A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola y = 11 − x 2 . What are the dimensions of such a rectangle with the greatest possible area?

Answer:

[tex] f ( 2 x ) = 4 x ^ 2 - 8 [/tex]

Step-by-step explanation:

We are given the following expression and we are to simplify the given function:

[tex] f ( x ) = x ^ 2 - 8 [/tex]

Applying the function [tex]f(2x)[/tex] on [tex] f ( x ) = x ^ 2 - 8 [/tex] to get:

[tex] f ( 2 x ) = ( 2 x ) ^ 2 - 8 [/tex]

[tex] f ( 2 x ) = 4 x ^ 2 - 8 [/tex]

Answer:

13/4

Step-by-step explanation:

The slope of the line between 2 points is found by

m = (y2-y1)/(x2-x1)

   = (6--7)/(-3--7)

   = (6+7)/(-3+7)

   = 13/4

Answer:

The slope is 13/4.

Step-by-step explanation:

Slope formula:

[tex]\displaystyle \frac{y_2-y_1}{x_2-x_1}[/tex]

[tex]\displaystyle \frac{6-(-7)}{(-3)-(-7)}=\frac{13}{4}[/tex]

[tex]\huge\boxed{\frac{13}{4}}[/tex], which is our answer.

Answer:

The amount he should borrow is 19.25 hundreds dollars.

Step-by-step explanation:

Given : The cost (in hundreds of dollars) of tuition at the community college is given by [tex]T = 1.25c + 3[/tex], where c is the number of credits the student has registered for. If a student is planning to take out a loan to cover the cost of 13 credits.

To find : Use the model to determine how much money he should borrow?

Solution :

The model is given by [tex]T = 1.25c + 3[/tex]

Where,

c is the number of credits the student has registered.

T is the cost of tuition at the community college.

If a student is planning to take out a loan to cover the cost of 13 credits.

The amount he should borrow will get by putting the value of c in the model,

[tex]T = 1.25c + 3[/tex]

[tex]T = 1.25(13) + 3[/tex]

[tex]T = 16.25 + 3[/tex]

[tex]T =19.25[/tex]

Therefore, The amount he should borrow is 19.25 hundreds dollars.

Answer:

monthly payment is $377

C is the correct option.

Step-by-step explanation:

The formula for the monthly payment is given by

[tex]C=\frac{Prt(1+r)^n}{(1+r)^n-1}[/tex]

Given that,

P =  $20,000

n = 5 years = 60 months

r = 0.05

Substituting these values in the formula

[tex]C=\frac{20000\cdot \frac{0.05}{12}(1+\frac{0.05}{12})^{60}}{(1+\frac{0.05}{12})^{60}-1}[/tex]

On simplifying, we get

[tex]C=\$377.425\\\\C\approx \$377[/tex]

Therefore, the monthly payment is $377

C is the correct option.

Answer:

[tex]y(x)=c_1e^{-\frac{5}{2}x}+c_2xe^{-\frac{5}{2}x}[/tex]

Step-by-step explanation:

The given differential equation is 4y"+20y'+25y = 0

The characteristics equation is given by

[tex]4r^2+20r+25=0[/tex]

Now, solve the equation for r

Factor by middle term splitting

[tex]4r^2+10r+10r+25=0\\\\2r(2r+5)+5(2r+5)=0[/tex]

Factored out the common term

[tex](2r+5)(2r+5)=0[/tex]

Use Zero product property

[tex](2r+5)=0,(2r+5)=0[/tex]

Solve for r

[tex]r_{1,2}=-\frac{5}{2}[/tex]

We got the repeated roots.

Hence, the general equation for the differential equation is

[tex]y(x)=c_1e^{-\frac{5}{2}x}+c_2xe^{-\frac{5}{2}x}[/tex]

Final answer:

The general solution to the differential equation 4y"+20y'+25y = 0 is y(x) = (A + Bx)e^(-5/2x), where A and B are constants determined by initial conditions.

Explanation:

The general solution to the differential equation 4y"+20y'+25y = 0 can be found by looking for solutions in the form of y = ekx, where k is a constant. Substituting y into the differential equation, we get a characteristic equation of (ak² +bk+c)y= 0, which simplifies to (4k² + 20k + 25)y = 0. This is a quadratic equation in k that can be factored as (2k + 5)². Therefore, the two values of k that satisfy this equation are both -5/2, giving us a repeated root.

The general solution for a second-order linear homogeneous differential equation with repeated roots is y = (A + Bx)ekx, where A and B are constants determined by the initial conditions. In this case, k = -5/2, hence the general solution is y(x) = (A + Bx)e-5/2x.

Answer:  The required number of different possible committees is 81172.

Step-by-step explanation:    Given that a department contains 9 men and 15 women.

We are to find the number of different committees of 6 members that are possible if the committee must have strictly more women than men.

