The bottom of a ske slope is 6,500 feet above sea level,the top of the slope is 11,000 feet above sea level, and the slope drops 5 feet vertically for every 11 feet traveled in the horizontal direction. From the top of the slope, Kayla skis down at an average speed of 30 miles per hour. Which of the following function gives the best estimate for the distance above sea level, d, Kayla is t seconds after she begins her ski run where `6,500ltdlt11,000`?

Options

A. `d(t)=11,000-((150)/(11))t`
B. `d(t)=11,000-2.2t`
C. `d(t)=11,000-20t`
D. `d(t)=4,500-1,200t`

Answer

To solve the problem step by step, we will determine the function that estimates the distance above sea level, \( d \), that Kayla is at \( t \) seconds after she begins skiing down the slope.

### Step 1: Understand the height of the slope
The bottom of the ski slope is at 6,500 feet above sea level, and the top is at 11,000 feet above sea level. This means that Kayla starts skiing from a height of 11,000 feet.

### Step 2: Determine the slope of the ski run
The slope drops 5 feet vertically for every 11 feet traveled horizontally. This gives us a slope ratio of:
\[ \text{slope} = -\frac{5}{11} \]
The negative sign indicates that the height is decreasing as she skis down.

### Step 3: Convert Kayla's speed to feet per second
Kayla skis down at an average speed of 30 miles per hour. To convert this speed to feet per second:
\[ 30 \text{ miles/hour} = 30 \times 5280 \text{ feet/mile} \div 3600 \text{ seconds/hour} \]
Calculating this gives:
\[ 30 \times 5280 = 158400 \text{ feet/hour} \]
\[ \frac{158400}{3600} = 44 \text{ feet/second} \]

### Step 4: Calculate the descent per second
To find out how much Kayla descends vertically each second, we multiply her speed in feet per second by the slope:
\[ \text{Descent per second} = 44 \text{ feet/second} \times \frac{5 \text{ feet}}{11 \text{ feet}} = 20 \text{ feet/second} \]

### Step 5: Determine the total descent after \( t \) seconds
After \( t \) seconds, the total descent will be:
\[ \text{Total descent} = 20t \text{ feet} \]

### Step 6: Write the function for height above sea level
Starting from 11,000 feet, the height above sea level after \( t \) seconds will be:
\[ d(t) = 11000 - 20t \]

### Conclusion
Thus, the function that gives the best estimate for the distance above sea level, \( d \), Kayla is at \( t \) seconds after she begins her ski run is:
\[ d(t) = 11000 - 20t \]