The formula for R(N) for the function f(x) = (5x)2 on [1, 5] in terms of N is R(N) = [tex]100+400\times\frac{(N+1)}{N^2}+\frac{800}{3}\times\frac{(N+1)(2N+1)}{N^3}[/tex] and area is 767(approx).
In the given question, we have to find a formula for R(N) for the function f(x) = (5x)^2 on [1, 5] in terms of N.
The given function is f(x) = (5x)^2 on [1, 5].
∆x = (5-1)/N
∆x = 4/N
X = 1+i∆x
X = 1+i4/N
f(x) = 25X^2
f(x) = 25(1+i4/N)^2
f(x) = 25(1+i8/N+16/N^2 i^{2})
R(N) = [tex]\sum_{i=1}^{N}f(X)\Delta x[/tex]
Now putting the value of f(x) and ∆x
R(N) = [tex]\sum_{i=1}^{N}\frac{4}{N}\left[25\left(1+\frac{8}{N}i+\frac{16}{N^2}i^{2}\right)\right][/tex]
R(N) = [tex]\frac{100}{N}\sum_{i=1}^{N}\left(1+\frac{8}{N}i+\frac{16}{N^2}i^{2}\right)[/tex]
R(N) = [tex]\frac{100}{N}\left(\sum_{i=1}^{N}1+\sum_{i=1}^{N}\frac{8}{N}i+\sum_{i=1}^{N}\frac{16}{N^2}i^{2}\right)[/tex]
R(N) = [tex]\frac{100}{N}\left(N+\frac{8}{N^2}\sum_{i=1}^{N}i+\frac{16}{N^3}\sum_{i=1}^{N}i^{2}\right)[/tex]
R(N) = [tex]\frac{100}{N}\left(N+\frac{8}{N^2}\frac{N(N+1)}{2}+\frac{16}{N^3}\frac{N(N+1)(2N+1)}{6}\right)[/tex]
R(N) = [tex]\frac{100}{N}\times N+\frac{100}{N}\times\frac{8}{N^2}\times\frac{N(N+1)}{2}+\frac{100}{N}\times\frac{16}{N^3}\times\frac{N(N+1)(2N+1)}{6}[/tex]
R(N) = [tex]100+400\times\frac{(N+1)}{N^2}+\frac{800}{3}\times\frac{(N+1)(2N+1)}{N^3}[/tex]
Now finding the area
[tex]\text{lim}_{N\rightarrow \infty}[/tex]R(N) = [tex]\text{lim}_{N\rightarrow \infty}\left[100+400\times\frac{(N+1)}{N^2}+\frac{800}{3}\times\frac{(N+1)(2N+1)}{N^3}\right][/tex]
[tex]\text{lim}_{N\rightarrow \infty}[/tex]R(N) = 100 + 400 + 800/3
[tex]\text{lim}_{N\rightarrow \infty}[/tex]R(N) = 766.67
[tex]\text{lim}_{N\rightarrow \infty}[/tex]R(N) = 767(approx)
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Mae Ling earn a weekly alary of $360 plu a 7. 5% commiion on ale at a gift hop. How much would he make in a work week if he old $4,500 worth of merchandie?
He will make $697.5 in a work week if he sold $4500 worth of merchandise.
Mea Ling earn a weekly salary of $360 which means salary per week is $360.
Then plus a 7.5% commission on sales at a gift shop if she sold $4,500 worth of merchandise, which means additional.
Divide by 100. Most interest rates are quoted and advertised in terms of a percentage.
7.5%= 0.075
Multiplication In mathematics, multiplication (denoted by ‘ × ’) is a method of finding the product of two or more values.
In arithmetic, multiplication of two numbers represents the repeated addition of one number with respect to another.
0.075×4500= 337.5
Then total she make will be
337.5+360
= $697.5
Therefore, he will make $697.5 in a work week.
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A giraffe heart weighs about 4×102 ounces. For comparison, a human heart weighs about 1×101 ounces.
Use these numbers to complete the sentence below. Write your answer in standard form, without using exponents.
Answer:
The heart weighs 404 ounces and a human heart compared to that is 303
Step-by-step explanation:
4 x 102 = 404 - 101
please help thank you!
