Use logarithmic differentiation to find the derivative of the function. y = (x3 + 2)2(x4 + 4)4
Answers
The derivative of y = (x³ + 2)²(x⁴ + 4)⁴ using logarithmic differentiation is;
y' = {[6x²In(x³ + 2)]/(x³ + 2)} + {[16x³In(x⁴ + 4)⁴]/(x⁴ + 4)} × [(x³ + 2)²(x⁴ + 4)⁴]
We are given the function;
y = (x³ + 2)²(x⁴ + 4)⁴
We want to find the derivative using logarithmic differentiation;
Step 1; Take the natural log of both sides;
In y = In[(x³ + 2)²(x⁴ + 4)⁴]
Step 2;
Using log of a product property on this, we have;
In y = In(x³ + 2)² + In(x⁴ + 4)⁴
Step 3; We will now differentiate both sides with chain rule to get;.
y'/y = [6x²In(x³ + 2)]/(x³ + 2) + [16x³In(x⁴ + 4)⁴]/(x⁴ + 4)
Step 4; Using multiplication property of equality, multiply both sides by y to get;
y' = {[6x²In(x³ + 2)]/(x³ + 2) + [16x³In(x⁴ + 4)⁴]/(x⁴ + 4)} × y
Step 5; Plug in the value of y = (x³ + 2)²(x⁴ + 4)⁴ to get;
y' = {[6x²In(x³ + 2)]/(x³ + 2)} + {[16x³In(x⁴ + 4)⁴]/(x⁴ + 4)} × [(x³ + 2)²(x⁴ + 4)⁴]
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Related Questions
At the city museum, child admission is $5.80 and adult admission is $9.00 . On Monday, three times as many adult tickets as child tickets were sold, for a total sales of $984.00 . How many child tickets were sold that day?
Answers
Answer:
30
Step-by-step explanation:
A bundle of 3 adult tickets and 1 child ticket sells for $32.80, so there were ...
... $984/$32.80 = 30 . . . . bundles sold.
The number of child tickets sold was 30.
_____
Using an equation
Let c represent the number of child tickets sold. Then 3c is the number of adult tickets sold. The total revenue is ...
... 5.80c + 9.00·(3c) = 984.00
... 32.80c = 984.00 . . . . . . . . . . simplify
... c = 984.00/32.80 = 30 . . . . . divide by the coefficient of c
Jay is cutting a roll of biscuit dough into slices that are 3/8 inch thick. If the roll is 10 1/2 inches long, how many slices
can he cut?
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21/2 * 8/3 =
84 / 3 =
28 <== he can cut 28 slices
The number of slices that Jay could make is equal to 28
What is a mixed fraction?A fraction represented with its quotient and remainder is a mixed fraction. For example, 2 1/3 is a mixed fraction, where 2 is the quotient, and 1 is the remainder. So, a mixed fraction is a combination of a whole number and a proper fraction.
Given here: Dimension of the roll= 10¹/₂ inches and length of each slice=3/8 inch
Thus the number of slices that Jay could make is = 10+ 1/2 /3/8
=21/2 × 8/3
=28
Hence The number of slices that Jay could make is equal to 28
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Kwan's parents bought a home for $50,000 in 1997 just as real estate values in the area started to rise quickly. Each year, their house was worth more until they sold the home for $309,587. Model the growth of the home's value from 1997 to 2007 with both linear and an exponential equation. Graph the two models below.
Answers
A. First let us start with the linear model.
The equation is in the form of y = m x + b
Calculating for the slope m:
m = (309,587 – 50,000) / (2007 – 1997)
m = 25,958.7
Subsituting:
y = 25,958.7 x + b
Taking x = 1997, y = 50,000. Solve for b:
50,000 = 25,958.7 (1997) + b
b = -51,789,523.9
The complete equation is therefore:
y = 25,958.7 x - 51,789,523.9
B. The exponential model has the following form:
y = a b^x
where a and b are constants
Taking x1= 1997, y1 = 50,000; x2 = 2007, y2 = 309,587
50,000 = a b^1997
309,587 = a b^2007
Combining in terms of a:
50,000 / b^1997 = 309,587 / b^2007
b^2007 / b^1997 = 309,587 / 50,000
b^10 = 6.19174
b = 1.2
Substituting:
y = a 1.2^x
Solving for a:
50,000 = a 1.2^1997
a = 3.75 x 10^-154
The complete equation is:
y = 3.75 x 10^-154 * 1.2^x
A baseball team has a goal of hitting more than 84 home runs this season. They average 7 home runs each game and have already hit 35 home runs so far. How many more games, x, will it take the baseball team to reach its home run hitting goal if they continue to average 7 home runs per game?