Since we need committees of 6 members, so the possible combinations are

(4 women, 2 men), (5 women, 1 men) and (6 women).

Therefore, the number of different committees of 6 members is given by

[tex]n\\\\\\=^{15}C_4\times ^9C_2+^{15}C_5\times ^9C_1+^{15}C_6\\\\\\=\dfrac{15!}{4!(15-4)!}\times \dfrac{9!}{2!(9-2)!}+\dfrac{15!}{5!(15-5)!}\times \dfrac{9!}{1!(9-1)!}+\dfrac{15!}{6!(15-6)!}\\\\\\\\=\dfrac{15\times14\times13\times12\times11!}{4\times3\times2\times1\times11!}\times\dfrac{9\times8\times7!}{2\times1\times7!}+\dfrac{15\times14\times13\times12\times11\times10!}{5\times4\times3\times2\times1\times10!}\times\dfrac{9\times8!}{1\times8!}+\dfrac{15\times14\times13\times12\times11\times10\times9!}{6\times5\times4\times3\times2\times1\times9!}\\\\\\=1365\times36+3003\times9+5005\\\\=49140+27027+5005\\\\=81172.[/tex]

Thus, the required number of different possible committees of 6 members is 81172.

Answer:

C

Step-by-step explanation:

The melons are 3 for 8 dollars, this means each melon is worth 8/3 dollars, this is the first proportion in the expression.  In this proportion the numerator is the value of money so 25 will be the numerator in the other proportion.

8/3=25/x, C

a. Recall that

[tex]\displaystyle\int\frac{\mathrm dx}{1-x}=-\ln|1-x|+C[/tex]

For [tex]|x|<1[/tex], we have

[tex]\displaystyle\frac1{1-x}=\sum_{n=0}^\infty x^n[/tex]

By integrating both sides, we get

[tex]\displaystyle-\ln(1-x)=C+\sum_{n=0}^\infty\frac{x^{n+1}}{n+1}[/tex]

If [tex]x=0[/tex], then

[tex]\displaystyle-\ln1=C+\sum_{n=0}^\infty\frac{0^{n+1}}{n+1}\implies 0=C+0\implies C=0[/tex]

so that

[tex]\displaystyle\ln(1-x)=-\sum_{n=0}^\infty\frac{x^{n+1}}{n+1}[/tex]

We can shift the index to simplify the sum slightly.

[tex]\displaystyle\ln(1-x)=-\sum_{n=1}^\infty\frac{x^n}n[/tex]

b. The power series for [tex]x\ln(1-x)[/tex] can be obtained simply by multiplying both sides of the series above by [tex]x[/tex].

[tex]\displaystyle x\ln(1-x)=-\sum_{n=1}^\infty\frac{x^{n+1}}n[/tex]

c. We have

[tex]\ln2=-\dfrac\ln12=-\ln\left(1-\dfrac12\right)[/tex]

[tex]\displaystyle\implies\ln2=\sum_{n=1}^\infty\frac1{n2^n}[/tex]

The power series of f(x) = ln(1 - x) is  [tex]\rm -\sum^{\infty}_{n=1}\dfrac{x^{n} }{n}[/tex], the power series of xln(1 - x) is  [tex]\rm -\sum^{\infty}_{n=1}\dfrac{x^{n+1} }{n}[/tex] and the value of ln(2) is [tex]\rm \sum^{\infty}_{n=0}\dfrac{1}{n2^n}[/tex].

Given :

f(x) = ln (1−x)

a) The integration of 1/(1 - x) is given by:

[tex]\rm \int \dfrac{1}{1-x}dx=-ln|1-x| + C[/tex]

When |x| >1 :

[tex]\dfrac{1}{1-x} = \sum^{\infty}_{n=0} x^n[/tex]

Now, integrate on both sides in the above equation.

[tex]\rm -ln(1-x) = C+\sum^{\infty}_{n=0}\dfrac{x^{n+1} }{n+1}[/tex]   --- (1)

Now, at (x = 0) the above expression becomes:

[tex]\rm -ln(1-0) = C+\sum^{\infty}_{n=0}\dfrac{0^{n+1} }{n+1}[/tex]

By simplifying the above expression in order to get the value of C.

C = 0

Now, substitute the value of C in expression (1).

[tex]\rm ln(1-x) =-\sum^{\infty}_{n=0}\dfrac{x^{n+1} }{n+1}[/tex]

Now, by shifting the index the above expression becomes:

[tex]\rm ln(1-x) =-\sum^{\infty}_{n=1}\dfrac{x^{n} }{n}[/tex]

b) Now, multiply by 'x' in the above expression in order to get the power series of (x ln(1 - x)).

[tex]\rm xln(1-x) =-\sum^{\infty}_{n=1}\dfrac{x^{n+1} }{n}[/tex]

c) Now, substitute the value x = 1/2 in the expression (1).

