2.2 KiB
Rational Expressions
Question 1 a)
Let \(`S_r`\) be Ron’s speed and \(`S_m`\) be Mack’s speed. Let \(`t`\) be the time.
\(`t_a = \dfrac{30}{S_r}`\)
\(`t_b = \dfrac{20}{S_m}`\)
\(`t_{\text{total}} = \dfrac{30}{S_r} + \dfrac{20}{S_m} = \dfrac{30S_m + 20S_r}{(S_m)(S_r)}`\)
Question 1 b)
Here, \(`S_r = 35, S_m = 25`\) (Because we want the faster guy to travel more distance to save time). We plug it into our formula above:
\(`t_{\text{tota}} = \dfrac{30(25) + (20)(35)}{(35)(25)} = 1.657`\)
Therefore the minimum amount of time it will take to complete the race is \(`1.657`\) hours, or about \(`99.42`\) minutes or \(`99`\) minutes and \(`25.2`\) seconds.
Question 2 a)
I would first flip the second fraction and then cross-cross out the common factors like so:
\(`\dfrac{(x+3)(x-6)}{(x+4)(x+5)} \times \dfrac{(x+4)(x-7)}{(x-6)(x+8)}`\)
We can cross out the \(`(x+4)`\) and \(`(x-6)`\) since they cancel each other out.
The final fraction is therefore \(`\dfrac{(x+3)(x-7)}{(x+5)(x+8)}`\)
For restrictions, at each step, I would mark down the restrictions. Such without doing anthing, we know that \(`x =\not -4, -5, 7`\), then after we flip the fraction, we know that \(`x =\not 6, -8`\), and at the final step, we know that \(`x=\not -5, -8`\).
Therefore the final restrictions on \(`x`\) would be: \(`x=\not -4, -5,-8, 6, 7`\)
Question 2 b)
By using the restrictions and the final product, we know that the 2 fraction can only have the following as its denominator: \(`(x+4), (x-2), (x-1)`\)
And since we know there is a \(`(x-2)`\) as the denominator and \(`(x+5)`\) as the numerator for the final product, we just need one of the fractions to cancel out the denominators \(`(x+4), (x-1)`\).
Thus, 2 fractions such as below would work:
\(`\dfrac{(x+5)}{(x-4)(x-1)} \times \dfrac{(x-4)(x-1)}{(x-2)}`\)
Question 2 c)
The student forgot to multiply the numerator by the same number he used to multiply the denomiator.
Question 3 a)
Let \(`V, SA`\) be the volume and surface area respectively.
\(`V = \pi r^2 h`\)
\(`SA = 2r\pi h + 2\pi r^2 \implies 2r \pi (h + r)`\)
The ratio of \(`V : SA`\) is equal to:
\(`\dfrac{\pi r^2 h}{2r\pi (h + r)}`\)
\(` = \dfrac{rh}{2(h+r)}`\)
Question 3 b)
The restrictions by looking at the fraction are \(`h =\not -r, r =\not -h`\), also \(`r =\not 0`\).