147.59 is the total amount given to each brother after all purchases were made.
Step-by-step explanation:
x = mass of apples
y = mass of strawberries
z = mass of the box
x = 3y
x + z = 16240
y + z = 6350
let's subtract the second from the first equation
x + z = 16240
- y + z = 6350
-----------------------
x - y + 0 = 9890
using the first identity in that result gives us
3y - y = 9890
2y = 9890
y = 4,945 g
x = 3y = 3×4945 = 14,835 g = the mass of the apples
This is quite a complex problem. I wrote out a really nice solution but I can't work out how to put it on the website as the app is very poorly made. Still, I'll just have to type it all in...
Okay so you need to use a technique called logarithmic differentiation. It seems quite unnatural to start with but the result is very impressive.
Let y = (x+8)^(3x)
Take the natural log of both sides:
ln(y) = ln((x+8)^(3x))
By laws of logarithms, this can be rearranged:
ln(y) = 3xln(x+8)
Next, differentiate both sides. By implicit differentiation:
d/dx(ln(y)) = 1/y dy/dx
The right hand side is harder to differentiate. Using the substitution u = 3x and v = ln(x+8):
d/dx(3xln(x+8)) = d/dx(uv)
du/dx = 3
Finding dv/dx is harder, and involves the chain rule. Let a = x+ 8:
v = ln(a)
da/dx = 1
dv/da = 1/a
By chain rule:
dv/dx = dv/da * da/dx = 1/a = 1/(x+8)
Finally, use the product rule:
d/dx(uv) = u * dv/dx + v * du/dx = 3x/(x+8) + 3ln(x+8)
This overall produces the equation:
1/y * dy/dx = 3x/(x+8) + 3ln(x+8)
We want to solve for dy/dx, achievable by multiplying both sides by y:
dy/dx = y(3x/(x+8) + 3ln(x+8))
Since we know y = (x+8)^(3x):
dy/dx = ((x+8)^(3x))(3x/(x+8) + 3ln(x+8))
Neatening this up a bit, we factorise out 3/(x+8):
dy/dx = (3(x+8)^(3x-1))(x + (x+8)ln(x+8))
Well wasn't that a marathon? It's a nightmare typing that in, I hope you can follow all the steps.
I hope this helped you :)
Answer:
A. t=4n+950
B. h=9.5n
C. d=9.5n-(4n+950)
D. The tortoise will finish first.
Step-by-step explanation:
To find the first two equations, we put the rate at which they are traveling in front of the n, and then add where they start. Since the tortoise gets a 950 meter head start, we add 950. The hare is starting at the start line so we don't need to add anything.
For answer c, we need to subtract the two equations from each other to figure out when they would pass each other. d=(equation h)-(equation t).
For the final answer, all you need to do is plug the first two equations into a graph, and then find where each line is at the point of y 1,600. (To make the graphs I replaced the n with x and the t and h with y.) Using this, we are able to see who got there first, which was the tortoise.
