each other at 1:30 pm. 12 pm to 1:30 pm is 1 hour 30 minutes(1.5 hours). Both of them will cover a total distance of 60km.
[tex]\begin{gathered} \text{speed}=\frac{dis\tan ce}{\text{time}} \\ \text{speed}\times time=dis\tan ce \end{gathered}[/tex]Therefore,
[tex]\begin{gathered} 1.5x+1.5(4+x)=60 \\ 1.5x+6+1.5x=60 \\ 3x=60-6 \\ 3x=54 \\ x=\frac{54}{3} \\ x=18\text{ km/hr} \end{gathered}[/tex]speed of wade = 18km/hr
speed of Randy = 4 + 18 = 22km/hr
identify the amplitude and period of the function then graph the function and describe the graph of G as a transformation of the graph of its parent function
Given the function:
[tex]g(x)=cos4x[/tex]Let's find the amplitude and period of the function.
Apply the general cosine function:
[tex]f(x)=Acos(bx+c)+d[/tex]Where A is the amplitude.
Comparing both functions, we have:
A = 1
b = 4
Hence, we have:
Amplitude, A = 1
To find the period, we have:
[tex]\frac{2\pi}{b}=\frac{2\pi}{4}=\frac{\pi}{2}[/tex]Therefore, the period is = π/2
The graph of the function is shown below:
The parent function of the given function is:
[tex]f(x)=cosx[/tex]Let's describe the transformation..
Apply the transformation rules for function.
We have:
The transformation that occured from f(x) = cosx to g(x) = cos4x using the rules of transformation can be said to be a horizontal compression.
ANSWER:
Amplitude = 1
Period = π/2
Transformation = horizontal compression.
write each phrase as an algebraic expression:1) n times 72) 4 minus y3) 13 added to x
1) n times 7
times means multiplication
7n
PLEASE READ BEFORE ANSWERING: ITS ALL ONE QUESTION HENCE "QUESTION 6" THEY ARE NOT INDIVIDUALLY DIFFERENT QUESTIONS.
First, lets note that the given functions are polynomials of degree 2. Since the domain of a polynomial is the entire set of real numbers, the domain for all cases is:
[tex](-\infty,\infty)[/tex]Now, lets find the range for all cases. In this regard, we will use the first derivative criteria in order to obtain the minimum (or maximim) point.
case 1)
In the first case, we have
[tex]\begin{gathered} 1)\text{ }\frac{d}{dx}f(x)=6x+6=0 \\ which\text{ gives} \\ x=-1 \end{gathered}[/tex]which corresponds to the point (-1,-8). Then the minimum y-value is -8 because the leading coefficient is positive, which means that the curve opens upwards. So the range is
[tex]\lbrack-8,\infty)[/tex]On the other hand, the horizontal intercept (or x-intercept) is the value of the variable x when the function value is zero, that is,
[tex]3x^2+6x-5=0[/tex]which gives
[tex]\begin{gathered} x_1=-1+\frac{2\sqrt{6}}{3} \\ and \\ x_2=-1-\frac{2\sqrt{6}}{3} \end{gathered}[/tex]Case 2)
In this case, the first derivative criteria give us
[tex]\begin{gathered} \frac{d}{dx}g(x)=2x+2=0 \\ then \\ x=-1 \end{gathered}[/tex]Since the leading coefficient is positive, the curve opens upwards so the point (-1,5) is the minimum values. Then, the range is
[tex]\lbrack5,\infty)[/tex][tex]\lbrack5,\infty)[/tex]and the horizontal intercepts do not exists.
Case 3)
In this case, the first derivative criteris gives
[tex]\begin{gathered} \frac{d}{dx}f(x)=-2x=0 \\ then \\ x=0 \end{gathered}[/tex]Since the leading coeffcient is negative the curve opens downwards and the maximum point is (0,9). So the range is
[tex](-\infty,9\rbrack[/tex]and the horizontal intercepts occur at
[tex]\begin{gathered} -x^2+9=0 \\ then \\ x=\pm3 \end{gathered}[/tex]Case 4)
In this case, the first derivative yields
[tex]\begin{gathered} \frac{d}{dx}p(t)=6t-12=0 \\ so \\ t=2 \end{gathered}[/tex]since the leading coefficient is postive the curve opens upwards and the point (2,-12) is the minimum point. Then the range is
[tex]\lbrack-12,\infty)[/tex]and the horizontal intercetps ocurr when
[tex]\begin{gathered} 3x^2-12x=0 \\ which\text{ gives} \\ x=4 \\ and \\ x=0 \end{gathered}[/tex]Case 5)
In this case, the leading coefficient is positive so the curve opens upwards and the minimum point ocurrs at x=0. Therefore, the range is
[tex]\lbrack0,\infty)[/tex]and thehorizontal intercept is ('0,0).
