Find the area of the region bounded by the parabola y=2x^2 , the tangent line to the parabola at (5,50), and the x-axis

Answers

Answer 1
check the picture below.

so.. the graph looks like so, since the tangent line is at 5,50, it touches the parabola when y = 50, x = 5.

so, let's get the tangent line at that point,

[tex]\bf y=2x^2\implies \left. \cfrac{dy}{dx}=4x \right|_{x=5}\implies 20\leftarrow m \\\\\\ y-y_1=m(x-x_1)\implies y-50=20(x-5)\implies y=20x-50\\ \left. \qquad \right. \uparrow\\ \textit{point-slope form}[/tex]

since I don't see an Up/Down function, just a Right/Left one... so, we'll use the function in "y-terms" then, the second one in the graph for each.

now... to get the bounds.... we could make them equal to each other, however, we know that the x-axis( y = 0) is the boundary line at the bottom, and that the tangent line is touching the parabolat a y = 50, thus, our bounds are from 0 to 50.

[tex]\bf \begin{cases} \cfrac{y+50}{20}=x\impliedby \textit{right function}\\\\ \sqrt{\cfrac{y}{2}}=x\impliedby \textit{left function} \end{cases} \\\\\\ \displaystyle \int\limits_{0}^{50}~\left[ \left( \cfrac{y+50}{20} \right)~-~\left( \sqrt{\cfrac{y}{2}}\right) \right]dy \\\\\\ \displaystyle \cfrac{1}{20}\int\limits_{0}^{50}~y\cdot dy+\int\limits_{0}^{50}~\cfrac{5}{2}\cdot dy-\cfrac{1}{\sqrt{2}}\int\limits_{0}^{50}~y^{\frac{1}{2}}\cdot dy[/tex]

[tex]\bf \left. \cfrac{y^2}{40}+\cfrac{5y}{2}-\cfrac{2\sqrt{y^3}}{3\sqrt{2}} \right]_{0}^{50}\implies \left[ \cfrac{250}{4}+125-\cfrac{500\sqrt{2}}{3\sqrt{2}} \right]-[0] \\\\\\ \cfrac{375}{2}-\cfrac{500}{3}\implies \cfrac{125}{6}[/tex]
Find The Area Of The Region Bounded By The Parabola Y=2x^2 , The Tangent Line To The Parabola At (5,50),
Answer 2
Final answer:

The area of the region bounded by the parabola[tex]y=2x^2[/tex]angent line at (5,50), and the x-axis is found by integrating the function of the parabola, finding the x-intercept of the tangent line, and subtracting the area under the tangent line from the area under the parabola between x=0 and x=5.

Explanation:

The area of the region bounded by the parabola , the tangent line at the point (5,50), and the x-axis can be found using integral calculus. First, we find the equation of the tangent line to the parabola at (5,50). The derivative of y with respect to x is given by [tex]y=2x^2[/tex]= 4x. At x=5, the slope of the tangent line is 20. Thus, the equation of the tangent line is y - 50 = 20(x - 5). To find the bounded area, we integrate the area under the parabola from the x-intercept of the tangent line to x=5, and subtract the area under the tangent line in the same interval.

Let's call the x-intercept of the tangent line x1. To find x1, we set the y-value of the tangent line equation to 0 and solve for x. The integration itself uses the antiderivative of  which is [tex]2x^2,[/tex] and the antiderivative of the linear tangent line equation. Subtracting the integral of the tangent line from the integral of the parabola gives us the exact area under the parabola.


Related Questions

Fifteen people volunteer for a park cleanup. The number of volunteers increases by 7 people each month for several months. After how many months will there be 50 volunteers

Answers

It will take 5 months

You can buy DVDs at a local store for $15.49 each. You can buy them at an online store for $13.99 each plus $6 for shippimg. How many DVDs can you buy for the same amount at the two stores

Answers

15.49x = 13.99x + 6
15.49x - 13.99x = 6
1.50x = 6
x = 6 / 1.50
x = 4 <=== u can buy 4 DVD's for the same amount

15.49 * 4 = 61.96
13.99(4) + 6 = 61.96
Final answer:

To find out how many DVDs you can buy for the same price at both stores, you create a formula incorporating the cost per DVD and the shipping cost at the online store. Solving this equation will give you the number of DVDs where the cost will be equal at both stores. However, this might not always result in a whole number, indicating that there is no exact count of DVDs resulting in the same cost.

