Site names | Your School | School #2 | |
Latitude | . | . | The latitude is how far north of the equator you are.(You can find this value from a world map.) |
shadow length | . | . | The lenth of the shadow your meter stick casts |
sun angle | . | . | The angle formed by the meter stick and the sun rays |
N/S distance | . | . | This is the perpendicular distance between the latitude your schoolis on and the latitude of school #2. |
Circumference | . | . | The circumference is the distance around the earth. This value willbe based on the data above. |
% error | . | . | This tells you how close you came to approximating the circumferenceof the earth. |
Side BC is 1.24 times longer than side AC. If that is true, what mustangle A be? |
The ratio of the gnomon to the shadow in New Jersey is100 / 80.5 or 1.24. So we want the spreadsheet to find the measure of angle A for a tangent value of 1.24. A is the base angle. The Sun angle is B.How would you find that value if you know angle A? (Hint: Angle C is a right angle.) |
To find Angle A in the language of a spreadsheets, we use this formula: A = ATAN 1.24 which means that the computer is finding the base angle measure for a right triangle where the opposite (gnomon) side is 1.24 times longer than its adjacent (shadow) side. (Since most spreadsheets use radian measure instead of angle measurements, we have to convert from one to the other in our formula.)
The formula that goes into cell D2 is
=90 - ATAN (100/80.5) * 57.3
Finding the North/South Distance
If you travel along a line of longitude, each 1 degreeof latitude is approximately 111.13 km long. Since Manasquan, NJ is latitudinalposition is 40 degrees and Puerto Rico is at 18 degrees, they are separatedby 22 degrees. Since each degree is 111 km, then the north/south distancebetween them is 22 x 111 or approximately 2, 442 km. This translates intothe following spreadsheet formula (in E2 and E3)
=ABS(B2-B3) * 111.133 whereABS stands for absolute value
Finding the Circumference
The difference in the shadow angles tell us what kindof slice Manasqan and Puerto Rico make. The difference is 38.85 - 18.58= 20.27 which is not only the difference of their sun angles, but is alsothe measure of the central angle. Since a complete circle is 360 degrees,there are 17.7 slices with an edge of . So the estimate for the circumferenceis . The spreadsheet formula looks like this:
=(E3 * 360 ) / ABS (D2-D3)
Here's what the formula looks like.
A | B | C | D | E | 6 | 7 | |
1 | Site | Latitude | Shadow | Sun Angle | N/S Distance | Circum-
ference |
% error |
2 | NJ | 40.13 | 80.5 | 38.85 | |||
3 | PR | 18.28 | 33.7 | 18.58 | 2421.59 |