Alan

After 1 minute the temperature will be 0.96*150

After 2 minutes the temperature will be 0.96²*150

After 3 minutes the temperature will be 0.96³*150

...

After t minutes the temperature will be 0.96^t*150

What is t when the temperature is 100?

0.96^t*150 = 100

Take logs of both sides

log(0.96^t)+log(150) = log(100)

t*log(0.96) = log(100)-log(150) = log(100/150) = log(2/3)

t = log(2/3)/log(0.96)

$${\mathtt{t}} = {\frac{{log}_{10}\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}{{log}_{10}\left({\mathtt{0.96}}\right)}} \Rightarrow {\mathtt{t}} = {\mathtt{9.932\: \!515\: \!862\: \!422\: \!638\: \!3}}$$

t ≈ 9.93 minutes

Here's a starter for a mathematics sub-category list

Arithmetic

Geometry

Trigonometry

Algebra

Calculus

Differential equations

Logic

Probability and statistics

I wonder if this is asking about how to express the position and velocity in terms of the unit vectors, i, j and k.

If so, then, using p for position and v for velocity:

p = sin(t)i + (t+1)j + cos(t)k

v = cos(t)i + j - sin(t)k

You can enter it just as you have written it.

Note that the area should have units of in² not just in

Multiply both sides by 8.

$$\frac{x}{8}\times 8=2\times 8\\\\
x = 16$$

The left-hand side is NOT equal to the right-hand side in general.  For example, suppose A = 37°

$${\frac{{cot}{\left({\mathtt{37}}\right)}{\mathtt{\,\times\,}}{csc}{\left({\mathtt{37}}\right)}}{\left({cot}{\left({\mathtt{37}}\right)}{\mathtt{\,\small\textbf+\,}}\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{cos}}{\left({\mathtt{37}}^\circ\right)}\right)}} = {\mathtt{1.037\: \!348\: \!331\: \!178\: \!024\: \!9}}$$

$${\frac{\left({cot}{\left({\mathtt{37}}\right)}{\mathtt{\,-\,}}\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{cos}}{\left({\mathtt{37}}^\circ\right)}\right)}{\left({cot}{\left({\mathtt{37}}\right)}{\mathtt{\,\times\,}}{csc}{\left({\mathtt{37}}\right)}\right)}} = {\mathtt{0.239\: \!633\: \!701\: \!060\: \!557\: \!7}}$$

"I think there is an error in one of the equations there in the book. "

No, I think the error in the sign is a bug in the website calculator.  Mathcad gets the sign the same as the text book.

See if the following help:

To solve these equations using this website's calculator enter the following in the vector/matrix section (Note, you only enter the values in the table part, not in the top line.  You also select the ^-1 next to the A):

Press the equals sign to get the following:

Hmmm!  Seems to be a sign error for I₂ here.