Explanation:

To determine how many DVDs you can buy for the same amount at both stores, you can set up a mathematical equation accounting for the cost of each DVD and the shipping cost in the online store. The equation would look like this:

15.49 x = 13.99 y + 6

where x and y represent the number of DVDs bought from the local and online store respectively. By solving this equation we're finding out for which number of DVDs the total costs at both stores are equal.

To find the exact number of DVDs where the total cost will be the same, the equation needs to be solved for a whole number. This may result in no exact integer solution, showing that there's no exact number of DVDs for which the cost will be the same at both stores.

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In 2012, the average donation to the help way was $225 with a standard deviation of $45. in 2013, the average donation was $400 with a standard deviation of $60. in which year did the donations show a more dispersed distribution?

Answers

Final answer:

The donations in 2012 had a more dispersed distribution than in 2013, as indicated by a higher coefficient of variation of 20% compared to 15% in 2013.

Explanation:

To determine which year had a more dispersed contribution distribution, we examine the coefficient of variation (CV), which is calculated by dividing the standard deviation by the mean and then multiplying by 100 to express it as a percentage. A higher CV indicates a more dispersed distribution.

For 2012:

Mean = $225Standard Deviation = $45CV = ($45 / $225) x 100 = 20%

For 2013:

Mean = $400Standard Deviation = $60CV = ($60 / $400) x 100 = 15%

Comparing the coefficients of variation, we can conclude that the donations in 2012 had a more dispersed distribution than in 2013 since the CV in 2012 was higher.

What is 25 divided by 28

Answers

0.8928571 that's the answer welcome

The solution after divide is,

⇒ 0.893

What is Division method?

Division method is used to distributing a group of things into equal parts. Division is just opposite of multiplications. For example, dividing 20 by 2 means splitting 20 into 2 equal groups of 10.

We have to given that;

To divide the numbers,

⇒ 25 divided by 28

Now, We can divide the numbers as;

⇒ 25 divided by 28

⇒ 25 ÷ 28

⇒ 25 / 28

⇒ 0.893

Thus, The solution after divide of the numbers is,

⇒ 0.893

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What is the equation of the circle shown in the graph

Answers

the answer is x² + (y + 3)² = 4 C. i just did the problem

Answer: its the third option

Explain how you can find 4 × 754 using two different methods

Answers

Method 1:

4(754)

=4(700+50+4)

=2800 + 200 + 16

=3000+16

=3016

-----------------

Method 2:

4(754)

=4(250+250+250+4)

=1000+1000+1000+16

=3016

Mitchell deposited $651.37 in a savings account that earns 3.9% simple interest what is Mitchell's account balance after 15 years?

Answers

itd be 1,032.20 after 15 years
The formula is
A=p (1+rt)
A future value?
P present value 651.37
R interest rate 0.039
T time 15 years
A=651.37×(1+0.039×15)
A=1,032.42

Hope it helps!

When does the proportion form of newton's second law take the form of an equation?

Answers

The proportion form of Newton’s second law is when finding the acceleration of a problem as this is the equation suitable of finding the acceleration and to be able to compute the missing answer to the problem. In the form of the equation, it is ‘a + f / m’.

What is the contrapositive of the conditional statement? If two variables are directly proportional, then their graph is a linear function.

Answers

To find the contrapositive, you negate the hypothesis and the conclusion of the statement, and then switch their positions in the statement such that the conclusion becomes the hypothesis and vice versa:

If two variables are directly proportional, then their graph is a linear function. 

If two variables are not directly protional, then their graph is not a linear function.

what is 5/6 divided by 1/12

Answers

5/6 / 1/12

5/6 * 12

10

Hope this helps!
You have to make 5/6 have a common denominator (bottom number) of 1/12, so 5/6 would become 10/12 (5/6 multiplied by 2 is 10/12), and so you would get 10/12 divided by 1/12 equals 10.

Estimate square root of 250 to one decimal place

Answers

this is probably 15.8113... or 15.8113883008

please tell me hown in thw world in the answer key this is 0.45

Answers

450=c(10^3)
450=1000c
450/1000=c
9/20=c
c=0.45


If the manager at first city bank surveys a sample of 100 customers to determine how many miles they live from the​ bank, is the mean travel distance for this sample considered a parameter or a​ statistic? explain.

Answers

Final answer:

The mean travel distance for the sample of 100 customers surveyed by the manager at First City Bank is considered a statistic.

Explanation:

The mean travel distance for the sample of 100 customers surveyed by the manager at First City Bank is considered a statistic.