Answer:
...…...................c = 50
HELPPP PLS!!!!Which of the following is equivalent to sum from n equals 1 to infinity of 10 times quantity negative seven eighths end quantity to the power of n question mark
Answer:
-14/3
Step-by-step explanation:
this is a geometric series with r = -7/8
The infinite sum of geometric series with |r| < 1 is
a1/(1-r)We already have r, but we need to find a1 which is the first term of the series.
To find it, only plug in 1 for n, 10(-7/8) and this = -35/4
Now plug in r and a1 in the formula above:
[tex]\frac{-35/4}{1+7/8}[/tex]
after simplifying it, you will get the infinite sum = -14/3
A baker sells boxes of cookies this table shows the total cast, in dollars, for different numbers of boxes of cookies based on the table what is the total cost in dollars of 7 boxes of cookies
The total cost in dollars of 7 boxes of cookies is; $24.50
How to interpret Function Tables?
From the attached table, we see the following coordinates that represents number of boxes (n) and total cost (c in dollars);
(3, 10.5)
(4, 14)
(5, 17.5)
(9, 31.50)
Now, let us find the slope with two coordinates using the formula;
slope = (y₂ - y₁)/(x₂ - x₁)
Let us use the first two coordinates to get;
Slope = (14 - 10.5)/(4 - 3)
Slope = $3.5
Thus, this is the cost of each box
Thus, for 7 boxes, the cost = 3.5 * 7 = $24.5
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Which of the following propositions is tautology?
a. (p v q)→q b. p v (q→p) c. p v (p→q) d. Both (b) & (c)
From the truth table, the values for p v (p→q) are all true. Therefore, this proposition is considered a tautology.
A statement or formula that is true under every scenario is referred to as a tautology. For example, the bag is red or it is not. Whatever the color of the bag, these two statements are true.
Construct a truth table for the given propositions. From the table, we can tell that the proposition p v (p→q) is in tautology. Because whatever the value of p and q, the result is true for this proposition.
Therefore, the correct option is option c.
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The surface area in square units of a cylinder with a volume of 18 cubic units is a function of its radius r in units where s(r)=2pi r^2 + 36/r . What is the surface area of a cylinder with a volume of 18 cubic units and a radius of 3 units?
The surface area of a cylinder with a volume of 18 cubic units and a radius of 3 units is 18π +12.
What is the surface area of the cylinder?
The quantity of space enclosing a three-dimensional shape's exterior is its surface area. The area of the cylinder is determined by adding the areas of its two circular bases and curving surface.
Here, we have
Given
The surface area in square units of a cylinder with a volume of 18 cubic units is a function of its radius r in units
where s(r)=2πr² + 36/r ....(1)
Now, we put the value of r = 3 in equation (1) and we get
s(r)=2π(3)² + 36/3
s(r) = 18π + 12
Hence, the surface area of a cylinder with a volume of 18 cubic units and a radius of 3 units is 18π +12.
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What is the circumference of the following circle?
Use 3.14 for \piπpi and enter your answer as a decimal.
Answer:
43.96
Step-by-step explanation:
C = 2πr
C = 2 × 3.14 × 7
C = 43.96
Step-by-step explanation:
given in fig :
radius = 7
so,
circumference of the given figure= 2πr
=2×3.14×7
= 43.96
find the area of the region enclosed by one loop of the curve. r = sin(8θ)
π/32 is the area enclosed by the curve r= sin(8θ)
The given curve is polar curve and hence the area of the polar curve is given by:
Let A be the area of the curve so,
A = [tex]\int\limits^a_b {\frac{1}{2} r^2 } \, d\theta[/tex]
where a and b is the boundary at which r=0
so after equation r=0
sin(8θ) =0
=> sin(8θ) =0
=> 8θ = 0,π
=> θ = 0, π/8
so a=0 , b= π/8
now
A = [tex]\int\limits^a_b {\frac{1}{2} r^2 } \, d\theta[/tex] ------(i)
so [tex]r^2[/tex] = (sin(8θ))^2
=> [tex]sin^2[/tex] ( 8θ )
ans we know that
cos(2α) = 1 - 2[tex]sin^2[/tex] α
so [tex]r^2[/tex] = (1- cos(16θ) )/2
putting the value of r in the equation (i) we get :-
A = [tex]\int\limits^a_b {\frac{1}{4} *(1-cos(16\alpha ) } \, d\alpha[/tex]
=> 1/4* [tex]\int\limits^a_b {(1-cos(16\alpha ) } \, d\alpha[/tex]
here a=0 and b=π/8
after putting the value and solving the integral
A = π/32
so A is the area enclosed by r=sin(8θ) is π/32
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Piper invested $990 in an account paying an interest rate of 33% compounded
continuously. Sadie invested $990 in an account paying an interest rate of 3%
compounded monthly. After 14 years, how much more money would Piper have in
her account than Sadie, to the nearest dollar?