Answers
Final answer:
The baseball team needs to play 7 more games to exceed its goal of 84 home runs, based on their current average of 7 home runs per game.
Explanation:
The baseball team has hit 35 home runs and wants to hit more than 84. To find how many more games it will take, we first calculate the total number of home runs needed to surpass 84, which is 84 - 35 = 49 home runs. Since the team averages 7 home runs per game, we divide the remaining home runs needed by the average per game to find x, the number of games needed: 49 home runs / 7 home runs per game = 7 games.
Therefore, the baseball team needs to play 7 more games to reach its goal, assuming the average stays constant.
Evaluate the line integral ∫cf⋅dr∫cf⋅dr, where f(x,y,z)=−5xi+yj+zkf(x,y,z)=−5xi+yj+zk and c is given by the vector function r(t)=⟨sint,cost,t⟩, 0≤t≤3π/2.
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[tex]=\displaystyle\int_0^{3\pi/2}(-6\sin t\cos t+t)\,\mathrm dt=\frac{9\pi^2}8-3[/tex]
If g(x) = x3 - 5 and h(x) = 2x - 2, find g(h(3))
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mathematically represented as:
h(3)=2(3)-2 which gives 4
you then substitute 4 with X in g(x)
we then have g(h(3)) where h(3)=4 and X is also =h(3) as
g(4)=4^3-5
=59
•
• • g(h(3))=59
The required simplified value of the g(h(3)) is given as 59.
What is simplification?The process in mathematics to operate and interpret the function to make the function or expression simple or more understandable is called simplifying and the process is called simplification.
Here,
g(x) = x³ - 5 and h(x) = 2x - 2
h(3) = 2(3) - 2
h(3) = 4
g(h(3)) = 4³ - 5
= 64 - 5
= 59
Thus, the required simplified value of the g(h(3)) is given as 59.
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one pumpkin vine produce 5pumpkins I harvest 30 how many vines do I have
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Twice the difference between a number and 10 is equal to 6 times the number plus 16. What is the number?
Answers
Evaluate the upper and lower sums for f(x) = 1 + x2, −1 ≤ x ≤ 1, with n = 3 and 4.
Answers
Answer:
92/27 and 58/27 for n=3 and 13/4 and 9/4 for n=4
Step-by-step explanation:
n = 3: First let’s find ∆x:
∆x = (b − a)/n = 1 − (−1)3 = 2/3
We will have three intervals: −1 ≤ x ≤ −1/3, −1/3 ≤ x ≤1/3, 1/3 ≤ x ≤ 1.
Upper sum: On the first interval, the highest point occurs at f(−1) = 2. On the second interval, the highest point occurs at f(1/3) = f(−1/3) = 1 + ( 1/3)^2=1 + 1/9 = 10/9
On the third interval, the highest point occurs at f(1) = 2. So A ≈ A upper = 3Σ i=1 f(xi)∆x = [f(−1) + f (1/3)+ f(1)]∆x = (2 +10/9+ 2)*2/3 = 92/27
Lower sum: On the first interval, the lowest point occurs at f(−1/3) = 10/9
On the second interval, the lowest point occurs at f(0) = 1. On the third interval, the lowest point occurs at f(1/3) = 10/9. SoA ≈ A lower =3Σ i=1 f(xi)∆x = f(−1/3) + f(0) + f(1/3) . ∆x = (10/9+ 1 +10/9)· 2/3= 58/27
Apply the same technique for n=4
which statements are true about the regular polygon? check all that apply.
Answers
its 1, 3, and 5 I think
Answer:
1) False
2) True
3) True
4) False
5) True
Step-by-step explanation:
We are given the following information in the question:
1) False
The sum of measures of interior angle is given by:
[tex](n-2)\times 180^\circ\\\text{where n is the number of sides in regular polygon}[/tex]
Putting n = 5
Sum of interior\r angle = [tex](5-2)\times 180 = 3\times 180 = 540^\circ[/tex]
2) True
Each interior angle measure =
[tex]\displaystyle\frac{\text{Sum of interior angles}}{\text{Number of sides}} = \frac{540}{5} = 108^\circ[/tex]
3) True
All the angles in a regular polygon are equal.
