Which equation best represents this situation? the number 84 increased by an unknown number is equal to 115.

Find the solution of the initial value problem<br><br> dy/dx=(-2x+y)^2-7 ,y(0)=0

Leokris [45]

Substitute $v(x)=-2x+y(x)$ , so that $\dfrac{\mathrm dv}{\mathrm dx}=-2+\dfrac{\mathrm dy}{\mathrm dx}$ . Then the ODE is equivalent to

$\dfrac{\mathrm dv}{\mathrm dx}+2=v^2-7$

which is separable as

$\dfrac{\mathrm dv}{v^2-9}=\mathrm dx$

Split the left side into partial fractions,

$\dfrac1{v^2-9}=\dfrac16\left(\dfrac1{v-3}-\dfrac1{v+3}\right)$

so that integrating both sides is trivial and we get

$\dfrac{\ln|v-3|-\ln|v+3|}6=x+C$

$\ln\left|\dfrac{v-3}{v+3}\right|=6x+C$

$\dfrac{v-3}{v+3}=Ce^{6x}$

$\dfrac{v+3-6}{v+3}=1-\dfrac6{v+3}=Ce^{6x}$

$\dfrac6{v+3}=1-Ce^{6x}$

$v=\dfrac6{1-Ce^{6x}}-3$

$-2x+y=\dfrac6{1-Ce^{6x}}-3$

$y=2x+\dfrac6{1-Ce^{6x}}-3$

Given the initial condition $y(0)=0$ , we find

$0=\dfrac6{1-C}-3\implies C=-1$

so that the ODE has the particular solution,

$\boxed{y=2x+\dfrac6{1+e^{6x}}-3}$

5 0

3 years ago

Which function below is the inverse of f(x) = x^2-9

USPshnik [31]

To find the inverse
replace f(x) with y
switch all x and y
solve for y
replace y with f⁻¹(x)

f(x)=x^2-9
y=x^2-9
x=y^2-9
x+9=y^2
sqrt both sides
$\sqrt{x+9}$
replace
f⁻¹(x)= $\sqrt{x+9}$

that is the inverse

5 0

3 years ago

Jeremy is having his basketball team over for an end-of-season party. He has $80 to spend on pizza and soda. Pizzas cost $13.99

Lapatulllka [165]

Answer:

13.99p + 4.49s ≤ 80

Step-by-step explanation:

Pizza is p and soda is s. If he has only $80 to spend on a combination of both, he cannot obviously spend more than that. Therefore, the inequality sign is "less than or equal to". He can spend 80, but he can also spend less than 80. He cannot spend anything over that, since he doesn't have it to spend!

If pizzas are represented by p, and we know the cost of each one is 13.99, we represent one pizza as 13.99p; if the cost of sida is 4.49 per case, we represent one case as 4.49s.

The sum of these cannot exceed 80, so the inequality, then, is:

13.99p + 4.49s ≤ 80

5 0

3 years ago

Question #3 math help please

LiRa [457]

Answer:

x = 45.6°

Step-by-step explanation:

As we can see, this is a right triangle, so there are known relationships

well to start we have to know the relationship between angles, legs and the hypotenuse

α = x°

a: adjacent

o: opposite = 5

h: hypotenuse = 7

sin α = o/h

cos α = a/h

tan α = o/a

we see that it has (angle, hypotenuse, opposite)

is the sine

sin α = o/h

sin x = 5/7

x = sin^-1 (5/7)

x = 45.58°

if we want to round to the nearest tenth

x = 45.58° = 45.6°

4 0

3 years ago

A museum groundskeeper is creating a semicircular statuary garden with a diameter of 29 feet. There will be a fence around the g

Jlenok [28]

Answer:

The fence will cost $330.09

Step-by-step explanation:

First we have to calculate the perimeter of the semicircle for that we use the following formula:

d = diameter = 29ft

s = perimeter of semicircle

π = 3.14

s = d * 1/2 * π

we replace the values that we know

s = 29ft * 1/2 * 3.14

s = 45.53ft

Now we multiply the number of feet by the price of 1 foot of fence

45.53 * $7.25 = 330.0925

Rounding to the nearest hundredth

$330.0925 = $330.09

8 0

3 years ago