[tex]\rm ln2 = \sum^{\infty}_{n=0}\dfrac{1}{n2^n}[/tex]

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19 and 42 are coprime, so we can use the CRT right away. Start with

[tex]x=19+42[/tex]

Taken mod 42, we're left with a remainder of 19. Notice that

[tex]19\cdot3\equiv57\equiv15\pmod{42}[/tex]

so we need to multiply the first term by 3 to get the remainder we want.

[tex]x=19\cdot3+42[/tex]

Next, taken mod 19, we're left with a remainder of 4. Notice that

[tex]42\cdot6\equiv252\equiv5\pmod{19}[/tex]

so we need to multiply the second term by 6.

Then by the CRT, we have

[tex]x\equiv19\cdot3+42\cdot6\equiv309\pmod{42\cdot19}\implies x\equiv309\pmod{798}[/tex]

so that any solution of the form [tex]x=798n+309[/tex] is a solution to this system.

We are to find the surface integral of [tex]\mathbf{\int \int _S \ F.ds}[/tex]

where;

∴

The plane x + y + z = 1

z = 1 - x - y

[tex]\dfrac{\partial z}{\partial x} = -1[/tex]

[tex]\dfrac{\partial z}{\partial y} = -1[/tex]

Since the surface is oriented downward

∴

[tex]dS = \Big( \dfrac{\partial z}{\partial x}i + \dfrac{\partial z}{\partial y}j - k) dxdy[/tex]

[tex]dS = (-i-j-k) dxdy[/tex]

However,

Flux = [tex]\mathbf{\int \int _S \ F.ds}[/tex]

[tex]= \int \int_R F. \Big( \dfrac{\partial z}{\partial x}i + \dfrac{\partial z}{\partial y}j - k) dxdy \\ \\ \\ = \int^1_{ x=0} \int ^{1-x}_{y=0} ( y^3i + 8j -xk)*(-i-j-k) dx dy \\ \\ \\ =\int^1_{ x=0} \int ^{1-x}_{y=0} (-y^3 -8+x) dxdy \\ \\ \\ = \int^1_{ x=0} \Big( \int ^{1-x}_{y=0} \Big(x-y^3 -8\Big) dy \Big) dx \\ \\ \\ = \int^1_{ x=0} \Bigg [xy - \dfrac{y^4}{4}-8y \Bigg]^{1-x}_{y=0} \ dx \\ \\ \\[/tex]

[tex]= \int^1_{ x=0} \Bigg [x(1-x) - \dfrac{1}{4}(1-x)^{4} - 8(1-x) \Bigg ] dx \\ \\ \\ = \int^1_{ x=0} \Bigg [(x-x^2) - \dfrac{1}{4}(x-1)^4+8(x-1)\Bigg] dx[/tex]

[tex]= \Bigg [ \dfrac{x^2}{2}-\dfrac{x^3}{3} - \dfrac{1}{4} \dfrac{(x-1)^5}{5}+(x-1)^2\Bigg] ^1_0[/tex]

[tex]= \Bigg [ \dfrac{1}{2}-\dfrac{1}{3} -0+0-0+0+ \dfrac{1}{4}\times \dfrac{(-1)^5}{5}-(0-1)^2\Bigg][/tex]

[tex]= \Bigg [ \dfrac{1}{2}-\dfrac{1}{3} - \dfrac{1}{20}-1\Bigg][/tex]

[tex]= \Bigg [ \dfrac{30-20-3-60}{60}\Bigg][/tex]

[tex]= \Bigg [ \dfrac{-53}{60}\Bigg][/tex]

Therefore, we can conclude that the surface integral is [tex]\mathbf{-\dfrac{53}{60}}[/tex]

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To compute a surface integral, take the antiderivatives of both dimensions defining the area with the surface edges as the bounds of the integral. The net flux through the surface can be found using the open surface integral formula.

A surface integral over the given oriented surface can be computed by taking the antiderivatives of both dimensions defining the area, with the edges of the surface as the bounds of the integral. The net flux through the surface can be found using the open surface integral formula.

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Answer:

The required amount of mice is 11.82 gram.

Step-by-step explanation:

Given : Assuming that it is known from previous studies that σ = 4.5 grams. If we wish to be 95% confident that the mean weight of the sample will be within 3 grams of the population mean for all mice subjected to this protein diet.

To find : How many mice should be included in our sample?

Solution :

The formula used in the situation is

[tex]z=\frac{x-\mu}{\sigma}[/tex]

Where, z value at 95% confidence interval is z=1.96

[tex]\mu=3[/tex] gram is the mean of the population

[tex]\sigma=4.5[/tex] gram is the standard deviation of the sample

Substituting the value, to find x sample mean

[tex]1.96=\frac{x-3}{4.5}[/tex]

[tex]1.96\times 4.5=x-3[/tex]

[tex]8.82=x-3[/tex]

[tex]x=8.82+3[/tex]

[tex]x=11.82[/tex]

Therefore, The required amount of mice is 11.82 gram.