In statistics, a parameter is a numerical characteristic of the whole population that can be estimated by a statistic. Since the manager is surveying only a sample of customers, the mean travel distance calculated from this sample is a statistic, not a parameter.

For example, if the manager had surveyed all customers of First City Bank, then the mean travel distance calculated from that data would be a parameter.

The s/n ration for a set of 50 measurements is 10. what number of measurements would make the s/n ratio 5

Answers

5 to 1 the process is quite simple

Tony says that 4,400 is halfway between 4,000 and 5,000.is he correct?why or why not?

Answers

Final answer:

Tony is not correct. The midway point between 4,000 and 5,000 is found by calculating the average of the two numbers, which is 4,500. Therefore, the halfway point is 4,500, not 4,400.

Explanation:

Tony states that 4,400 is halfway between 4,000 and 5,000. To determine if Tony is correct, we need to find the midpoint between the two numbers. The midpoint can be found by adding the two numbers together and dividing by 2:

(4,000 + 5,000) / 2 = 9,000 / 2 = 4,500.

Therefore, the exact halfway point between 4,000 and 5,000 is 4,500. This means that Tony's assertion that 4,400 is halfway between these two numbers is not correct; the correct halfway point is actually 4,500.

Final answer:

No, Tony is not correct. Halfway between 4,000 and 5,000 would be the average of the two numbers, which is 4,500.

Explanation:

No, Tony is not correct. Halfway between 4,000 and 5,000 would be the average of the two numbers. To find the average, we add the two numbers together and divide by 2. In this case, the average of 4,000 and 5,000 would be (4,000 + 5,000) / 2 = 9,000 / 2 = 4,500. Therefore, 4,400 is not halfway between 4,000 and 5,000.

Stacy made a deposit of $8,500.00 into an account. After ten years, the account had earned $5,525.00 in interest. What simple interest rate did the account earn?

Answers

The answer is 6.5 interest.

what all fractions are equivalent to 1.27 repeating

Answers

1.2727 * 100 = 127.272

127.272-1.2727 = 126

126/99 = 14/11

Hope this helps! :)


Find the slope at tangent line f(x)=(3/2)x+1 at (-2,-2)

Answers

[tex]\bf f(x)=\cfrac{3}{2}x+1\implies \cfrac{df}{dx}=\cfrac{3}{2}\qquad \stackrel{\textit{point-slope form}}{y-{{ y_1}}={{ m}}(x-{{ x_1}})} \\\\\\ \stackrel{(-2,-2)}{y-(-2)=\cfrac{3}{2}[x-(-2)]}\implies y+2=\cfrac{3}{2}(x+2)\implies y+2=\cfrac{3}{2}x+3 \\\\\\ y=\cfrac{3}{2}x+1[/tex]

you might notice the equation kinda went for a full circle runaround, ended at square one, well, that's because, the equation is linear and therefore a line and the tangent line at such point, will be the same as the line's, and it's equation will also be the same.

It takes 23 minutes for 6 people to paint 6 walls. How many minutes does it take 9 people to paint 9 walls?

Answers

Since the proportion is the same, it is also 23 minutes.

Answer:

23

Step-by-step explanation:

Imagine that each person is assigned one wall, and all 6 people begin painting at the same time.

Since everyone will finish painting their assigned wall after 23 minutes, it takes one person 23 minutes to paint one wall.

If we have 9 people and 9 walls, we can again assign one wall to each person.

Everyone will take 23 minutes to paint their assigned wall.

In other words, it takes 23 minutes for 9 people to paint 9 walls.

Solve (1-sin(x))(1+sin(x))

Answers

Since these are conjugates (a conjugate is when we have anything a - anything b multiplied by anything a + anything b for the most part - this excludes things such as infinity) Since sin(x) and 1 both fit under 'anything', we have a special way of multiplying these - it's a^2-b^2, or
1^2-sin^2x=1-sin^2x=cos^2x using trigonometric properties

xm-2m if x=-2 and m=-3

Answers

Xm-2m =6+6=12.Too easy !

xm-2m

x=-2, m = -3

xm = -2 *-3 = 6

2m = 2*-3 = -6

6- -6 = 6+6 = 12

How do you write 15 cubed in exponential notation

Answers

15^3
expanded form 15x15x15

estimate the answer to 109 divided by 96

Answers

109 divided by 96=1.13541666666

The answer is 1.14

If an equation is an identity, then how many solutions does it have?

Answers

Hello,

If an equation is an identity, then every value of the variable makes the egality true.

