A road sign at the top of a mountain indicates that for the next 4 miles the grade is 12%. Find the angle of the grade and the change in elevation for a car descending the mountain.

The grade of an incline is its vertical rise in some units per every 100 horizontal run in the same units.

so
rise/run = tan slope = .12
so slope = tan^-1 .12 = 6.8 degrees
height in miles = .12 run in miles
( the question is ambiguous. I am assuming the 4 miles is on the map, horizontal, rather than on the odometer , hypotenuse)
height change = .12*4 miles = .48 miles down
or .48*5280 = 2534 feet down

Ah, the perils of mountain-driving! Let me see if I can humorously assist you with this one.

To find the angle of the grade, we can imagine it as a triangle. Now, a triangle is like a slice of pizza – it always comes with three angles. Since we have the length of the side (4 miles) and the percentage of the grade (12%), we can use some mathematical wizardry to work this out.

Now, assuming that the road sign isn't playing any practical jokes on us, we can determine that the change in elevation for the car descending the mountain is equal to 12% of the 4 miles. In other words, it's a 12% discount on the uphill struggle!

To calculate the angle, we need to convert the percentage to a decimal. So, 12% in decimal form is 0.12. Now, remember, the opposite side of the triangle is the change in elevation and the adjacent side is the distance. So, using some trigonometric magic (specifically, tanθ = opposite/adjacent), we discover that the angle of the grade is approximately 6.87 degrees.

So, there you have it – the grade is the punchline to this mountainous riddle, and the change in elevation is the discount your car receives as it descends. Drive safely and enjoy the downhill journey!

To find the angle of the grade, you can use trigonometry. The grade is represented by the tangent of the angle, which is given by the formula:

tangent(angle) = (change in elevation) / (horizontal distance)

Given that the grade is 12%, we need to convert it to a decimal by dividing it by 100:

12% = 12/100 = 0.12

Now we can use the formula to find the angle:

tangent(angle) = 0.12

To find the angle, we can take the inverse tangent (arctan) of both sides:

angle = arctan(0.12)

Using a calculator or a trigonometric table, we can find that the angle is approximately 6.87 degrees.

To find the change in elevation for a car descending the mountain, we can use the formula:

change in elevation = (grade) * (horizontal distance)

Given that the grade is 12% and the horizontal distance is 4 miles, we can calculate:

change in elevation = 0.12 * 4

The change in elevation is equal to 0.48 miles.

Therefore, the angle of the grade is approximately 6.87 degrees and the change in elevation for a car descending the mountain is 0.48 miles.

To find the angle of the grade and the change in elevation for a car descending the mountain, we can use the formula:

Change in elevation = Grade * Distance

Given that the grade is 12% and the distance is 4 miles, we can proceed with the calculation.

Step 1: Convert the grade to decimal form.
To convert a percentage to a decimal, divide it by 100. In this case, the grade is 12%, so we can convert it to a decimal by dividing 12 by 100:
Grade = 12/100
Grade = 0.12

Step 2: Calculate the change in elevation.
Change in elevation = Grade * Distance
Change in elevation = 0.12 * 4 miles
Change in elevation = 0.48 miles

Step 3: Convert the change in elevation from miles to feet.
Since the grade is given in percentage, and the change in elevation is in miles, it's a good idea to convert it to feet. There are 5,280 feet in a mile, so:
Change in elevation = 0.48 miles * 5280 feet/mile
Change in elevation = 2,534.4 feet

So, the change in elevation for a car descending the mountain is approximately 2,534.4 feet.

Step 4: Calculate the angle of the grade.
The angle of the grade can be calculated using trigonometry. The tangent of an angle is equal to the opposite side (change in elevation) divided by the adjacent side (distance). Therefore, we have:
Tangent of angle = Change in elevation / Distance

Rearranging the formula, we get:
Angle = arctan(Change in elevation / Distance)

Angle = arctan(2,534.4 feet / 4 miles)

To find the angle using a scientific calculator, press the 'arctan' or 'inverse tangent' button and enter the value of (2,534.4 / 4), and convert the result to degrees.

After performing the calculation, the approximate angle of the grade for the car descending the mountain is 14.04 degrees.

A sign on the roadway at the top of a mountain indicates that for the next 4 miles the grade is 9.5° (see figure). Find the change in elevation for a car descending the 4-mile stretch. (Round your answer to two decimal places.)