Answer:
55
Step-by-step explanation:
that’s if you meant the interest rate is 3 3/8 for compounded continuously and 3 1/8 for compounded monthly
the width of a rectangle is two feet less than the length. the perimeter is 52 feet. find the length and the width.
If the width of a rectangle is two feet less than the length and the perimeter is 52 feet, then the length and width of the rectangle is 14 feet and 12 feet respectively
The width of a rectangle is two feet less than the length
Consider the length of the rectangle as x
Then the width of the rectangle = x - 2
The perimeter of the rectangle = 52 feet
The perimeter of the rectangle = 2(l + w)
Where l is the length and w id the width
Substitute the values in the equation
2(x + x - 2) = 52
2(2x - 2) = 52
2x - 2 = 52/2
2x - 2 = 26
2x = 28
x = 14 feet
Width of the rectangle = 14 - 2
= 12 feet
Therefore, the length and width of the rectangle is 14 feet and 12 feet respectively
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two groups of tourists are visiting a skyscraper, and they are currently on the ground floor. the first group boards an elevator that is moving up at a speed of 66 meters per minute. when it has traveled 200 meters, the second group gets on an express elevator that goes up at a speed of 200 meters per minute. how long will it be before the second elevator catches up to the first?
Answer:
1 minute
because the first elevator has traveled 200 meters and the second elevator goes 200 meters in one minute so they will catch up with the first group in 1 minute.
Gracie is trying to read 48 books this year. So far, she’s read 16 book. How many books does she need to read per week if there are only 18 weeks left in the year?
Answer: The answers is, 1.77777, repeated 7s
Step-by-step explanation: The explanation if you need to show your work is: Start with the goal of (48 books). Subtract the number of books already read (16 books read).
(48 - 16) = 32 books left to read.
You must then take the other given of 18 weeks left to complete said goal of (48 books) and divide the number of books left (32 books to read) by the amount of time left (18) weeks to complete the desired goal.
(32 / 18) = 1.77777
Therefore, the answer to your question is that Gracie must read 1.777 books per week to complete her goal on time.
Good Luck, Hope this helps.
HELPPPPP For a given recipe, 2 cups of flour are mixed with 4 cups of sugar. How many cups of sugar should be used if 6 cups of flour are used?
Answer: 12 cups of sugar
Step-by-step explanation:
you can set it up as a ratio
2:4 and this can be simplified to 1:2, so for every cup of flour there is 2 cups of sugar, you could count it by hand but with a ratio of 1:2 you can just multiply by 2 so 6x2=12
Answer:
Step-by-step explanation:
y - 5 = 3 ( x + 9) in slope intercept
Answer:y=3x+32
Step-by-step explanation:Hope this helps
Earth’s distance from the sun is 1.496 x 108 km. Saturn’s distance from the sun is 1.4246 x 109 km. How many times further from the sun is Saturn?
Saturn's distance is 9.5 farther from the Sun than the earth.
What is an expression?An expression is a way of writing a statement with more than two variables or numbers with operations such as addition, subtraction, multiplication, and division.
Example: 2 + 3x + 4y = 7 is an expression.
We have,
Earth's distance from the sun = 1.496 x [tex]10^8[/tex] km
Saturn's distance from the sun = 1.4246 x [tex]10^9[/tex] km
Divide Saturn's distance by Earth's distance.
= 1.4246 x [tex]10^9[/tex] / 1.496 x [tex]10^8[/tex]
= 9.5
We see that,
Saturn's distance from the sun x 0.95 = Earth's distance from the sun.
Thus,
Saturn's distance is 0.95 farther from the sun.
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everytime adam shoots a free throw there is a 70% chance he'll make it. if adman shoots 20 free thrwos, how many do you expect him to make?
The number of free throws that adam can expect to make if he attempts 20 free throws is 14.
Probability is a metric used to express the possibility or chance that a particular event will occur. Both proportions ranging from 0 to 1 and percentages ranging from 0% to 100% can be used to describe probabilities.
Given the probability that the person making a free throw is 70%. The question asks to estimate how many of his 20 free throw attempts will be successful.