Answer: Mean = 760

Standard deviation = 6.16

Step-by-step explanation:

Given : The number of trials: [tex]n=800[/tex]

The probability that  a certain protection system will be recovered :[tex]p=0.95[/tex]

We know that the mean and standard deviation of binomial distribution is given by :_

[tex]\text{Mean}=np[/tex]

[tex]\text{Standard deviation}=\sqrt{np(1-p)}[/tex], where n is the number of trials and p is the probability of success.

Now, the mean and standard deviation of the number of cars recovered after being stolen is given by :-

[tex]\text{Mean}=800\times0.95=760[/tex]

[tex]\text{Standard deviation}=\sqrt{800\times0.95(1-0.95)}\\\\=6.164414002\approx6.16[/tex]

Hence, the mean is 760 and standard deviation is 6.16 .

The mean and standard deviation of the number of cars recovered after being stolen can be found using the properties of the binomial distribution.

To find the mean and standard deviation of the number of cars recovered after being stolen, we can use the properties of the binomial distribution. In this case, the probability of recovering a car is 0.95, and the number of stolen cars is 800.

The mean can be calculated by multiplying the number of trials (800) by the probability of success (0.95), giving us a mean of 760 cars.

The standard deviation can be calculated using the formula:

standard deviation = sqrt(n * p * (1 - p))

Substituting in the values, we get:

standard deviation = sqrt(800 * 0.95 * (1 - 0.95))

standard deviation ≈ 8.72 cars.

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To find the cost of resurfacing the path, we first calculate the area of the path which is 300 square feet. We then multiply this by the unit cost of resurfacing which comes out to be $600.

This is a problem in area calculation and application of unit cost. Firstly, we need to calculate the area for the path surrounding the pool. The total area of the pool and the path is (14+2*3) feet by (30+2*3) feet = 20 feet by 36 feet, which equals 720 square feet. The area of the pool itself is 14 feet by 30 feet = 420 square feet. So, the area of just the path is 720-420 = 300 square feet. With a cost of $2 per square foot to resurface the path, the total cost would be 300*$2 = $600.

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The fourth point that will complete the quadrilateral is a rhombus is -3.

In a rhombus, two parallel sides must be equal.

            slope(m)=tan∅=y axis/x axis.

Calculation:-

1st coordinate- side=[tex]\sqrt{10}[/tex]

2nd coordinate- side= [tex]\sqrt{18}[/tex]

3rd coordinate  must be [tex]\sqrt{10}[/tex]

so,y=-3

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Answer:

Consider the three points graphed. What is the value of y in the fourth point that will complete the quadrilateral as a rhombus?

(–1, -5 )

Step-by-step explanation:

Answer:

1680 ways

Step-by-step explanation:

Total number of integers = 10

Number of integers to be selected = 6

Second smallest integer must be 3. This means the smallest integer can be either 1 or 2. So, there are 2 ways to select the smallest integer and only 1 way to select the second smallest integer.

2 ways   1 way                                      

Each of the line represent the digit in the integer.

After selecting the two digits, we have 4 places which can be filled by 7 integers. Number of ways to select 4 digits from 7 will be 7P4 = 840

Therefore, the total number of ways to form 6 distinct integers according to the given criteria will be = 1 x 2 x 840 = 1680 ways

Therefore, there are 1680 ways to pick six distinct integers.

Answer:

70 total selections

Step-by-step explanation:

The set: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

You know that that 3 is definitly a part of the set, so you can ignore it. If 3 is the second smallest, the smallest number in the set is either 1 or 2, not both.

The number of ways to choose between 1 and 2 is [tex]2^{C}1[/tex] ways which is equal to 2, so all that's left is choosing from the group of the set between 4 and 10.

Since you've already chosen 2 numbers (3 and 1 or 2) you need to find out how many ways can you choose 4 out of the numbers between 4 and 10. Since there are 7 numbers from 4 to 10, you need to figure out [tex]7^{C}4[/tex] which is equal to 35.

Since you are looking to find the cross between the 2, multiply 2 by 35 = 70, the answer.

[tex]_{12}C_5=\dfrac{12!}{5!7!}=\dfrac{8\cdot9\cdot10\cdot11\cdot12}{120}=792[/tex]

792>20, so yeah, he will be able.

Answer:

792>20, so yeah, he will be able.

Step-by-step explanation:

Answer:

$16.583

Step-by-step explanation:

Given :Your friend, Isabel, has a credit card with an APR of 19.9%!

To Find : How many dollars would she pay as a finance charge for just 1 month on a $1000 charge?

Solution:

We are given that finance charge for just 1 month on a $1000 charge.