In summary, by rounding to the nearest tenth, the answers are:
Mai made $192 for 12 hours of work at the same rate how many hours would she have to work to make $128? Please help
We were told that Mai made $192 for 12 hours of work. This means that the amount that she made per hour is
192/12 = $16
Given that her constant rate is $16 per hour,
let x = the number of hours would she have to work to make $128. Then, we have the following equations
1 = 16
x = 128
By crossmultiplying, we have
16x = 128
x = 128/16
x = 8
She has to work for 8 hours
model and solve. 3/5 ÷ 1/2 =
Solution:
Consider the following diagram
extremes and means are multiplied in the diagram. Then we have that:
[tex]\frac{\frac{3}{5}}{\frac{1}{2}}\text{ = }\frac{3\text{ x 2}}{5\text{ x1}}\text{ = }\frac{6}{5}\text{ = 1.2}[/tex]and this number is represented on the real line as follows:
es26. Name the relationship between the anglesand solve for x.12x + 417x + 2
From the given figure, we can see that the angle (12x+4) and the angle (17x+2) lie on the same straight line.
Theerfore, they are supplimentary to each other.
3x - 2y < 10 ) 5x - 3-15 부 다. N
Part 1
we have the inequality
3x-2y < 10
isolate the variable y
-2y< -3x+10
Multipli by -1 both sides
2y > 3x-10
the solution is the shaded area above the dashed line 2y=3x-10
using a graphing tool
see the attached figure
please wait a minute to draw the solution
[tex]5x-3y\leq-15[/tex]isolate the variable y
[tex]\begin{gathered} 5x-3y\leq-15 \\ -3y\leq-5x-15 \\ \text{Multiply by -1 both sides} \\ 3y\ge5x+15 \end{gathered}[/tex]the solution is the shaded area above the solid line 3y=5x+15
using a graphing tool
see the attached figure
Part 7 we have the inequality
Question 17. 4 pts
In 98 years of football, Loudon has averaged 296 points per season and the standard deviation is 14. What percent of the years has Loudon scored between 254 and 338 points per season?
Answer:
Over 98 years, London scored 75.14% per season between 254 and 338 points.
Step-by-step explanation:
Use the slope formula to find the slope of the line that passes through the points (5,2) and (13,3)A)m=7B)m=-2/11C)m=1/8D)m=3/11
Given the word problem, we can deduce the following information:
1. The line that passes through the points (5,2) and (13,3).
We can get the slope of the line using the slope formula:
Based on the given points, we let:
We plug in what we know:
[tex]\begin{gathered} m=\frac{y_2-y_1}{x_2-x_1} \\ =\frac{3-2}{13-5} \\ \text{Simplify} \\ m=\frac{1}{8} \end{gathered}[/tex]Therefore, the answer is c. m=1/8.
The zookeeper records how many scoops of peanuts she feeds the elephant for several days . Tuesday 21 Wednesday 19 5/8.
Explanation:
We want to know the difference between the amount of scoops she fed the elephant on Wednesday and on Tuesday:
[tex]21-19\frac{5}{8}[/tex]We can write the second number as an improper fraction:
[tex]21-(19\cdot\frac{8}{8}+\frac{5}{8})=21-(\frac{152}{8}+\frac{5}{8})=21-\frac{157}{8}[/tex]And now substract the two numbers:
[tex]\begin{gathered} 21-\frac{157}{8}=\frac{21\cdot8}{8}-\frac{157}{8} \\ 21-\frac{157}{8}=\frac{168}{8}-\frac{157}{8} \\ 21-\frac{157}{8}=\frac{168-157}{8}=\frac{11}{8} \end{gathered}[/tex]Answer:
She fed the elephant 11/8 scoops of peanuts more on Tuesday than on Wednesday
Slove for p 14 = -(p - 8)
Solve:
[tex]\begin{gathered} 14=-(p-8) \\ -14=p-8 \\ -14+8=p \\ p=-14+8 \\ p=-6 \end{gathered}[/tex]p=-6
What is the product of 3√6 and 5√12 in simplest radical form?