Then, the number of free throws is calculated as follows,
[tex]\begin{aligned}\text{70\% of 20}&=\frac{70\times20}{100}\\&=14\end{aligned}[/tex]
Then, the number of successful free throws is 14.
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What is the equation of the line that passes through the point (-4, 5) and has a
slope of o?
Answer:
y = 5
Step-by-step explanation:
A line with a slope of 0 is a horizontal line parallel to the x- axis with equation
y = c ( c is the value of the y- coordinates the line passes through )
the line passes through (- 4, 5 ) with y- coordinate 5 , then
y = 5 ← equation of line
At center street middle school, 240 scholars voted for their middle school play. the ratio of scholars who voted for annie to scholars who voted for wicked was 5 : 2, and the ratio of scholars who voted for wicked to scholars who voted for the wizard of oz was 2 : 3. how many scholars voted for wicked?
The number of scholars who voted for the play named Wicked will be equal to 48.
This problem can be solved by using the concept of Ratio. It is given that the total number of students are 240. Now, the ratio of Annie to Wicked is equal to 5:2 and ratio of Wicked to Wizard Oz is equal to 2:3. If we assume that the number of people who voted for Annie is 'x', the number of people who voted for Wicked is 'y' and number of people who voted for Wizard Oz is 'z'. Now, we write the expressions as
x/y = 5:2; from this we get x = 5y/2 ....(1)
y/z = 2:3; from this we get z = 3y/2 ....(2)
x + y + z = 240 ....(3)
We use the above values of equation (1) and (2) in equation (3) as
5y/2 + y + 3y/2 = 240
4y + y = 240
5y = 240
y = 48
So, the number of students who voted for Wicked will be 48.
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last question finally!
Answer:
v=116
u=137
Step-by-step explanation:
hope it's helpful for you
The midpoint of AB is M(3,-3). If the coordinates of A are (2, -1), what are the
coordinates of B?
Answer:
The coordinates of B is (4, 7)
Step-by-step explanation:
To find the answer for B, you have to do the inverse of the midpoint formula, which is M equals (x1 multiplied by x2, over 2; y1 multiplied by y2, over 2).
If you substitute in A's coordinates for x1 and y1, you come up with (3, 3) = (2+x, over 2; and -1+y, over 2)
If the x coordinate answer for AB was 3, you can create an equation like this to solve for B's x coordinate: 2+x over 2 equals 3.
you multiply 2 by 3 to get 6---> 2+x=6-----> x=4
If the y coordinate for AB was also three, you can create another equation to solve for B's y coordinate: -1+y over 2 equals 3.
you multiply 2 by 3 to get 6 again---->so -1+y=6---->y=7
so the answer for B's coordinates is (4, 7).
Answer:
B(4, - 5)--------------------------
Given:
A = (2, - 1), Midpoint M = (3, - 3),Find the coordinates of the other end point B.
Use midpoint equation, considering point B as (x, y):
3 = (2 + x)/2 ⇒ 2 + x = 2*3 ⇒ 2 + x = 6 ⇒ x = 4,- 3 = (- 1 + y)/2 ⇒ -1 + y = 2*(-3) ⇒ -1 + y = - 6 ⇒ y = - 5.Therefore the point B has coordinates of (4, - 5).
How do you measure the size of an angle using a protractor or a compass? Explain the steps and show your work.
The size of the angles is measured in the form of degrees by using a protractor and compass.
The symbol for degrees is a little circle °.
The FULL CIRCLE is 360° (360 degrees). A half circle or a straight angle is 180°.A quarter circle or a right angle is 90°A protractor is a tool that measures the number of degrees in an angle.
To measure an angle with a protractor:
Place the midpoint of the protractor on the VERTEX of the angle.Line up one side of the angle with the zero line of the protractor.Read the protractor to see where the other side of the angle crosses the number scale.Most protractors have two number scales. It's important to use the same number scale for both sides of the angle.
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Am cla ha collected 52 can in a food drive. They plan to ort the can into a x bag, with an equal number of can in each bag. Write an expreion to how how many can there will be in each bag
Answer:
Math Expert Answer
Step-by-step explanation:
52 cans in total separated into x number of bags. 52/x with x being the number of bags.
Answer:
c = 52/x
Step-by-step explanation:
We have 52 total cans and x bags of with an equal amount of cans in each.
Thus, we would have c = 52/x cans per bag. This is probably the answer you are looking for.
