SOLUTION: I keep getting the incorrect answer to the following problem, I'm not sure where my error was, can you help me? One method for determining the depth of a well is to drop a ston

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Question 196647: I keep getting the incorrect answer to the following problem, I'm not sure where my error was, can you help me?
One method for determining the depth of a well is to drop a stone into it and then measure the time it takes until the splash is heard. If d is the depth of the well (in feet) and t1 the time (in seconds) it takes for the stone to fall, then d = 16t1^2, so t1 = √d/4. Now if t2 is the time it takes for the sound to travel back up, then d = 1090t2 because the speed of sound is 1090 ft/s. So t2 = d/1090. Thus, the total time elapsed between dropping the stone and hearing the splash is given by the following equation.
t1 + t2 = √d/4 + d/1090
How deep is the well if this total time is 2.3 s? (Round the answer to the nearest whole number.)
thanks!
Found 2 solutions by jim_thompson5910, Alan3354:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Since the "total time is 2.3 s", this means that t%5B1%5D%2Bt%5B2%5D=2.3


t%5B1%5D%2Bt%5B2%5D=sqrt%28d%29%2F4%2Bd%2F1090 Start with the given equation.


2.3=sqrt%28d%29%2F4%2Bd%2F1090 Plug in t%5B1%5D%2Bt%5B2%5D=2.3


Multiply EVERY term by the LCD 2180 to clear out the fractions.


5014=545sqrt%28d%29%2B2d Multiply and simplify


5014-2d=545sqrt%28d%29 Subtract 2d from both sides.


%285014-2d%29%5E2=%28545sqrt%28d%29%29%5E2 Square both sides


%285014-2d%29%5E2=297025d Square 545sqrt%28d%29 to get 297025d


25140196-20056d%2B4d%5E2=297025d FOIL the left side


25140196-20056d%2B4d%5E2-297025d=0 Get all terms to the left side.


4d%5E2-317081d%2B25140196=0 Combine like terms.


Notice we have a quadratic in the form of Ad%5E2%2BBd%2BC where A=4, B=-317081, and C=25140196


Let's use the quadratic formula to solve for d


d+=+%28-B+%2B-+sqrt%28+B%5E2-4AC+%29%29%2F%282A%29 Start with the quadratic formula


Plug in A=4, B=-317081, and C=25140196


d+=+%28317081+%2B-+sqrt%28+%28-317081%29%5E2-4%284%29%2825140196%29+%29%29%2F%282%284%29%29 Negate -317081 to get 317081.


d+=+%28317081+%2B-+sqrt%28+100540360561-4%284%29%2825140196%29+%29%29%2F%282%284%29%29 Square -317081 to get 100540360561.


d+=+%28317081+%2B-+sqrt%28+100540360561-402243136+%29%29%2F%282%284%29%29 Multiply 4%284%29%2825140196%29 to get 402243136


d+=+%28317081+%2B-+sqrt%28+100138117425+%29%29%2F%282%284%29%29 Subtract 402243136 from 100540360561 to get 100138117425


d+=+%28317081+%2B-+sqrt%28+100138117425+%29%29%2F%288%29 Multiply 2 and 4 to get 8.


d+=+%28317081+%2B-+545%2Asqrt%28337137%29%29%2F%288%29 Simplify the square root (note: If you need help with simplifying square roots, check out this solver)


d+=+%28317081%2B545%2Asqrt%28337137%29%29%2F%288%29 or d+=+%28317081-545%2Asqrt%28337137%29%29%2F%288%29 Break up the expression.


So the answers are d+=+%28317081%2B545%2Asqrt%28337137%29%29%2F%288%29 or d+=+%28317081-545%2Asqrt%28337137%29%29%2F%288%29


which approximate to d=79190.884 or d=79.366


Notice that if the time for the stone to fall was 2.3 seconds, then the depth would be AT MOST d=16%282.3%29%5E2=84.64. In other words, the MAX depth that is possible in this problem is about 85 feet.


Since the value d=79190.884 is too large, this means that d=79.366 is the only answer.


So to the nearest whole number, the depth of the well is about 79 feet.

Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
t1 + t2 = √d/4 + d/1090
How deep is the well if this total time is 2.3 s? (Round the answer to the nearest whole number.)
--------------
2.3 = sqrt(d)/4 + d/1090
sqrt(d)/4 = 2.3 - d/1090
d/16 = 5.29 - 4.6d/1090 + d^2/1090^2
d^2 + 5.29*1090^2 - 4.6d*1090 - d*1090^2/16 = 0
d^2 - d*(5014 + 74256.25) + 6285049 = 0
d^2 - 79270.25d + 6280549 = 0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-79270.25x%2B6280549+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-79270.25%29%5E2-4%2A1%2A6280549=6258650339.0625.

Discriminant d=6258650339.0625 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--79270.25%2B-sqrt%28+6258650339.0625+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-79270.25%29%2Bsqrt%28+6258650339.0625+%29%29%2F2%5C1+=+79190.9410674966
x%5B2%5D+=+%28-%28-79270.25%29-sqrt%28+6258650339.0625+%29%29%2F2%5C1+=+79.3089325033652

Quadratic expression 1x%5E2%2B-79270.25x%2B6280549 can be factored:
1x%5E2%2B-79270.25x%2B6280549+=+%28x-79190.9410674966%29%2A%28x-79.3089325033652%29
Again, the answer is: 79190.9410674966, 79.3089325033652. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-79270.25%2Ax%2B6280549+%29

79 feet looks right.