In order to calculate and simplify this product, we need to use the following properties:
[tex]\begin{gathered} \sqrt[]{a}\cdot\sqrt[]{b}=\sqrt[]{a\cdot b} \\ \sqrt[c]{a^b}=a\sqrt[c]{a^{b-c}} \end{gathered}[/tex]So we have that:
[tex]\begin{gathered} 3\sqrt[]{6}\cdot5\sqrt[]{12} \\ =(3\cdot5)\cdot(\sqrt[]{6}\cdot\sqrt[]{2\cdot6}) \\ =15\cdot\sqrt[]{2\cdot6^2} \\ =15\cdot6\cdot\sqrt[]{2} \\ =90\sqrt[]{2} \end{gathered}[/tex]So the result in the simplest radical form is 90√2.
Describe a situation that can be represented by the expression –15 + 8.
Answer:
-7
Step-by-step explanation:
Tiger Woods was 15 under par after the third round of a golf tournament, but was 8 over par for the fourth round. So, his score for the entire tournament was -15 + 8 = -7 (That is, 7 under par).
What is the greatest common factor of 28y^2 and 49y^2?A. 196y^2B. 7y^2C. 21y^2D. 7y
the value is 7 and keep the y^2
so is
[tex]7y^2[/tex]Seth earns $25 a day and $3 for each ticket he sells at the local theatre. Write and solve aninequality that can be used to find how many tickets he must sell in a day to earn at least $115.Solve.
Seth earns $25 a day and also she earns $3 for each ticket he sells at the local theatre.
Therefore $25 is the independent value and $3 is the dependent value because it depends on how many tickets are sold.
We can write the next expression:
[tex]25+3x[/tex]Now, we need to make an inequality about he must sell at least $115 in a day.
The word "at least" means greater than or equal to, therefore:
[tex]25+3x\ge115[/tex]Now, let's solve the inequality:
Subtract both sides by 25:
[tex]25-25+3x\ge115-25[/tex][tex]3x\ge90[/tex]Then, divide both sides by 3:
[tex]\frac{3x}{3}\ge\frac{90}{3}[/tex]Simplify:
[tex]x\ge30[/tex]If R is between G and Z, GZ = 12in., and RG =3in., then RZ =
Given R is between G and Z.
GZ=12 inches
RG=3 inches.
Since, R is between G and Z,
[tex]GZ=GR+RZ[/tex]It follows
[tex]\begin{gathered} RZ=GZ-GR \\ =12-3 \\ =9 \end{gathered}[/tex]So, RZ is 9 inches.
pls help????? –2x = –20y + 18
Answer:
y = 1/10x + 9/10
Step-by-step explanation:
Find slope intercept form: –2x = –20y + 18
slope intercept form: y = mx + b
_______________________________
–2x = –20y + 18
add 20y to both sides:
–2x + 20y = –20y + 18 + 20y
–2x + 20y = 18
add 2x to both sides:
–2x + 20y + 2x = 18 + 2x
20y = 18 + 2x
divide all terms by 20:
20y/20 = 18/20 + 2x/20
y = 9/10 + 1/10x
reorder terms for slope intercept form:
y = 1/10x + 9/10
Answer:
[tex]y=\dfrac{1}{10}x+\dfrac{9}{10}[/tex]
Step-by-step explanation:
[tex]\boxed{\begin{minipage}{6.3 cm}\underline{Slope-intercept form of a linear equation}\\\\$y=mx+b$\\\\where:\\ \phantom{ww}$\bullet$ $m$ is the slope. \\ \phantom{ww}$\bullet$ $b$ is the $y$-intercept.\\\end{minipage}}[/tex]
Given equation:
[tex]-2x=-20y+18[/tex]
To write the given equation in slope-intercept form, make y the subject.