However, this can evaluate to a fractional number of cans, so you may be looking to round that answer down to an integer. In that case, the answer would be c = ⌊52/x⌋, which is solving for the greatest integer less than the value of 52/x
1. When renting a vehicle from Schrader car rental, a customer pays a fee up front, and is
also charged for miles driven when they return. Henry drove 85 miles and his rental bill
was $77. Marcia drove 115 miles and her rental bill was $83.
The up front payment costs $34.5 while the payment per miles driven is $0.5
How to solve Simultaneous equation word problems?We are told that when renting a vehicle from Schrader car rental, a customer pays a fee up front, and is also charged for miles driven when they return.
Let the amount paid up front be x and let the amount paid per mile driven be y.
Thus, the equation for Henry will be;
x + 85y = 77 -------(1)
The equation for Marcia will be;
x + 115y = 83 ------(2)
Subtract eq 1 from eq 2 to get;
30y = 6
y = 6/30
y = $0.5
Thus;
x + 85(0.5) = 77
x = 77 - 42.5
x = $34.5
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Complete question is;
When renting a vehicle from Schrader car rental, a customer pays a fee up front, and is
also charged for miles driven when they return. Henry drove 85 miles and his rental bill
was $77. Marcia drove 115 miles and her rental bill was $83.
What is the up fronmt cost and the cost per miles driven
Solve the equation by using the quadratic formula. 3 x squared minus 1 = 5 x a. startfraction 5 startroot 13 endroot over 6 endfraction, startfraction 5 minus startroot 13 endroot over 6 endfraction b. startfraction negative 5 startroot 13 endroot over 6 endfraction, startfraction negative 5 minus startroot 13 endroot over 6 endfraction c. startfraction negative 5 startroot 37 endroot over 6 endfraction, startfraction negative 5 minus startroot 37 endroot over 6 endfraction d. startfraction 5 startroot 37 endroot over 6 endfraction, startfraction 5 minus startroot 37 endroot over 6 endfraction
The solution to the quadratic equation 3x²-1 = 5x by formula method is
What is quadratic equation?A quadratic equation is a second-order polynomial equation in a single variable of x . It is expressed as ax²+bx+c=0. with a ≠ 0 . Because it is a second-order polynomial equation, the fundamental theorem of algebra guarantees that it has at least one solution. The solution may be real or complex.
In the equation 3x²-1= 5x
we put the equation in standard form. i.e 3x²- 5x -1 = 0
therefore a = 3 , b = -5 and c = -1
formula method is a method of solving the root of a quadratic equation. It is given by x = (-b±√b²-4ac)/2a
=(-(-5±√-5²-4×3×-1)/2×3
=(5±√25+12)/6
=(5±√37)/12
x = 5+6.1/12 or
x= 5-6.1/12
therefore x = 11.1/12 or -1.1/12
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Answer: it's d.
StartFraction 5 + StartRoot 37 EndRoot Over 6 EndFraction, StartFraction 5 minus StartRoot 37 EndRoot Over 6 EndFraction just took the test
Graph the equations y = -1/2x + 3 and y = 2x - 2
1After you graph the two lines, be sure to take a screenshot of your work and include it in your answer below.
2) What is the solution to the system of equations?
3) Prove that the answer you provided in part 2 is correct, by substituting the solution into the original equations. Be sure to show all your work.
a menorah holds one candle for each night of hanukkah and and extra candle called the shamash that is used to light the other candles. how many candles does a hanukkah menorah hold?
Answer: There is one candle for each night of Hanukkah, and the ninth candle, which sits in the middle, is known as the shamash, which translates to attendant or servant candle. This candle is lit first and is used to light the others.
Step-by-step explanation:
consider the equation 1 (x−1)2 + y2 + 4 (x+1)2 +y2 =2 . the graph of this curve looks like this:
The positive intercept y is [tex]\sqrt{ }(3 / 2)[/tex], the negative y intercept is y= [tex]-\sqrt{ }\left(\frac{3}{2}\right)[/tex], the equation of the tangent line at positive intercept is y=-0.48989x+1.2247, the equation of the tangent line negative intercept is y=0.48989x-1.2247.