Generaly we resolve an equation in R.

In this case Sol=R

Infinite is the answer

Write 15 is no more than a number T divided by 5 as an inequality ( great than less than...)

Answers

The inequality representing '15 is no more than a number T divided by 5' is written as 15 <= T/5, which means T divided by 5 should be greater than or equal to 15.

The phrase '15 is no more than a number T divided by 5' can be translated into an inequality. When we interpret 'no more than', it means 'less than or equal to'. Therefore, the statement can be translated into the inequality 15 <= T/5. This expresses that when you divide the number T by 5, the result should be at least 15 or any value less than 15. The interpretation of "no more than" as "less than or equal to" clarifies the inequality's meaning, emphasizing that the resulting quotient from dividing T by 5 should not exceed 15. Thus, the inequality encapsulates the relationship between the number 15 and T when considering their division by 5.

What's 307, 407 rounded to the nearest thousand?

Answers

307,407 rounded to the nearest thousand is 307,000.... because it is closer to 307,000 then it is to 308,000
since the hundreds place is lower than 5 you would round down so it is: 307,000

Each side of a square is increasing at a rate of 3 cm/s. at what rate is the area of the square increasing when the area of the square is 16 cm2?

Answers

Answer:

The rate of the area increasing for square is 24cm²/ s

            Rate of increasing side ?

    Changing the length of side with respect to time is the rate of increasing side .

    Given each side of square is increasing at a rate 3cm/s.

          Area of square = sides²

           Side length = a

         

       (dA/dt) = (dA/da)×(da/dt) { chain rule of differentiation}

     

             Differentiation of area w.r.t. t is

                dA/da= 2a (da/dt)

                dA/dt = 2a *(da/dt)

                 da/dt = 3 cm/s

              When A=16 cm²

                      16 = a²

                         a= 4

                (dA/dt) = 2x4 x 3

                           = 24 cm²/ s

Hence, the rate of the area increasing for square is 24cm²/s

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What percent of the first 10 natural numbers are prime numbers?

Answers

Alright, so the first 10 are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Since a prime number is only divisible by 1 and itself and greater than 1, our answers are 2, 3, 5, and 7. That's 4 numbers, or 4/10=0.4 of the 10. Since percentage is based out of 1, we can multiply 0.4 by 100 to get 40%

[tex]\boxed{{\mathbf{40\% }}}[/tex] of the first 10 natural numbers are prime numbers.

Further explanation:  

The prime numbers are the numbers that is only divisible by 1 and itself and greater than 1.

Natural numbers are the numbers that is easily countable as their successors can be defined and its starts from 1.

The natural number can be expressed as,

[tex]1,2,3,4,5,6,7,8,9,10 \ldots[/tex]  

Step by step explanation:

Step 1:

First determine the first ten natural numbers.

[tex]1,2,3,4,5,6,7,8,9,10[/tex]  

Step 2:

Now determine the prime numbers among the first ten natural numbers.

The prime numbers among the first ten natural number are [tex]2,3,5,7[/tex].

Therefore, there are 4 prime numbers.

Step 3:

Now, find the percentage of the first 10 natural numbers are prime numbers.

In general the percentage of any number can be calculated as,

[tex]{\text{percent of  }}x = \dfrac{x}{{{\text{total number}}}} \times 100[/tex]  

The percentage of the prime numbers can be calculated as,

[tex]\begin{aligned}{\text{percent of  prime number}}& = \frac{4}{{{\text{10}}}} \times 100\\ &= 40\% \\\end{aligned}[/tex]  

Therefore, [tex]40\%[/tex] of the first 10 natural numbers are prime numbers.

Learn more:  

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Answer details:

Grade: Middle school

Subject: Mathematics

Chapter: Percentage

Keywords: Integers, expression, natural numbers, prime numbers, percentage, denominator, numerator, fractions, divisible, total number, the required number

find the GFC of 18a and 20ab

Answers

GCF for 18a and 20ab is 2a

Nia has number cards for 2,3,8,9 and 5. What are all the different ways Nia can arrange the cards to show only even numbers with the 2 in the ten- thousands place?

Answers

So basically with the limits (even numbers with 2 in the ten thousands place) you would be left with:

2 _ _ _ 8

(two is in the ten thousands place and the only even number is in the ones places to make it an even number)

So since there are three other numbers and three other spaces, do three factorial, or 3 * 2 * 1.

The reason is because if you look at the first space, there are three choices, two for the second space, and only one for the last one.

So you get 6 different ways.
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