Question5. A hot air balloon is at an altitude of $100 \frac{1}{5}$ yards. The balloon's altitude decreases by $10 \frac{4}{5}$ yards every minute. Determine the number of minutes it will take the balloon to reach an altitude of 57 yards. (Example 2)

Studdy Solution

STEP 1

1. The initial altitude of the hot air balloon is $100 \frac{1}{5}$ yards.
2. The balloon's altitude decreases by $10 \frac{4}{5}$ yards every minute.
3. We need to determine the number of minutes it takes for the balloon to reach an altitude of 57 yards.

STEP 2

1. Convert mixed numbers to improper fractions.
2. Set up an equation to model the altitude change over time.
3. Solve the equation for the number of minutes.

STEP 3

Convert mixed numbers to improper fractions.
Convert $100 \frac{1}{5}$ to an improper fraction: $100 \frac{1}{5} = \frac{501}{5}$
Convert $10 \frac{4}{5}$ to an improper fraction: $10 \frac{4}{5} = \frac{54}{5}$

STEP 4

Set up an equation to model the altitude change over time.
Let $t$ be the number of minutes. The altitude after $t$ minutes is given by: $\frac{501}{5} - t \cdot \frac{54}{5} = 57$

STEP 5

Solve the equation for the number of minutes.
First, multiply the entire equation by 5 to eliminate the fractions: $501 - 54t = 285$
Subtract 501 from both sides: $-54t = 285 - 501$ $-54t = -216$
Divide both sides by -54 to solve for $t$ : $t = \frac{-216}{-54}$ $t = 4$
The number of minutes it will take for the balloon to reach an altitude of 57 yards is:
$\boxed{4}$

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