Question

Emma is planning her summer and would like to work enough to travel and buy a new laptop.

Answer 1

Answer:

(a) The required equation is $90x-150y=700$.

(b) The graph of given relation is shown below.

(c) $Domain=\{x|x\in Z,0\leq x\leq 40\}$ and $Range=\{y|y\in Z,0\leq x\leq \frac{58}{3}\}$.

(d) She needs to work for 18 days.

Step-by-step explanation:

Let x be the number of days she can work and y is the number of days she can travel.

She can earn $90 each day. She expects each day of travel will cost her $150 and the laptop she hopes to  buy costs $700.

$90x-150y=700$

Therefore the required equation is 90x-150y=700.

(b)

The given relation is

$90x-150y=700$             .... (1)

Rewrite the above equation in slope intercept form.

$150y=90x-700$

Divide both sides by 150.

$y=\frac{90x-700}{150}$

$y=\frac{3}{5}x-\frac{14}{3}$

It is a straight line with slope 3/5 and y-intercept (14)/3. The graph of given relation is shown below.

(c)

The relation is

$y=\frac{3}{5}x-\frac{14}{3}$

Here, x is the number of days she can work and she can work a maximum of 40 days. So the domain of the function is

$Domain=\{x|x\in Z,0\leq x\leq 40\}$

At x=0,

$y=\frac{3}{5}(0)-\frac{14}{3}=-\frac{14}{3}$

At x=40,

$y=\frac{3}{5}(40)-\frac{14}{3}=\frac{58}{3}$

y is the number of days she can travel, so y cannot be negative.

The range of the relation is

$Range=\{y|y\in Z,0\leq x\leq \frac{58}{3}\}$

(d)

We need to find the number of days she need to work if she plans to travel for 6 days.

Substitute y=6 in equation (1) to find the value of x.

$90x-150(6)=700$

$90x-900=700$

$90x=700+900$

$90x=1600$

$90x=\frac{1600}{90}$

$x=17.77\approx 18$

Therefore she needs to work for 18 days.

Answer 2

For the answer to the question above, f x is the number of days she works, she'll earn $90x 
After buying the laptop, she'll have $90x - $700 left over, which will pay for ($90x - $700) / $150 days of travel. So we have y = ($90x - $700) / $150 = (9x - 70) / 15 = 0.6x - (14/3) 
Note that y can't be negative. Also, if y = 0, then Emma doesn't get to travel at all, so we should avoid that. So we have: 
0.6x - (14/3) > 0 
0.6x > 14/3 
x > (14/3) / 0.6 
x > 70/9 
The question says that x can be up to 40, so the domain is 70/9 < x <= 40 
That's approximately 7.777... < x <= 40 
Multiply those numbers by 0.6 and then subtract 700 to get the range: 
0 < y <= 58/3 
That's approximately 0 < y <= 19.333