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3 years ago

The perimeter of a rectangle is 70 cm. The ratio of length to width is 2:5. Find the length and width of the rectangle.

Mathematics

1 answer:

olya-2409 [2.1K]3 years ago

5 0

Answer:

length = 10 cm; width = 25 cm

Step-by-step explanation:

Let's call the length 2x and the width 5x. Since perimeter can be calculated by multiplying the sum of the length and width by 2 we can write:

2 * (2x + 5x) = 70

2 * (7x) = 70

7x = 35

x = 5 which means the length is 2 * 5 = 10 and the width is 5 * 5 = 25.

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Find two power series solutions of the given differential equation about the ordinary point x = 0. y'' + xy = 0

nalin [4]

Answer:

First we write y and its derivatives as power series:

y=∑n=0∞anxn⟹y′=∑n=1∞nanxn−1⟹y′′=∑n=2∞n(n−1)anxn−2

Next, plug into differential equation:

(x+2)y′′+xy′−y=0

(x+2)∑n=2∞n(n−1)anxn−2+x∑n=1∞nanxn−1−∑n=0∞anxn=0

x∑n=2∞n(n−1)anxn−2+2∑n=2∞n(n−1)anxn−2+x∑n=1∞nanxn−1−∑n=0∞anxn=0

Move constants inside of summations:

∑n=2∞x⋅n(n−1)anxn−2+∑n=2∞2⋅n(n−1)anxn−2+∑n=1∞x⋅nanxn−1−∑n=0∞anxn=0

∑n=2∞n(n−1)anxn−1+∑n=2∞2n(n−1)anxn−2+∑n=1∞nanxn−∑n=0∞anxn=0

Change limits so that the exponents for x are the same in each summation:

∑n=1∞(n+1)nan+1xn+∑n=0∞2(n+2)(n+1)an+2xn+∑n=1∞nanxn−∑n=0∞anxn=0

Pull out any terms from sums, so that each sum starts at same lower limit (n=1)

∑n=1∞(n+1)nan+1xn+4a2+∑n=1∞2(n+2)(n+1)an+2xn+∑n=1∞nanxn−a0−∑n=1∞anxn=0

Combine all sums into a single sum:

4a2−a0+∑n=1∞(2(n+2)(n+1)an+2+(n+1)nan+1+(n−1)an)xn=0

Now we must set each coefficient, including constant term =0 :

4a2−a0=0⟹4a2=a0

2(n+2)(n+1)an+2+(n+1)nan+1+(n−1)an=0

We would usually let a0 and a1 be arbitrary constants. Then all other constants can be expressed in terms of these two constants, giving us two linearly independent solutions. However, since a0=4a2 , I’ll choose a1 and a2 as the two arbitrary constants. We can still express all other constants in terms of a1 and/or a2 .

an+2=−(n+1)nan+1+(n−1)an2(n+2)(n+1)

a3=−(2⋅1)a2+0a12(3⋅2)=−16a2=−13!a2

a4=−(3⋅2)a3+1a22(4⋅3)=0=04!a2

a5=−(4⋅3)a4+2a32(5⋅4)=15!a2

a6=−(5⋅4)a5+3a42(6⋅5)=−26!a2

We see a pattern emerging here:

an=(−1)(n+1)n−4n!a2

This can be proven by mathematical induction. In fact, this is true for all n≥0 , except for n=1 , since a1 is an arbitrary constant independent of a0 (and therefore independent of a2 ).

Plugging back into original power series for y , we get:

y=a0+a1x+a2x2+a3x3+a4x4+a5x5+⋯

y=4a2+a1x+a2x2−13!a2x3+04!a2x4+15!a2x5−⋯

y=a1x+a2(4+x2−13!x3+04!x4+15!x5−⋯)

Notice that the expression following constant a2 is =4+ a power series (starting at n=2 ). However, if we had the appropriate x -term, we would have a power series starting at n=0 . Since the other independent solution is simply y1=x, then we can let a1=c1−3c2, a2=c2 , and we get:

y=(c1−3c2)x+c2(4+x2−13!x3+04!x4+15!x5−⋯)

y=c1x+c2(4−3x+x2−13!x3+04!x4+15!x5−⋯)

y=c1x+c2(−0−40!+0−31!x−2−42!x2+3−43!x3−4−44!x4+5−45!x5−⋯)

y=c1x+c2∑n=0∞(−1)n+1n−4n!xn

Learn more about constants here:

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1 year ago

Show All Work

Angelina_Jolie [31]

The unit rate you're trying to find is pages per day(or p/d), so the equation needs to have both a unit for pages and for days.
The equation we have is:
$5,249 = 29d$
If they read 5,249 pages, then we can include the unit for pages in the equation.
Since we also know that d is the number of days it took, you can replace d with days.
The equation becomes:
$5,249\ pages=29*(x\days)$
Now that we have one variable, we can solve for p/d:
$\frac{5,249\ pages}{29}=x\ days\\\\181 = x$
Thus it took them 181 days to read it all.

If Johnny read 5,249 pages over 181 days and the unit rate is pages per day(p/d), then the equation for finding p/d is:
$\frac{p\ pages}{d\ days}=\frac{5,249\ pages}{181\ days}=\frac{29\ pages}{1\ day}\\\\29\ \frac{pages}{day}$
Johnny read 29 pages per day.

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Kruka [31]

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Step-by-step explanation:

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