Add 20y to both sides:
[tex]\implies 20y-2x=-20y+18+20y[/tex]
[tex]\implies 20y-2x=18[/tex]
Add 2x to both sides:
[tex]\implies 20y-2x+2x=2x+18[/tex]
[tex]\implies 20y=2x+18[/tex]
Divide both sides by 20:
[tex]\implies \dfrac{20y}{20}=\dfrac{2x+18}{20}[/tex]
[tex]\implies \dfrac{20y}{20}=\dfrac{2x}{20}+\dfrac{18}{20}[/tex]
[tex]\implies y=\dfrac{1}{10}x+\dfrac{9}{10}[/tex]
Therefore, the given equation in slope-intercept form is:
[tex]\boxed{y=\dfrac{1}{10}x+\dfrac{9}{10}}[/tex]
A typical soda can has a diameter of 5.3 centimeters and height of 12 centimeters. How many square centimeters of aluminum is needed to make the can? My answer is 244. I am confused how I got the answer.
The can is made up of aluminium.
So the area of the can must be equal to the area of the Aluminium sheet.
The can is in the form of a cylinder with diameter (d) 5.3 cm, and height (h) 12 cm.
Then its area is calculated as,
[tex]\begin{gathered} A=\pi d(\frac{d}{2}+h) \\ A=\pi(5.3)(\frac{5.3}{2}+12) \\ A=243.9289 \\ A\approx244 \end{gathered}[/tex]Thus, the area of the Aluminium sheet required is 244 square centimeters.
Jamie paid the rent well past the due date for the months of April, May and June. As a result, he had been charged a total of $75 as a late fee. Howmuch did he pay as late fee per month?Use 'f to represent the late fee $$ per month.
Total fee = $75
Number of months = 3
Divide the total fee by the number of months
75/3 = $25 per month
an airplane flew for one hour and landed 100 miles north and 80 miles east from its origin. what was the distance traveled, speed and angle of direction from its origin?
The distance traveled by airplane is 180 miles.
The speed of the airplane is 3 miles per minute and the angle of direction from the origin is 51.34°
The airplane landed 100 miles north and 80 miles east from its origin and it flew for one hour.
Then, the total distance traveled by airplane will be:
= 100 miles + 80 miles = 180 miles.
The speed can be defined as the distance traveled by the total time taken.
Speed = distance/time
Speed = 180 miles/ 1 hour
Speed = 180 miles/60 minutes
Speed = 3 miles per minute
The angle of direction from its origin will be:
tan (x) = 100 miles/80 miles
x = tan⁻¹ ( 100/80)
x = tan⁻¹ ( 10/8) = tan⁻¹ ( 5/4)
x = 51.34°
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NO LINKS!! Describe the domain and range (in BOTH interval and inequality notation) for each function shown part 1a
Answer:
Domain as an inequality: [tex]\boldsymbol{\text{x} < 6 \ \text{ or } \ -\infty < \text{x} < 6}[/tex]
Domain in interval notation: [tex]\boldsymbol{(-\infty, 6)}[/tex]
Range as an inequality: [tex]\boldsymbol{\text{y} \le 6 \ \text{ or } \ -\infty < \text{y} \le 6}[/tex]
Range in interval notation: [tex]\boldsymbol{(-\infty, 6]}[/tex]
=========================================================
Explanation:
The domain is the set of allowed x inputs. For this graph, the right-most point is when x = 6. This endpoint is not part of the domain due to the open hole. The graph goes forever to the left to indicate [tex]\text{x} < 6[/tex] but I think [tex]-\infty < \text{x} < 6[/tex] is far more descriptive.
The second format directly leads to the interval notation of [tex](-\infty, 6)[/tex]
Always use parenthesis for either infinity. We use a parenthesis for the 6 to tell the reader not to include it as part of the domain.
------------------------
The range is the set of possible y outputs.
The highest y can get is y = 6
Therefore, y = 6 or y < 6
The range can be described as [tex]\text{y} \le 6 \ \text{ or } \ -\infty < \text{y} \le 6[/tex] where the second format is better suited to lead directly to the interval notation [tex](-\infty, 6][/tex]
Use a square bracket to include the 6 as part of the range. We don't have any open holes at the peak mountain point.