4) False
The polygon is a regular pentagon.
5) True
The sum of measures of interior angle is given by:
[tex]180^\circ(5-2)[/tex]
A bouncing ball reaches a height of 27 feet at its first peak, 18 feet at its second peak, and 12 feet at its third peak. Describe how a sequence can be used to determine the height of the ball when it reaches its fourth peak.
Answers
18-12 = 6
12- n = 3
-n = 3-12
-n = -9 ( cancel the negative sign by simple divide both sides with -1 )
n = 9
Therefore the height of the ball is 9 feet when it reaches fourth peak .
Answer:
so on the fourth bounce the ball will reach a height of 8 feet
Step-by-step explanation:
There is a common ratio of 2/3 between the height of the ball at each bounce so the bounce height from a geometric sequence 27,18,12 two-thirds of 12 is 8 so on the fourth bounce the ball will reach a height of 8 feet
1. The expressionx^3+3x^2+x-5 represents the total length across the front of the mansion. Find the length of side I. Show all work. X will stay a variable and your answer will be a polynomial
2. 2. The owners of the mansion want to install a new security system to protect the outside of their house. To get an estimate for the cost of the new security system they need to know the total distance around the outside of the mansion.
a. How many sides does the Mansion have?
b. What side would be the same length as side L?
c. What side would be the same length as side D?
d. Find an expression for the perimeter of the mansion. Show all work. X will stay a variable and your answer will be a polynomial.
3. The owners of the mansion also want to install new AC units. You must know the amount of air on the first floor of the mansion to determine the size of AC units necessary to function properly. The first floor of the mansion covers, or has a base, 15x^4+20x^2+45x+590 square feet. If the height of the ceilings on the first floor is 5x-3, find the total volume of air on the first floor. Use the formula V=Bh where Bis the area of the base and his the height. Show all work. X will stay a variable and your answer will be a polynomial.
4. The area of the Gothic Room in the mansion’s layout is2x^3+25x^2+169, what are the dimensions (length and width) for the room? Side E is x + 13. Use synthetic division to find the other side width. Show all work. X will stay a variable and your answer will be a polynomial.
Answers
Given that the expression [tex]x^3+3x^2+x-5[/tex] represents the total length across the front of the mansion. Let the length of side I be a, then
[tex]x^4-2x^3+x-10+a-x^3+5x^2+2=x^3+3x^2+x-5 \\ \\ \Rightarrow a=x^3+3x^2+x-5-x^4+2x^3-x+10+x^3-5x^2-2 \\ \\ =\bold{-x^4+4x^3-2x^2+3}[/tex]
Part 2:
From the given figure it can be seen that the mansion has 12 sides labelled A - L.
Side F will be of the same length as side L.
Side B will be of the same length as side D.
The perimeter of figure is the sum of all the lengths of the outlines of the figure.
The perimeter is given by:
[tex]P=2(x^3+3x^2+x-5)+2(-2x^4+24x^2-10)+2(4x)+x^2-x+6 \\ \\ =2x^3+6x^2+2x-10-4x^4+48x^2-20+8x+x^2-x+6 \\ \\ =\bold{-4x^4+2x^3+55x^2+9x-24}[/tex]
Part 3:
The volume of an object is given by: V = Area of base x height.
Given that the first floor of the mansion has a base, [tex]15x^4+20x^2+45x+590\ square\ feet[/tex]. If the height of the ceilings on the first floor is 5x-3, then, the total volume of air on the first floor is given by:
[tex]V=(5x-3)(15x^4+20x^2+45x+590) \\ \\ =75x^5+100x^3+225x^2+2950-45x^4-60x^2-135x-1770 \\ \\ =\bold{75x^5-45x^4+100x^3+165x^2-135x+1180}[/tex]
Part 4:
Given that the area of the Gothic Room in the mansion’s layout is [tex]2x^3+25x^2+169[/tex], with the length of one of the sides of the room as x + 13. Then, the length of the other side is given by:
-13 | 2 25 0 169
|
| -26 13 -169
|______________
2 -1 13 0
Therefore, the length of the other side of the length is [tex]-26x^2+13x-169[/tex]
Josh is twenty-five years older than his son Carlos. In ten years Josh will be twice as old as Carlos. How old is Carlos now?
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Under good weather conditions, 80% of flights arrive on time. during bad weather, only 30% of flights arrive on time. tomorrow, the chance of good weather is 60%. what is the probability that your flight will arrive on time?
