The bottom of a ski 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 12 feet traveled in the horizontal direction. From the top of the slope, Kayla skis at an average speed of 30 miles per hour down the slope. Which of the following functions gives the best estimate for the distance above sea level, d, Kayla is t seconds after she begins her ski run where 6,500 < d < 11,000? [Note: 5,280 feet = 1 mile]
A. d(t) = 11,000 - (572/5)
B. d(t) = 11,000 - 8.8t
C. d(t) = 11,000 - 44t
D. d(t) = 4,500 - 1,200t
thank you
To find the best estimate for the distance above sea level, d, Kayla is t seconds after she begins her ski run, we need to consider the vertical drop per horizontal distance traveled.
We know that the slope drops 5 feet vertically for every 12 feet traveled horizontally. This can be represented as a slope of -5/12.
We also know that the bottom of the slope is 6,500 feet above sea level, and the top of the slope is 11,000 feet above sea level.
Since Kayla is skiing down the slope, the distance above sea level, d, will be equal to the top of the slope minus the vertical distance she has traveled.
Now, let's analyze the given options:
A. d(t) = 11,000 - (572/5)
B. d(t) = 11,000 - 8.8t
C. d(t) = 11,000 - 44t
D. d(t) = 4,500 - 1,200t
Option A, d(t) = 11,000 - (572/5), does not take into account the vertical drop per horizontal distance traveled. It is not a correct representation.
Option B, d(t) = 11,000 - 8.8t, does not consider the vertical drop either. It assumes a linear decrease in elevation and does not account for the slope's specific characteristics. It is not a correct representation.
Option C, d(t) = 11,000 - 44t, does not accurately represent the vertical drop per horizontal distance traveled. It assumes a constant decrease of 44 feet per second, which is not correct. It is not a correct representation.
Option D, d(t) = 4,500 - 1,200t, does not take into account the starting elevation of 11,000 feet and only considers a constant vertical decrease. It is not a correct representation.
Therefore, none of the given options provide the best estimate for the distance above sea level, d, Kayla is t seconds after she begins her ski run.
To find the best estimate for the distance above sea level, we need to consider the information given:
- The bottom of the slope is 6,500 feet above sea level, and the top is 11,000 feet above sea level.
- The slope drops 5 feet vertically for every 12 feet horizontally.
We can first calculate the vertical drop per horizontal distance by dividing the vertical drop (5 feet) by the horizontal distance (12 feet):
Vertical drop per unit horizontal distance = 5/12 feet
Now, we know that Kayla skis down the slope at an average speed of 30 miles per hour. We need to convert this speed to feet per second to be consistent with the units used in the question:
Speed = 30 miles per hour
1 mile = 5,280 feet
1 hour = 60 minutes = 60 seconds
Speed = 30 * 5,280 / (1 * 60 * 60) feet per second
Next, we can calculate the vertical distance Kayla covers in t seconds using the derived vertical drop per unit horizontal distance and the speed:
Vertical distance covered in t seconds = (Speed) * (Vertical drop per unit horizontal distance) * t
Finally, since Kayla starts at the top of the slope, we need to subtract the calculated vertical distance from the initial height:
d(t) = 11,000 - (Vertical distance covered in t seconds).
Now we can simplify the options and choose the best estimate based on this calculation:
A. d(t) = 11,000 - (572/5)
B. d(t) = 11,000 - 8.8t
C. d(t) = 11,000 - 44t
D. d(t) = 4,500 - 1,200t
Comparing the options to our derived function, it can be observed that the correct answer should be:
B. d(t) = 11,000 - 8.8t
over the snow her speed is
30 mi/hr * 5280 ft/mi * 1 hr/3600 sec = 44 ft/second
for 5 down, 12 horizontal, hypotenuse is 13
so speed down = 44 * 5/13 = 17 ft/s
so I get
11,000 - 17 t