Answer:
[tex]\textsf{Domain}: \quad (-\infty, 6) \quad -\infty < x < 6[/tex]
[tex]\textsf{Range}: \quad (-\infty,6] \quad -\infty < y\leq 6[/tex]
Step-by-step explanation:
The domain of a function is the set of all possible input values (x-values).
The range of a function is the set of all possible output values (y-values).
An open circle indicates the value is not included in the interval.
A closed circle indicates the value is included in the interval.
An arrow show that the function continues indefinitely in that direction.
Interval notation
( or ) : Use parentheses to indicate that the endpoint is excluded.[ or ] : Use square brackets to indicate that the endpoint is included.Inequality notation
< means "less than".> means "more than".≤ means "less than or equal to".≥ means "more than or equal to".From inspection of the given graph, the function is not continuous and so the domain is restricted.
There is an open circle at x = 6.
Therefore, the domain of the function is:
Interval notation: (-∞, 6)Inequality notation: -∞ < x < 6From inspection of the given graph, the maximum value of y is 6.
The function continues indefinitely to negative infinity.
Therefore, the range of the function is:
Interval notation: (-∞, 6]Inequality notation: -∞ < y ≤ 6Austin and carly despoit 500.00 into a savings account which earns 1% interest compounded monthly they want to use the money in the account to go on a trip in 2 years how much will they be able to spend
EXPLANATION
Let's see the facts:
Austin and Carly deposit: $500
Interest rate= 1%
Compounding period = monthly
Total number of years = 2
Given the Compounding Interest Rate formula:
[tex]\text{Compound amount = P (1+r/n)\textasciicircum{}nt}[/tex]n is the compounding period
t is the number of years
r is te interest rate in decimal form
Replacing the given values will give us:
[tex]\text{Compound amount = 500 (1+}\frac{0.01}{12})^{12\cdot2}[/tex]Solving the power:
[tex]\text{Compound amount = 500 }\cdot1.020192843[/tex][tex]\text{Compound amount = \$510.09}[/tex]Answer: Austin and Carly will be able to spend $510.09.
Your family decides to go out to dinner to celebrate your brothers graduationfrom high school. The family's meal cost $75. Your waitress did a great job andyour parents decide to leave her a 20% tip. How much tip money should yourparents leave her if they leave her 20%? And what is the total cost of the meal? *
$75 -----> 100%
x ---------> 20%
[tex]\begin{gathered} x\times100=75\times20 \\ 100x=1500 \\ \frac{100x}{100}=\frac{1500}{100} \\ x=15 \end{gathered}[/tex]asnwer 1: they leave her $ 15
answer 2: the total cost of the meal is
[tex]75+15=90[/tex]$ 90
Select the postulate that is illustrated for the real numbers.
2(x + 3) = 2x + 6
A. The multiplication inverse
B. The addition inverse postulate
C. The commutative postulate for multiplication
D. Multiplication identity
E. The distributive postulate
F. The addition of zero postulate
G. Commutative postulate for addition
The postulate that is illustrated for the real numbers 2(x + 3) = 2x + 6 is The Distributive postulate , the correct option is (E) The Distributive postulate .
The Distributive Postulate states that for any three numbers a,b and c ,
a(b+c) = a*b + a*c
For Example : 5(6+1) = 5*6 + 5*1
5*7 = 30+5
35=35
In the question ,
it is given that
2(x + 3) = 2x + 6
On applying Distributive postulate in 2(x + 3)
we get
= 2*x + 2*3
= 2x + 6
hence Distributive postulate is applied in 2(x + 3) = 2x + 6 .
Therefore , the postulate that is illustrated for the real numbers 2(x + 3) = 2x + 6 is The Distributive postulate , the correct option is (E) The Distributive postulate .
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15. When x =9, which number is closest to the value of y on the line of best fit in the graph below? 121917
We have a scatter plot.
We have to find the closest value to y on the line of best fit when x = 9.
We can estimate a line of best fit by hand in the graph as:
Although we have a data point where x = 9 and y =9, the line of best fit is kind of in between the two groups of points.
When we draw the line like this, the estimated value from the line of best fit when x = 9 is y = 12, as we can see in the graph.
Answer: 12