So, Finance charge = [tex]\frac{19.9\% \times 1000}{12}[/tex]

Finance charge = [tex]\frac{\frac{199}{1000}\times 1000}{12}[/tex]

Finance charge = [tex]\frac{199\times 1000}{12}[/tex]

Finance charge = [tex]16.583[/tex]

Hence she pay $16.583 as a finance charge for just 1 month on a $1000 charge.

The solution of the given differential equation involves rearranging it into the standard form of a first-order linear differential equation, determining the integrating factor, and subsequently solving for the dependent variable y(x) via integration.

To solve the given differential equation, we can rewrite it in the form of dy/dx = f(x, y). That gives us (2(y-4x^2))/x = dy/dx. The resulting equation is a first-order linear differential equation, which can be solved using an integrating factor.

Here, the standard form of the differential equation is dy/dx + P(x)y = Q(x). Comparing this with our equation, we find P(x) = -2/x and Q(x) = -8x. We know that μ(x) = exp(∫P(x) dx) is the integrating factor. On solving we get μ(x) = 1/x2. We then multiply through our differential equation by μ(x) and integrate both sides to solve for y(x).

These steps on how to solve the differential equation involve certain knowledge in differential equation theory, namely about first-order linear differential equations, integrating factors, and the process of integration.

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Answer:

The given expression can be written as -1=19(-1)+18.

Step-by-step explanation:

According to quotient remainder theorem:

For any integer a and a positive integer d, there exist unique integers q and r such that

[tex]a=d\times q+r[/tex]

It can also written as

[tex]a\text{mod }d=r[/tex]

The given expression is

[tex]-1\text{mod }19[/tex]

Here a=-1 and d=19.

[tex]\frac{-1}{19}=19(-1)+18[/tex]

[tex]-1\text{mod }19=18[/tex]

If -1 is divides by 19, then the remainder is 18. The value of r is 18.

Therefore the given expression can be written as -1=19(-1)+18.

-1 mod 19 is equal to 18.

Finding -1 mod 19

To determine -1 mod 19, we use the formula a = dq + r, where a = -1, d = 19, and q and r are the quotient and remainder, respectively.

First, note that any number mod 19 will yield a remainder between 0 and 18.

Since -1 is a negative number, we rewrite it in terms of 19: -1 = -1 + 19k for some integer k.

To find a positive equivalent, we choose k such that -1 + 19k is positive and falls within the range of 0 to 18. When k = 1, we get -1 + 19(1) = 18.

Therefore, -1 mod 19 is equal to 18.

This means that if you divide -1 by 19, the remainder that falls within the range of 0 to 18 is 18.

The nominal value - without discounting the inflation rate - of income was $ 3000.

If the interest rate was 6%, a rule of three is enough to find the value of the original investment.

3000 - 6%

x - 100%

x = 50,000

The value of the investment was $ 50,000

In this case, the inflation rate also requires a simple calculation.

Inflation corroded $ 1000 dollars of income of $ 3000

Therefore the inflation rate will be 1000/3000 = 33.3%

Answer:

[tex]g(4) = 8 \pi h\\\\g(\frac{3}{2}) = 3 \pi h\\\\ g(2c) = 4 \pi ch\\\\g(c+3) = 2 \pi hc+6\pi h[/tex]

Step-by-step explanation:

You need to substitute [tex]r=4[/tex] into [tex]g(r) = 2 \pi r h[/tex]. Then:

[tex]g(4) = 2 \pi(4)h\\\\g(4) = 8 \pi h[/tex]

Substitute [tex]r=\frac{3}{2}[/tex] into [tex]g(r) = 2 \pi r h[/tex]. Then:

[tex]g(\frac{3}{2}) = 2 \pi(\frac{3}{2})h\\\\g(\frac{3}{2}) = 3 \pi h[/tex]

Substitute [tex]r=2c[/tex] into [tex]g(r) = 2 \pi r h[/tex]. Then:

[tex]g(2c) = 2 \pi(2c))h\\\\g(2c) = 4 \pi ch[/tex]

Substitute [tex]r=c+3[/tex] into [tex]g(r) = 2 \pi r h[/tex]. Then:

[tex]g(c+3) = 2 \pi (c+3)h\\\\g(c+3) = 2 \pi hc+6\pi h[/tex]

For this case we have the following function:

[tex]g (r) = 2 \pi * r * h[/tex]

We must evaluate the function for different values:

[tex]g (4) = 2 \pi * (4) * h = 8 \pi*h\\g (\frac {3} {2}) = 2 \pi * (\frac {3} {2}) * h = 3 \pi*h\\g (2c) = 2 \pi * (2c) * h = 4 \pi * c * h\\g (c + 3) = 2 \pi * (c + 3) * h = 2 \pi * c * h + 6 \pi * h[/tex]

Answer:

[tex]g (4) = 8 \pi*h\\g (\frac {3} {2}) =3 \pi*h\\g (2c) = 4 \pi * c * h\\g (c + 3) = 2 \pi * c * h + 6 \pi * h[/tex]

