To find the height of a lamppost (see accompanying figure), yo...

Question

To find the height of a lamppost (see accompanying figure), you stand a 6-ft pole 20 ft from the lamp and measure the length a of its shadow, finding it to be 15 ft, give or take an inch. Calculate the height of the lamppost using the value a = 15 and estimate the possible error in the result.

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Accepted Answer

Hello there. Today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A student wants to find the height of a pole. Using a 3 ft stick placed 12 ft from the flagpole. The student measures the shadow of the stick to be 9 ft long. If the measurement of the shadow has an uncertainty of plus or minus 0.25 inch, calculate the height of the pole and estimate the possible error. Awesome. So it appears for this particular problem we're asked to solve for two separate answers. Our first answer is we're trying to calculate the height of this flagpole means we do not know what the height of this flagpole is. And our second answer that we're trying to solve for is we're trying to estimate the possible error. So now that we know what we're ultimately trying to solve for, let's read off our multiple choice answers to see what our final answer pair might be. Noting that we'll read the height first, which will be in units of feet, and then we'll read the possible er error in height, which will also be in units of feet. So A is 6 and plus or minus 1 divided by 108, B is 6 and plus or minus 1 divided by 54, C is 7 and plus or minus 1 divided by 108, and D is 7 and plus or minus 1 divided by 54. OK. So, first off, in order to help us solve for this problem more effectively and efficiently, we need to create a diagram of what's happening in this problem, so we could better visualize what's going on. So, when we do that, we should produce a diagram that looks something along the lines of this. So, we're trying to find the height of this flagpole, and the height of the flagpole, which is going to be on the far left of our diagram, has some height H, so we do not know what the height is, that's what we're ultimately trying to solve for, so we do not know the value of H. And then we have a stick, which is 3 ft tall, which is stationed 12 ft away from the flight pole, which we have labeled in our diagram. And as you can see where the stick is at 3 ft tall, it's going to produce a shadow of 9 ft long. So S denotes the shadow of the stick or the length of the shadow, to be precise. So, S equals 9 ft, and once again this 9 ft. Represents the length of the shadow that is produced by the stick. So as you can see, we have our flagpole on the far left, and then going to the right, we have 12 ft away. So going from the height of our flagpole on the left, going 12 ft to the right, we have our 3 ft stick, which is standing upright. And then behind our 3 ft stick, we have a shadow that is 9 ft long that's produced from the stick. Fantastic. So once again, we need to define that the height of the flagpole is going to be H, and we do not know the value of H. And as you can see, we have triangles, which is going to make solving this problem simplistic, because we're just going to use geometric identities to help us solve for this problem. So using similar triangles, so thinking of our diagram as right triangles, we can state that H divided by 12 plus S is equal to 3 divided by S, and once again we can write this due to similar triangles. So H divided by 12 plus S is equal to 3 divided by S. Fantastic. So, we can therefore, we can also go ahead and write that H divided by 12 + 9 is equal to 3 divided by 9 when we plug in a value of 9 in for S. So when we rearrange this expression to isolate and solve for H, we will find that H is equal to 21, multiplied by 3 divided by 9, which is going to give us an answer of 7. So that's our first answer. Hooray, we did it. So our height is going to be 7 ft. So let's make a note of that. So, Our final answer for the height is going to be 7 ft, and once again, that is our value for the height. Fantastic. So now we need to determine the possible error when we measure the height, so we need to determine the possible error in height. So we need to recall and note that the length of the shadow has an uncertainty of plus or minus 0.25 inch. So since there are 12 inches and a foot, it follows that plus or minus 0.25 inch. is equal to + or minus 0.25 inch. Multiplied by 1 ft divided by 12 inches, so using dimensional analysis. is equal to plus or minus 1 divided by 48. Feet However, we need to consider the equation. H divided by 12 plus S is equal to 3 divided by S again. So when we do that, we will find that H is equal to, when we rearrange our expression for H divided by 12 plus S is equal to 3 divided by S, when we rearrange it to isolate and solve for S, you will find that H is equal to 12 plus S multiplied by 3 divided by S. Fantastic. So, Which when we like which when we start to solve, we will find because once again we know that S is going to be a value. Of 9, so when we plug in a value of 9 in for us, and we basically We're trying to simplify this expression. So when we do that, we will find that H is equal to 36 divided by S. Plus 3, but now we need to take the derivative, so we haven't plugged in a value of 9 in for S, so we can just note that S means that it's 9, but we haven't plugged in that value yet. Eventually we will. But now we need to take the derivative of this expression for H, so we need to take DH by DS is equal to negative 36 divided by Squared. So when we rearrange this expression to isolate and solve for DH, we'll find that DH is equal to -36 divided by Squared multiplied by DS. So that will mean that DH is equal to -36 divided by 9 to the power of 2, since we plug in a value of 9 in for S, multiplied by plus or minus 1 divided by 48. Which will give us an answer which is equal to -36 divided by 81, multiplied by plus or minus. So we're gonna be taking negative 36 divided by 81 multiplied by plus or minus. 1 divided by 48. So when you take 1 divided by 48, and then when we simplify and evaluate one last time, you'll get an answer of plus or minus. So it's gonna be equal to plus or minus 1 divided by 108 ft, and that is our final answer. Hooray, we did it. So, therefore, without a doubt, the possible error is plus or minus 1 divided by 108 ft. So that will mean looking at our multiple choice answers the correct answer, without a doubt has to be multiple choice answer letter C. Which states that the height is 7 ft and the possible error in height is plus or minus 1 divided by 108 ft. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Key Concepts

Similar Triangles

Proportional Relationships

Error Estimation

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