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The probability that your flight will arrive on time is 0.6, or 60%.
Explanation:To calculate the probability that your flight will arrive on time, we can use the concept of conditional probability. Let A represent the event of good weather and B represent the event of the flight arriving on time. We know P(A) = 0.6, P(B|A) = 0.8, and P(B|A') = 0.3 (where A' represents bad weather). We can use the formula for conditional probability: P(B) = P(A) * P(B|A) + P(A') * P(B|A'). Substituting the given values, we have P(B) = 0.6 * 0.8 + 0.4 * 0.3 = 0.48 + 0.12 = 0.6. Therefore, the probability that your flight will arrive on time is 0.6, or 60%.
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What is thirty-four thousand centimeters per second in scientific notation
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As a result, I got 3.4 x 10^4 per second. (Please check my method to see if it is correct)
Also, if you move the decimal to the right, the exponent will be negative.
The speed of thirty-four thousand centimeters per second in scientific notation is 3.4 x 10^2 meters per second, after converting centimeters to meters.
The speed of thirty-four thousand centimeters per second can be expressed in scientific notation by first converting it to the base unit of meters per second (since one meter is equal to one hundred centimeters) and then expressing it in the form of a significant figure followed by the power of 10.
First, convert centimeters to meters:
34000 centimeters = 34000 / 100 meters = 340 meters.Now, express this in scientific notation:
340 meters/second = 3.4 x 102 meters/second.This shows the speed in scientific notation with the unit meters per second, which is a combination of two SI units.
A spherical block of ice melts so that its surface area decreases at a constant rate: ds/dt = - 8 pi cm^2/s. calculate how fast the radius is decreasing when the radius is 3cm. (recall that s = 4 pi r^2.)
Answers
[tex] \frac{dS}{dt} =-8 \pi \, cm^{2}/s[/tex]
The surface area is
S = 4πr²
where r = the radius at time t.
Therefore
[tex] \frac{dS}{dt} = \frac{dS}{dr} \frac{dr}{dt} =8 \pi r \frac{dr}{dt} \\\\ -8 \pi = 8 \pi r \frac{dr}{dt} \\\\ \frac{dr}{dt} =- \frac{1}{r} [/tex]
When r = 3 cm, obtain
[tex] \frac{dr}{dt}]_{r=3} = - \frac{1}{3} \, cm/s [/tex]
Answer: -1/3 cm/s (or -0.333 cm/s)
George borrowed $1,895.50 for two years. The total amount he repaid was $2,189.38. How much interest did he pay for the loan?
a. $454.50
b. $293.88
c. $227.73
d. $455
Answers
You buy the same number of brushes, rollers, and paint cans. Write and expression in simplest form that represents the toral amount of money you spend for the painting supplies
Answers
X being the amount of supplies
B are the price of the brushes
R is the price of the rollers
C the price of the paint cans
M is the total price
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Three quarters of the batch of twenty cookies burned when you forgot to take them out of the oven how many cookies burned
Answers
"At a college, 4/5 of the students take an english class. Of these students, 5/6 take composition. Which fraction of the students at the college take compostion
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5/6 of the 4/5 of the college takes composition.
Thus (5/6)*(4/5) = 20/30 = 2/3 of the college takes composition.
Answer:2/3
Step-by-step explanation:
Translation of the graph. I can never get these right.
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Since you're finding the y value from the x value here, and you added 5 more to the x value in g(x), this means that it was translated 5 units up, because if you increase with the y value, it means that you're moving up on the coordinate graph.
So then your answer would be the third choice: The graph of g(x) is the graph of f(x) translated 5 units up.
which is the longer length 29ft or 9 yards
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9 yards, or 27 feet, is shorter than 29 feet.
29 feet is longer than 27 feet.
To compare 29 feet and 9 yards, convert 9 yards to feet, which equals 27 feet. Since 29 feet is greater than 27 feet, 29 feet is longer.
To determine which length is longer between 29 feet and 9 yards, we need to convert one of the measurements to the same unit. We know that 1 yard is equal to 3 feet. Therefore, to convert 9 yards to feet:
9 yards x 3 feet/yard = 27 feetComparing the two lengths, 29 feet is longer than 27 feet.
Thus, 29 feet is the longer length.
Arleen has a gift card for a local lawn and garden store. She uses the gift card to rent a tiller for 4 days. It cost $35 per day to rent the tiller. She also buys a rake for $9.