Answer:

[tex]4\sqrt{17}[/tex]

Step-by-step explanation:

Let's find the answer by using the arc length formula which is:

[tex]\int\limits^a_b {\sqrt{1+(\frac{dy}{dx})^{2} } } \, dx[/tex]

First, let's find dy/dx which is:

y=4x-5

y'=4*(1)-0

y'=4, now let's use the formula:

[tex]\int\limits^3_{-1} {\sqrt{1+4^{2}} } \, dx=\sqrt{17} *(3-(-1))=4\sqrt{17}[/tex]

Now, using the distance formula we have:

[tex]d=\sqrt{(x2-x1)^{2} +(y2-y1)^{2} }[/tex]

[tex]y(-1)=4*(-1)-5=-9 \\y(3)=4*(3)-5=7[/tex]

So we have two points (-1, -9) and (3, 7) so:

[tex]d=\sqrt{(3-(-1))^{2} +(7-(-9))^{2} }=4\sqrt{17}[/tex]

Notice both equations gave the same length [tex]4\sqrt{17}[/tex].

Final answer:

To find the length of the curve y = 4x - 5, -1 ≤ x ≤ 3 using the arc length formula, integrate sqrt(1 + (dy/dx)^2) from x = -1 to x = 3. The length of the curve is 4√17. The result can be confirmed by calculating the length using the distance formula.

Explanation:

To find the length of the curve y = 4x - 5, -1 ≤ x ≤ 3 using the arc length formula, we need to integrate the square root of 1 + (dy/dx)^2 from x = -1 to x = 3. The derivative of y = 4x - 5 is dy/dx = 4. Substituting this into the arc length formula, we have:

L = ∫sqrt(1 + (dy/dx)^2) dx = ∫sqrt(1 + 4^2) dx = ∫sqrt(17) dx = x√17 + C

Now, plugging in the limits of integration, we have:

L = [(3√17) + C] - [(-1√17) + C] = (3√17) - (-1√17) = 4√17

To check our answer, we can use the distance formula. The endpoints of the line segment are (-1, -9) and (3, 7). Using the distance formula:

D = sqrt((x2 - x1)^2 + (y2 - y1)^2) = sqrt((3 - (-1))^2 + (7 - (-9))^2) = sqrt(4^2 + 16^2) = sqrt(272) = 16√17

As we can see, the length of the curve obtained using the arc length formula (4√17) matches the length calculated using the distance formula (16√17), confirming our answer.

Answer:

The length of the ladder is 10 m.

Step-by-step explanation:

Let x shows the distance of the top of ladder from the bottom of base of the wall, y shows the distance of the bottom of ladder from the base of the wall and l is the length of the ladder,

Given,

[tex]\frac{dx}{dt}=-0.675\text{ m/s}[/tex]

[tex]\frac{dy}{dt}=0.9\text{ m/s}[/tex]

y = 6 m,

Since, the wall is assumed perpendicular to the ground,

By the pythagoras theorem,

[tex]l^2=x^2+y^2[/tex]

Differentiating with respect to t ( time ),

[tex]0=2x\frac{dx}{dt}+2y\frac{dy}{dt}[/tex]     ( the length of wall would be constant )

By substituting the value,

[tex]0=2x(-0.675)+2(6)(0.9)[/tex]

[tex]0=-1.35x+10.8[/tex]

[tex]\implies x=\frac{10.8}{1.35}=8[/tex]

Hence, the length of the ladder is,

[tex]L=\sqrt{x^2+y^2}=\sqrt{8^2+6^2}=\sqrt{64+36}=\sqrt{100}=10\text{ m}[/tex]

Answer:

The length of ladder=8m.

Step-by-step explanation:

Given

The rate at which the top of a ladder slides down a vertical wall,[tex]\frac{\mathrm{d}z}{\mathrm{d}t}[/tex]= 0.675m/s

The distance of bottom of ladder from the wall,x=6m

The rate at which it slides away from the wall ,[tex]\frac{\mathrm{d}x}{\mathrm{d}t}[/tex]=0.9m/s

Let length of ladder =z

Length of wall=y

Distance between foot of ladder and wall=x

By using pythogorous theorem

[tex]x^2+y^2=z^2[/tex]

Differentiate w.r.t time

[tex]x\frac{\mathrm{d}x}{\mathrm{d}t}=z\frac{\mathrm{d}z}{\mathrm{d}t}[/tex]

y does not change hence, [tex]\frac{\mathrm{d}y}{\mathrm{d}t}=0[/tex]

[tex]6\times 0.9=z\times 0.675[/tex]

[tex]z=\frac{5.4}{0.675}[/tex]

z=8 m

Hence, the length of ladder=8m.