(a). Here we have to find the positive y intercept
So letting x=0 in the given equation [tex]\frac{1}{(\mathrm{x}-1)^2+\mathrm{y}^2}+\frac{4}{(\mathrm{x}+1)^2+\mathrm{y}^2}=2[/tex]
x=0 implies
[tex]$$\begin{aligned}\frac{1}{(0-1)^2+y^2}+\frac{4}{(0+1)^2+y^2} & =2 \\\frac{1}{1+y^2}+\frac{4}{1+y^2} & =2 \\\frac{5}{1+y^2} & =2 \\5 & =2+2 y^2 \\5-2 & =2 y^2 \\\frac{3}{2} & =y^2 \\\mathrm{y} & =\sqrt{ }\left(\frac{3}{2}\right)\end{aligned}$$[/tex]
Therefore, the positive intercept y is [tex]\sqrt{ }\left(\frac{3}{2}\right)[/tex].
(b). Next we have to find the negative y intercept , the negative y intercept is y= [tex]-\sqrt{ }\left(\frac{3}{2}\right)[/tex]
(c). Next we have to find the equation of the tangent line at [tex]$(0, \sqrt{ }(3 / 2))$[/tex]using implicit differentiation Here
[tex]\frac{1}{(x-1)^2+y^2}+\frac{4}{(x+1)^2+y^2}=2 \\[/tex]
[tex]\left((x-1)^2+y^2\right)^{-1}+4 \times\left((x+1)^2+y^2\right)^{-1}=2[/tex]
Then differentiating using power rule
[tex](-1) \times\left((\mathrm{x}-1)^2+\mathrm{y}^2\right)^{-1-1} \times\left(2(\mathrm{x}-1)+2 \mathrm{y} \times \mathrm{y}^{\prime}\right)+4 \times(-1) \times\left((\mathrm{x}+1)^2+\mathrm{y}^2\right)^{-1-1} \times\left(2(\mathrm{x}+1)+2 \mathrm{yy} \mathrm{y}^{\prime}\right)=0 \\[/tex]
[tex](-1) \times\left((\mathrm{x}-1)^2+\mathrm{y}^2\right)^{-2} \times\left(2(\mathrm{x}-1)+2 \mathrm{y} \times \mathrm{y}^{\prime}\right)+(-4) \times\left((\mathrm{x}+1)^2+\mathrm{y}^2\right)^{-2} \times\left(2(\mathrm{x}+1)+2 \mathrm{yy} \mathrm{y}^{\prime}\right)=0 \\[/tex]
[tex](-1) \times\left((\mathrm{x}-1)^2+\mathrm{y}^2\right)^{-2} \times\left((\mathrm{x}-1)+\mathrm{yy} \mathrm{y}^{\prime}\right)+(-4) \times\left((\mathrm{x}+1)^2+\mathrm{y}^2\right)^{-2} \times((\mathrm{x}+1)+\mathrm{yy})=0[/tex]
Then Substituting x=0 and y= [tex]\sqrt{ }(3 / 2)$[/tex], then
[tex](-1) \times\left(1+\left(\frac{3}{2}\right)\right)^{-2} \times\left(-1+\sqrt{ }\left(\frac{3}{2}\right) \mathrm{y}^{\prime}\right. & =4\left(1+\left(\frac{3}{2}\right)\right)^{-2} \times\left(1+\sqrt{ }\left(\frac{3}{2}\right) \mathrm{y}^{\prime}\right) \\[/tex][tex]\left.-1 \times\left(\frac{2}{5}\right)^2\right) \times\left(-1+\sqrt{ }\left(\frac{3}{2}\right) \mathrm{y}^{\prime}\right) & =4\left(\frac{2}{5}\right)^2 \times\left(1+\sqrt{ }\left(\frac{3}{2}\right) \mathrm{y}^{\prime}\right) \\[/tex]
[tex]-\left(-1+\sqrt{ }\left(\frac{3}{2}\right) \mathrm{y}^{\prime}\right) & =4 \times\left(1+\sqrt{ }\left(\frac{3}{2}\right) \mathrm{y}^{\prime}\right) \\[/tex]
[tex]1-\sqrt{ }\left(\frac{3}{2}\right) \mathrm{y}^{\prime} & =4+4 \sqrt{ }\left(\frac{3}{2}\right) \mathrm{y}^{\prime} \\[/tex][tex]-3 & =5 \sqrt{ }\left(\frac{3}{2}\right) \mathrm{y}^{\prime} \\[/tex]
[tex]\mathrm{y}^{\prime} & =-0.48989[/tex]
Then the equation of the tangent line is y-y1=m(x-x1)
Here X1=0
[tex]& \mathrm{y} 1=\sqrt{ }(3 / 2)=1.2247 \\[/tex]
m=-0.48989
