**Question 1:** Will and Grace are canoeing on a lake. Will rows at $50$ meters per minute and Grace rows at $20$ meters per minute. Will starts rowing at $2$ p.m. from the west end of the lake, and Grace starts rowing from the east end of the lake at $2{:}05$ p.m. If they always row directly towards each other, and the lake is $3400$ meters across from the west side of the lake to the east side, at what time will the two meet?

**Question 2:** A plane flies from Penthaven to Jackson and then back to Penthaven. When there is no wind, the round trip takes $4$ hours and $40$ minutes, but when there is a wind blowing from Penthaven to Jackson at $50$ miles per hour, the trip takes $4$ hours and $48$ minutes. How many miles is the distance from Penthaven to Jackson?

**Question 3:** Find all values of $x$ such that $\sqrt{4x^2} + \sqrt{x^2} = 6$.

**Question 4:** For a certain value of $k$, the system \(\begin{align*} x + y + 3z &= 10, \\ 4x + 2y + 5z &= 7, \\ kx + z &= 3 \end{align*}\) has no solutions. What is this value of $k$?

ThreeLetters Apr 6, 2024

#2**+2 **

For number 1) Will started 5 minutes before Grace, so 50*5=250 meters

So they have 3400-250=3150 meters when they start approaching each other

.

They approach at 70m/minute, so 3150/70 = 45

So then they meet at 2:50 p.m

aboslutelydestroying Apr 7, 2024

#4**+2 **

For number 2)

Say plane speed is P mph

Distance is D miles.

Time for round trip is: D/P + D/P = 2D/P

4hr 40min = 14/3 hr

2D/P = 14/3

D/P = 7/3

=> P = 3D/7

Time taken (with wind) = D/(P+50) + D/(P-50)

Time taken: 4hr 48min = 24/5 hrs

=> D/(P+50) + D/(P-50) = 24/5

5D/(P+50) + 5D/(P-50) = 24

(5DP-250D)/(P^2-2500) + (5DP+250D)/(P^2-2500) = 24

10DP/(P^2-2500) = 24

10DP = 24P^2-24*2500

5DP = 12P^2-12*2500

P = 3D/7

=> 5D*3D/7 = 12*(9D^2/49 -2500)

Multiply 49 on both sides of the equation:

5D*21D = 108D^2-2500*12*49

105D^2 = 108D^2-2500*12*49

3D^2 = 2500*12*49

D^2 = 2500*4*49

D = 50*2*7 = 700

So the distance from Penthaven to Jackson is **700 miles**

aboslutelydestroying Apr 7, 2024

#5**+2 **

For number 3)

\(\sqrt{4x^2}+\sqrt{x^2}=6\)

\(|2x|+|x|=6\)

Say 2x+x = 6

3x = 6

x = 2

Then there is the other case:

2x+x = -6

3x = -6

x = -2

So the values of x would be **-2 and 2**

aboslutelydestroying Apr 7, 2024

#6**+2 **

For number 4)

\(x+y+3z=10\)

\(4x+2y+5z=7\)

\(kx+z=3\)

Multiply 2 on both sides of the 1st equation:

\(2x+2y+6z=20\)

Subtract it to the 2nd equation:

\(2x-z=-13\)

\(kx+z=3\)

multiply -1 to \(kx+z=3\) on both sides:

\(-kx-z=-3\)

in order for this system to have no solutions

k has to be -2 which gets us: \(2x-z=-3\)

This equation can't have 2 values since it is linear,

so this proof tells us, **k = 2**

aboslutelydestroying Apr 7, 2024