Help with 3, 4, 5

5. The next palindrome after 45954, is 46064, by calculating.

So, in 2 hours the car traveled 46064-45954=110 miles.

\(2h=110\)

\(\boxed{h=55}\)

3. 

Once we commonize the denominator, we achieve:

\(\frac{1}{\tan{x}}+\frac{1}{\cot{x}}=\frac{\tan{x}+\cot{x}}{\tan{x}\cdot\cot{x}}\)

Since, the problem gives us the numerical value:

\(\tan{x}+\cot{x}=\frac{144}{25}\),

we just need to solve for: 

\(\tan{x}\cdot\cot{x}\).

Since \(\tan{x}\) and \( \cot{x}\) are recipricals, \(\tan{x}\) is opposite over adjacent and \( \cot{x}\) is adjacent over opposite, their product is equal to 1. 

So our final expression becomes:

\(\frac{\frac{144}{25}}{1}\) , you can simplify. 

I hope this helped,

Gavin

4. 

y is only an integer when \(x\sqrt2\) is an integer. 

If x is an integer, \(x\sqrt2\), is only an integer when x = 0.

Therefore, there is only one lattice points on the graph.

Here is the line:https://www.desmos.com/calculator/mmw7vprxhx 

The one lattice point is: (0, -1)

\(y=0\cdot\sqrt2-1=-1\)

I hope this helped,

Gavin

The next palindrome after 45954 is 46064. 

Speed is distance divided by time. 

We already know the time, two hours, we just need to calculate the distance. 

46064 - 45954 = 110.

110 / 2 = 55

55 miles per hour, 

I hope this helped,

Gavin