solve the given applied problem invoving variation.

The distance d that can be seen from horizion to horizion from an airplane varies directly as the square root of the altitude h of the airplane. If d=133mi for h=12,000ft, find d for h=16,000ft

Question

solve the given applied problem invoving variation.

The distance d that can be seen from horizion to horizion from an airplane varies directly as the square root of the altitude h of the airplane. If d=133mi for h=12,000ft, find d for h=16,000ft

anonymous · Answer

@Psymon

anonymous · Answer

i get k=1.1 and then 1.1 square root 12,000 =120.5 however i might be wrong and have it backwards

psymon · Answer

SO varies directly means we have the proportion y = kx. With x being what y is varying directly to. In this case, d (basically y in this case) is varying directly to the square root of the altitude, meaning we must put y = ksqrt(x). So d (or y) is 133 miles and the height is 12000 feet. Another thing to be aware of, though, is we have d in miles but h in feet. We cant have that, we must have the same units. Because feet is a massive number, probably  best to turn feet into miles. No matter what conversion we do, if we put it in a calculator we just get decimals, so we'll have to keep this in terms of reduced fractions. So 12000/5280 reduces to 25/11. So since we now have everything in terms of miles, we can set up the proportion 
$$133 = k \sqrt{\frac{25}{11}} \implies 133 = k \frac{ 5\sqrt{11} }{ 11 }$$

So solving for k we get $$k = \frac{ 133 \sqrt{11} }{ 5 }$$

Really ugly, but not much that can be done. So now we can find y with the other info.

Of course the info is given, 16000 ft, in feet, so we need that into miles so it matches up with everything. 16000/5280 = 100/33 miles. Now we square root that and  we set up the proportion with k and 100/33 to find y

$$d =\sqrt{\frac{ 100 }{ 33 }}*\frac{ 133\sqrt{11} }{ 5 } = \frac{ 266\sqrt{3} }{ 3 } \approx 153.58mi$$
 Yeah, looks like its impossible to avoid decimals without large numbers. The main silly thing in this problem is converting from feet to miles and making sure you do it correctly. Sorry about the length of explanation, just weird and big numbers.

anonymous · Answer

no no it's fine i'm glad you explained thgis to me. Thank you.

psymon · Answer

Yeah, the key is just actually noticing you need to convert units. Otherwise the process is straightforward, just gotta be careful