Answer:

We know that equation of a circle with origin as it's center is given by

[tex]x^{2}+y^{2}=r^{2}\\\\\therefore x^{2}+y^{2}=2^{2}\\\\(\frac{x}{2})^{2}+(\frac{y}{2})^{2}=1\\\\[/tex]

Comparing with [tex]sin^{2}(\theta )+cos^{2}(\theta )=1[/tex] we get

[tex]\frac{x}{2}=sin(\theta )\\\\\therefore x=2sin(\theta )\\\\\frac{y}{2}=cos(\theta )\\\\\therefore y=cos(\theta )[/tex]

Since [tex]sin(\theta ),cos(\theta )[/tex] have a period of 2π but in the given question we need to increase the period to 4π  thus we reduce the argument by 2

[tex]\therefore x=2sin(\frac{\theta }{2})\\\\y=2cos(\frac{\theta }{2})[/tex]

The parametric equations for a clockwise circle of radius 2 centered at origin traced out once for a full revolution (0 ≤ t ≤ 4π) are x = 2 cos(-t/2), y = -2 sin(t/2). This can be confirmed by substituting -t/2 for t in Pythagorean Identity sin²(t) + cos²(t) = 1 which results in 1, proving the correctness of the equations.

The parametric equations for a circle of radius 2, centered at the origin, traced out once in the clockwise direction for 0 ≤ t ≤ 2π are x = 2 cos(t), y = 2 sin(t). However, as your question indicates the path traced out in the clockwise direction, the equations would be changed to x = 2 cos(-t) and y = 2 sin(-t). This is achieved by replacing t with -t in the original equations.

In the context of the question, parametric equations which are traced out once for a full revolution (0 ≤ t ≤ 4π in the negative or clockwise direction) are x = 2 cos(-t/2), y = -2 sin(t/2). This is because time is needed twice as much to make a full revolution, so 2t is replaced with t/2.

To verify these equations, you can use the Pythagorean Identity sin²(t) + cos²(t) = 1, substituting -t/2 for t in this identity equation, you will find that the result indeed equals 1, confirming these are the correct parametric equations.

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Let [tex]y_2(t)=tv(t)[/tex]. Then

[tex]{y_2}'=tv'+v[/tex]

[tex]{y_2}''=tv''+2v'[/tex]

and substituting these into the ODE gives

[tex]t^2(tv''+2v')+3t(tv'+v)-3tv=0[/tex]

[tex]t^3v''+5t^2v'=0[/tex]

[tex]tv''+5v'=0[/tex]

Let [tex]u(t)=v'(t)[/tex], so that [tex]u'(t)=v''(t)[/tex]. Then the ODE is linear in [tex]u[/tex], with

[tex]tu'+5u=0[/tex]

Multiply both sides by [tex]t^4[/tex], so that the left side can be condensed as the derivative of a product:

[tex]t^5u'+5t^4u=(t^5u)'=0[/tex]

Integrating both sides and solving for [tex]u(t)[/tex] gives

[tex]t^5u=C\implies u=Ct^{-5}[/tex]

Integrate again to solve for [tex]v(t)[/tex]:

[tex]v=C_1t^{-6}+C_2[/tex]

and finally, solve for [tex]y_2(t)[/tex] by multiplying both sides by [tex]t[/tex]:

[tex]tv=y_2=C_1t^{-5}+C_2t[/tex]

[tex]y_1(t)=t[/tex] already accounts for the [tex]t[/tex] term in this solution, so the other independent solution is [tex]y_2(t)=t^{-5}[/tex].

Answer:

64

Step-by-step explanation:

[tex]mean=\frac{sum\ of\ total\ number\ of\ score}{total\ number\ of\ students}[/tex]

we have given that mean of 9 students is 84

so total score of 9 students = mean×9

                                              =84×9=756

and we have given mean score of exam is 82 and there is total 10 students so the total score of 10 students =10×82

                                                      =820

so the unreadable score = score of 10 students -score of 9 students =820-756=64

Answer:

0.4

Step-by-step explanation:

Given

60 % wear neither ring nor a necklace

20 % wear a ring

30 % wear necklace

This question can be Solved by using Venn diagram

If one person is choosen randomly  among the given student the probability that this student is wearing a ring or necklace is

[tex]P\left ( wear \ ring\ or \ necklace )+P\left ( neither\ ring\ or\ necklace)=1[/tex]

[tex]P\left ( wear \ ring\ or\ necklace )=1-0.6=0.4[/tex]

The sum of probabilty is equal to 1 because it completes the set

Therefore the required probabilty is 0.4

Final answer:

The probability that a randomly chosen student is wearing either a ring or a necklace is 40%. This conclusion is based on the complement of the given percentage of students who wear neither, assuming that there is no overlap in the 20% and 30% who wear rings and necklaces respectively.

Explanation:

To find the probability that a randomly chosen student is wearing a ring or a necklace, we need to understand that 'or' in probability means either one or the other, or both. According to the question, 60% of the students wear neither, which means 40% of the students wear either a ring, a necklace, or both. Since 20% wear a ring and 30% wear a necklace, we might be tempted to add these percentages to get 50%. However, doing so could potentially double-count students who wear both a ring and a necklace.

