Question

time, t, required to empty a tank varies inversely as the rate, f, of pumping. if a pump can empty a tank hours at a rate of 400 gallons per minute, how long will it take to empty a tank at 500 gallons per minute?

mount of miles traveled is directly proportional to the amount of time. if pamela travels 110 miles in 2 how long will it take pamela to travel 385 miles?

etch the graph of the given rational function and then identify the key attributes.
\\(\frac{x^2 - 7x + 10}{x - 3}\\)

Explanation:

Step1: Fix first problem missing value

Note: The first problem is missing the time taken at 400 gallons per minute. Assuming the given time is 2.5 hours (a common value for this type of inverse variation problem) to solve it.

Step2: Inverse variation formula setup

Inverse variation: $t = \frac{k}{r}$, where $k$ is constant.
Substitute $t=2.5$, $r=400$:
$2.5 = \frac{k}{400}$

Step3: Solve for constant $k$

$k = 2.5 \times 400 = 1000$

Step4: Find time for $r=500$

$t = \frac{1000}{500} = 2$

Step5: Direct variation formula setup

Direct variation: $d = kt$, where $d$=distance, $t$=time.
Substitute $d=110$, $t=2$:
$110 = 2k$

Step6: Solve for speed $k$

$k = \frac{110}{2} = 55$ miles/hour

Step7: Find time for $d=385$

$t = \frac{385}{55} = 7$

Step8: Simplify rational function

Factor numerator: $x^2-7x+10=(x-2)(x-5)$
Function: $\frac{(x-2)(x-5)}{x-3}$

Step9: Find x-intercepts

Set numerator=0: $(x-2)(x-5)=0$

Step10: Find vertical asymptote

Set denominator=0: $x-3=0$

Step11: Determine horizontal asymptote

Degree of numerator > denominator: No horizontal asymptote (slant asymptote exists, but HA: None)

Step12: Check for holes

No common factors in numerator/denominator: No hole

Step13: Find domain

Exclude $x=3$: All real numbers except $x=3$

Step14: Find range

Since slant asymptote exists, range is all real numbers

Answer:

1. Time to empty at 500 gallons per minute: 2 hours
2. Time to travel 385 miles: 7 hours
3. Rational function key attributes:

• x-int: $x=2$, $x=5$
• VA: $x=3$
• HA: None
• Hole: None
• Domain: All real numbers where $x

eq 3$

• Range: All real numbers