A. Find the change to the value on the gift card. (Question #1)
B. The original amount on the gift card was $200. Does Arleen have enough to buy a Wheelbarrow for $50? (Question #2)
Answers
200 - 149 = 51 left on gift card. Yes she has enough to buy the wheelbarrow.
Answer:
Step-by-step explanation:
Arleen has a gift card of the amount = $200
The total cost to rent the tiller for 4 days and a rake for $9
= ($35 × 4) + $9
= 140 + 9 = $149
A. Now the balance on the gift card = $200 - $149
= $51
B. Arleen have $51 in her gift card so she has enough to buy a wheelbarrow for $50.00.
Myra borrowed $1,500 at 12.5% interest for three months. How much
does she have to repay under a single-payment plan?
a. $46.88
b. $1,562.50
c. $1,546.88
d. $187.50
**please explain how you got the answer, it's just so that I can understand
Answers
now, 3 months is not even a year, there are 12 months in 1 year, so 3 months is 3/12 years, or 1/4.
[tex]\bf \qquad \textit{Simple Interest Earned}\\\\ I = Prt\qquad \begin{cases} I=\textit{interest earned}\\ P=\textit{original amount deposited}\to& \$1500\\ r=rate\to 12.5\%\to \frac{12.5}{100}\to &0.125\\ t=years\to \frac{3}{12}\to &\frac{1}{4} \end{cases} \\\\\\ I=1500\cdot 0.125\cdot \cfrac{1}{4}\implies 46.875\approx 46.88[/tex]
Maria put $500 into a savings account. She made no more deposits or withdrawals. The account earns 2% simple interest per year. How much money will be in the account after 5 years?5,000,50,550,1000
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AB = 2 and AC = 11. Find m∠C to the nearest degree.
Answers
Answer:
10°
Step-by-step explanation:
To find this, we look at the sides of the triangle that we are given and their position in relation to ∠C.
AB is opposite ∠C and AC is adjacent to ∠C.
The ratio of opposite/adjacent is the ratio of tangent. This gives us the equation
tan C = 2/11
To solve this for C, we take the inverse tangent of each side:
tan⁻¹(tan C) = tan⁻¹(2/11)
C = 10.305 ≈ 10°
The measure of the angle ∠c will be 10°.
What is trigonometry?The branch of mathematics that sets up a relationship between the sides and the angles of the right-angle triangle is termed trigonometry.
The trigonometric functions, also known as circular, angle, or goniometric functions in mathematics, are real functions that link the angle of a right-angled triangle to the ratios of its two side lengths.
To find this, we look at the sides of the triangle that we are given and their position in relation to ∠C.
AB is opposite ∠C and AC is adjacent to ∠C.
The ratio of opposite/adjacent is the ratio of a tangent. This gives us the equation
tan C = 2/11
To solve this for C, we take the inverse tangent of each side:
tan⁻¹(tan C) = tan⁻¹(2/11)
C = 10.305 ≈ 10°
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Reuben bought a desktop computer and a laptop computer. Before finance charges, the laptop cost $150 more than the desktop. He paid for the computers using two different financing plans. For the desktop the interest rate was 7% per year, and for the laptop it was 9.5% per year. The total finance charges for one year were $303 . How much did each computer cost before finance charges?
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a = price for the desktop
b = price for the laptop
we know the laptop is 150 bucks more than the desktop, b = a + 150.
how much is 7% of a? (7/100) * a, 0.07a.
how much is 9.5% of b? (9.5/100) * b, 0.095b.
total interests for the financing add up to 303, 0.07a + 0.095b = 303.
[tex]\bf \begin{cases} \boxed{b}=a+150\\ 0.07a+0.095b=303\\ ----------\\ 0.07a+0.095\left(\boxed{a+150} \right)=303 \end{cases} \\\\\\ 0.07a+0.095a+14.25=303\implies 0.165a=288.75 \\\\\\ a=\cfrac{288.75}{0.165}\implies a=1750[/tex]
how much was it for the laptop? well b = a + 150.