Without additional information, we can simply state that the probability that a student is wearing a ring or a necklace is the complement of the probability of a student wearing neither, which is 40%. Here we're assuming that students either wear a ring or a necklace or both, as there is no mention of wearing neither in the probabilities given.

Answer:

Equation is 4x + 6 = 598, where x represents smaller number.

Numbers are 148, 152, 156 and 160

Step-by-step explanation:

Let x be the smaller natural number,

So, the other consecutive natural numbers are x+1, x+2, x+3,

According to the question,

Sum of x, x+1, x+2 and x+3 is 598,

⇒ x + x + 1 + x + 2 + x + 3 = 598

⇒ 4x + 6 = 598

Which are the required equation,

Subtract 6 on both sides,

4x = 592

Divide both sides by 4,

x = 148

Hence, the numbers are 148, 152, 156 and 160

Final answer:

The equation to find four consecutive natural numbers with a sum of 598 is 4x + 6 = 598. Solving for x gives the first number as 148, which leads to the sequence: 148, 149, 150, and 151.

Explanation:

The student is tasked with finding four consecutive natural numbers whose sum is 598.

To solve this problem, we introduce a variable to represent the first number in the sequence, and then express the following three numbers in terms of this variable.

Let's denote the first number as x. Then the next three numbers will be x+1, x+2, and x+3, respectively. Our equation to find the numbers is:

x + (x+1) + (x+2) + (x+3) = 598

Combining like terms, we get:

4x + 6 = 598

We then solve for x:

4x = 598 - 6

4x = 592

x = 592 / 4

x = 148

So the four consecutive numbers are 148, 149, 150, and 151.

Answer:

So since your point is (-8,6), then your vertical line is x=-8 and horizontal line is y=6.

Step-by-step explanation:

Horizontal lines are in the form y=b.

Vertical lines are in the form x=a.

a and b are just constant numbers.

So anyways, in general:

The horizontal line going through (a,b) is y=b.

The vertical line going through (a,b) is x=a.

So since your point is (-8,6), then your vertical line is x=-8 and horizontal line is y=6.

1) x-intercept:

x-intercept is the point where the graph of the equation crosses the x-axis. From the given figure, we can see that the line is crossing the x-axis at -10. Thus the x-intercept is -9

2) y-intercept:

y-intercept is the point where the graph of the equation crosses the y-axis. From the given figure, we can see that the line is crossing the y-axis at -10. Thus the y-intercept is -9

3) Slope:

Slope of a line is calculated as:

[tex]slope=m=\frac{\text{Difference in y coordinates}}{\text{Difference in x coordinates}}[/tex]

For calculating the slope we can use both intercepts. x-intercept is ordered pair will be (-9, 0) and y-intercept will be (0, -9). So the slope of the line will be:

[tex]m=\frac{-9-0}{0-(-9)}=-1[/tex]

Therefore, the slope of the line is -1.

4) Slope intercept form of the line:

The slope intercept form of the line is represented as:

[tex]y=mx+c[/tex]

where,

m = slope of line = -1

c = y-intercept = -9

Using these values, the equation becomes:

[tex]y=-x- 9[/tex]

Answer:

x-intercept: [tex]-9[/tex].

y-intercept: [tex]-9[/tex].

Slope: [tex]-1[/tex]

Equation: [tex]y=-x-9[/tex]

Step-by-step explanation:

We have been given a graph of a line on coordinate plane. We are asked to find the x-intercept, y-intercept, and slope.          

We know that x-intercept of a function is a point, where graph crosses x-axis.

Upon looking at our given graph, we can see that graph crosses x-axis at point [tex](-9,0)[/tex], therefore, x-intercept is [tex]-9[/tex].

We know that y-intercept of a function is a point, where graph crosses y-axis.

Upon looking at our given graph, we can see that graph crosses y-axis at point [tex](0,-9)[/tex], therefore, y-intercept is [tex]-9[/tex].

We have two points on the line. Let us find slope of line using points  [tex](-9,0)[/tex] and  [tex](0,-9)[/tex].

[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]

[tex]m=\frac{-9-0}{0-(-9)}=\frac{-9}{0+9}=\frac{-9}{9}=-1[/tex]

Therefore, the slope of the line is [tex]-1[/tex].

Now, we will substitute [tex]m=-1[/tex] and y-intercept [tex]-9[/tex] in slope form intercept of equation as:

[tex]y=mx+b[/tex], where,

m = Slope,

b = The y-intercept.

[tex]y=-1(x-(-9))[/tex]

[tex]y=-1(x+9)[/tex]

[tex]y=-x-9[/tex]

Therefore, the equation of the line would be [tex]y=-x-9[/tex].