y=-1.2247=-0.48989x
y=-0.48989x+1.2247
(d). Next we have to find the equation of the tangent line at[tex]$(0,-\sqrt{ }(3 / 2))$[/tex] using implicit differentiation
Here [tex]\left((x-1)^2+y^2\right)^{-1}+4 \times\left((x+1)^2+y^2\right)^{-1}=2[/tex]
Then differentiating using power rule
[tex](-1) \times\left((\mathrm{x}-1)^2+\mathrm{y}^2\right)^{-2} \times\left((\mathrm{x}-1)+\mathrm{yy^{ \prime } )}+(-4) \times\left((\mathrm{x}+1)^2+\mathrm{y}^2\right)^{-2} \times\left((\mathrm{x}+1)+\mathrm{y} \mathrm{y}^{\prime}\right)=0\right.[/tex]
Then Substituting x=0 and y=[tex]-\sqrt{ }(3 / 2)$[/tex], then
[tex](-1) \times\left(1+\left(\frac{3}{2}\right)\right)^{-2} \times\left(-1-\sqrt{ }\left(\frac{3}{2}\right) y^{\prime}\right) & =4\left(1+\left(\frac{3}{2}\right)\right)^{-2} \times\left(1-\sqrt{ }\left(\frac{3}{2}\right) y^{\prime}\right) \\[/tex]
[tex](-1) \times\left(-1-\sqrt{ }\left(\frac{3}{2}\right) y^{\prime}\right) & =4 \times\left(1-\sqrt{ }\left(\frac{3}{2}\right) y^{\prime}\right) \\[/tex]
[tex]\left(1+\sqrt{ }\left(\frac{3}{2}\right) y^{\prime}\right) & =4-4 \sqrt{ }\left(\frac{3}{2}\right) y^{\prime} \\[/tex]
[tex]-3 & =-5 \sqrt{ }\left(\frac{3}{2}\right) y^{\prime} \\y^{\prime} & =\frac{3}{5 \sqrt{ }\left(\frac{3}{2}\right)}=0.48989[/tex]
Then the equation of the tangent line is
y+1.2247 =0.48989x
y=0.48989x-1.2247.
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Enter the value of x below.
x + 9 = 12
Answer:
x = 3
Step-by-step explanation:
Enter the value of x below.
x + 9 = 12
x = 12 - 9
x = 3
-----------------------------------------------
check
3 + 9 = 12
12 = 12
the answer is good
Solve the system using substitution or elimination:
8x - 3y = -13
-3x + 5y = 1
Make sure you answer is an ordered pair (x, y).
Answer:
{x = -2, y = -1
Step-by-step explanation:
Solve the following system:
{-3 y + 8 x = -13
5 y - 3 x = 1
Choose an equation and a variable to solve for.
In the first equation, look to solve for x:
{-3 y + 8 x = -13
5 y - 3 x = 1
Isolate terms with x to the left-hand side.
Add 3 y to both sides:
{8 x = 3 y - 13
5 y - 3 x = 1
Hint: | Solve for x.
Divide both sides by 8:
{x = (3 y)/8 - 13/8
5 y - 3 x = 1
Perform a substitution.
Substitute x = (3 y)/8 - 13/8 into the second equation:
{x = (3 y)/8 - 13/8
5 y - 3 ((3 y)/8 - 13/8) = 1
Expand the left-hand side of the equation 5 y - 3 ((3 y)/8 - 13/8) = 1.
5 y - 3 ((3 y)/8 - 13/8) = 5 y + (-(9 y)/8 + 39/8) = (31 y)/8 + 39/8:
{x = (3 y)/8 - 13/8
((31 y)/8 + 39/8) = 1
Choose an equation and a variable to solve for.
In the second equation, look to solve for y:
{x = (3 y)/8 - 13/8
(31 y)/8 + 39/8 = 1
Isolate terms with y to the left-hand side.
Subtract 39/8 from both sides:
{x = (3 y)/8 - 13/8
(31 y)/8 = -31/8
Solve for y.
Multiply both sides by 8/31:
{x = (3 y)/8 - 13/8
y = -1
Hint: | Perform a back substitution.
Substitute y = -1 into the first equation:
Answer: {x = -2, y = -1