Find the solution to the linear system of differential equations {x′y′==−5x+3y−18x+10y satisfying the initial conditions x(0)=4 and y(0)=11
Answers
[tex]\begin{bmatrix}x\\y\end{bmatrix}'=\begin{bmatrix}-5&3\\-18&10\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}[/tex]
The coefficient matrix has eigenvalues [tex]\lambda=1,4[/tex], with corresponding eigenvectors [tex]\mathbf v=\begin{bmatrix}1\\2\end{bmatrix},\begin{bmatrix}1\\3\end{bmatrix}[/tex]. So the general solution is
[tex]\begin{bmatrix}x\\y\end{bmatrix}=C_1e^t\begin{bmatrix}1\\2\end{bmatrix}+C_2e^{4t}\begin{bmatrix}1\\3\end{bmatrix}[/tex]
Given that [tex]x(0)=4[/tex] and [tex]y(0)=11[/tex], we get
[tex]\begin{cases}4=C_1+C_2\\11=2C_1+3C_2\end{cases}\implies C_1=1,C_2=3[/tex]
so that the particular solution to the system is
[tex]\begin{bmatrix}x\\y\end{bmatrix}=e^t\begin{bmatrix}1\\2\end{bmatrix}+3e^{4t}\begin{bmatrix}1\\3\end{bmatrix}[/tex]
or in equivalent terms,
[tex]\begin{cases}x=e^t+3e^{4t}\\y=2e^t+9e^{4t}\end{cases}[/tex]
The problem given is a linear system of differential equations with initial conditions, but the system is not properly defined and appears to have a typo. Normally, such a system would be in the form x' = ax + by and y' = cx + dy, with initial conditions that would determine the particular solution.
Explanation:This question is about solving a linear system of differential equations with initial conditions, specifically the system {x′y′==−5x+3y−18x+10y with the initial conditions x(0)=4 and y(0)=11. However, there seems to be a typo, as the system is not properly defined and doesn't make sense as written. Under normal conditions, such a system would be in the form x' = ax + by and y' = cx + dy, where a,b,c,d are constants and x' and y' are the derivatives of x and y respectively.
Additionally, the initial conditions x(0)=4 and y(0)=11 would serve to define the particular solution for the system of differential equations. Correcting and resolving such typos would be the first step in progressing towards a solution.
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Find the value of c such that the line y = 9/4 x + 9 is tangent to the curve y = c x .
Answers
The value of c such that the line y = 9/4 x + 9 is tangent to the curve y = c x is 9/4. This is because a line is tangent to a curve when it touches the curve at exactly one point, hence the slope of the given line (9/4) will be equal to the value of c, which is the slope of the curve.
Explanation:The problem asks us to find the value of c such that the line y = 9/4 x + 9 is tangent to the curve y = c x. A line is tangent to a curve when it touches the curve at exactly one point. In this context, the slope of the given line (9/4) will be equal to the value of c since in the line equation y = c x, c is the slope. So the value of c is 9/4.
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A particle moving along a hyperbola xy =8. as it reaches the point (4,2), the y-coordinate is decreasing at a rate of 3cm/s. how fast is the x-coordinate of the point changing at that instant.
Answers
The x-coordinate of the point is changing at a rate of [tex]\( 6 \text{ cm/s} \)[/tex].
To determine how fast the x-coordinate of the particle is changing at the instant it reaches the point (4, 2), we start with the equation of the hyperbola and apply the related rates method.
Given:
[tex]\[ xy = 8 \][/tex]
Differentiate both sides of the equation with respect to time [tex]\( t \)[/tex]:
[tex]\[ \frac{d}{dt}(xy) = \frac{d}{dt}(8) \][/tex]
Using the product rule on the left side:
[tex]\[ x \frac{dy}{dt} + y \frac{dx}{dt} = 0 \][/tex]
We need to find [tex]\(\frac{dx}{dt}\)[/tex] when the particle is at the point [tex]\((4, 2)\)[/tex] and [tex]\( \frac{dy}{dt} = -3 \text{ cm/s} \)[/tex] (since the y-coordinate is decreasing).
Substitute [tex]\( x = 4 \)[/tex], [tex]\( y = 2 \)[/tex], and [tex]\( \frac{dy}{dt} = -3 \)[/tex] into the differentiated equation:
[tex]\[ 4 \left( -3 \right) + 2 \frac{dx}{dt} = 0 \][/tex]
Simplify and solve for [tex]\( \frac{dx}{dt} \)[/tex]:
[tex]\[ -12 + 2 \frac{dx}{dt} = 0 \][/tex]
[tex]\[ 2 \frac{dx}{dt} = 12 \][/tex]
[tex]\[ \frac{dx}{dt} = 6 \][/tex]