**************************************** Problem of the Week, February 7-11
Last week, Jill and Jimmy were attempting to figure out how wide a river was so that they could begin plans to build a bridge. They determined that the river was 46 meters wide. Now they have to give an estimate as to the cost of the bridge, which means they have to know how long the bridge will be. Here's the catch:
They want to build a bridge, starting 10 meters inland from the edge of the river, to the top of an already-existing abuttment on the other side of the river. The abuttment is 16 meters high and 7 meters from the edge of the river.
How long will the bridge be? Extra: What is the angle of elevation of the bridge?
As the bridge saga continues, we received correct responses from:
Nicole Dunlap Grade 10, Shaler Area High School rachel laughlin Grade 10, Shaler High School Ben Ladik, grade 12, and Katie Getsy, grade 9; Steel Valley High School, Pa. Ryan Ferchak and Bob Gallagher, grade 9, Steel Valley High School Derek Morrison and Jill Grobelski, grade 9, Steel Valley High School. Pa. Scott Pisula and Andy Fedoris, grade 9, Steel Valley High School, Pa. Kristie Tunney and Tonya Kosko, grade 9, Steel Valley Hogh School, Pa. Susan Quan, grade 7, Masterman School, Philadelphia Don Mahaney Grade 10, Shaler Area High School Tim Thiel Grade 10, Shaler Area High School Jen Eskra Grade 10, Shaler Area High School Jennifer Strong Grade 10, Shaler Area High School Mike Orehowsky Grade 10, Shaler High School Thomas Niebel Grade 10, Shaler High Dan Schrom Grade 10, Shaler Area High School Gino Perrotte Grade 10, Shaler Area High School Alyssa McGrath Grade 10, Shaler Area High School Matt Bouton Grade 9, Steel Valley High School, Pa. Tim Schnurr Grade 10, Fairfield HS, Connecticut Other solvers from Fairfield High School: Grade 9 Grade 10 Amanda Adams Elissa Colter Hilary Aleksa Matt Lucas Lindsey Becker Morgan May Amy Decrescenzo Natalie Painchaud Brendan Hogan Ryan Phelan Molly MacDonald Nghiem Vu Kathy Medlin Rebecca Naughton Agata Raszczyk-Lawska Phil Rossi Katy Ruff Allison Sullivan Heather Wilcox Jan Wilson Sandy Broten Grade 10, Edgerton High School, WI Jeremy Goede Grade 10, Edgerton High School, WI
If the abutment was in the river: | ? | 16m | | _________|____46m__________________________|______ 10m 7m
a^2+b^2=c^2 = 58m
If the abutment was on land: | ? | | 16m | _________|__________________________________|________| 10m 46m 7m
a^2+b^2=c^2 = 65m
This is all assuming that the bridge has a diagonal slope, lowest at the bank and highest at the abutment. if the bridge were straight across, or started at a slope then flattened out, the distance would change.
Rachel Laughlin Shaler Area High School 10th Grade
Submitted by Ben Ladik, grade 12, and Katie Getsy, grade 9; Steel Valley High School, Pa.
In order to start this problem, you must first know the distance from our start and end points, which is 63 meters. Next, we took the height, 16 m, of the abuttment and used it as one of our legs and 63 m as the other. We uaed the Pythagorean Theorem to find the length of the span, which was 65 m.
63 squared + 16 squared = x squared 4225 = x squared the square root of 4225 = x 65 = x
We also scaled down the meters to millimeters on our drawing, then measured,and still got 65.
Notice, the sum of the bases of the bridge is 63 meters. The abuttment is 16 meters high. From that knowledge, you can form the two legs of the triangle. Connect the new area for the abuttment to the existing one to find the hypotenuse. At this point the hypotenuse is unknown, but by using the Pythagorean Theorem you can find the hypotenuse's length. (a squared + b squared = c squared) Take 63 and square it; you should get 3969. Take 16 and square it; you should get 256. Take the two and add them. You now have 4225. Take the square root of that to get the measure of c. c = 65
Submitted by Derek Morrison and Jill Grobelski, grade 9, Steel Valley High School. Pa.
First, we added 46 meters, the width of the river, to 10 meters, the distance inland from the edge of the river, to 7 meters, the distance from the other edge of the river to the abuttment. The sum of the three was 63.
Second, we knew that the abuttment was 16 meters high and 7 meters from the edge of the river.
We now could form a triangle by using 16 meters as the shorter leg and 63 meters as the longer leg. Now all that we needed was to find the hypotenuse or length of the bridge. By using the Pythagorean Theorem, leg one squared + leg two squared = hypotenuse squared, we could determine that the hypotenuse or length of the bridge is 65 meters. 16 squared + 63 squared = h squared 256 + 3963 = 4225 the square root of 4225 = 65 Therefore, the length of the bridge is 65 meters.
Solution from Susan Quan, grade 7, Masterman School, Philadelphia
Assuming the bridge follows a straight line from a point 10 ft. from the river to the front edge of an abuttment 16 ft. high, the bridge would be 65 ft. long. The bridge is the hypotenuse of a right triangle with legs 63 ft. and 16 ft. so using the Pythagorean Theorem gives the length of the bridge.
16^2 + 63^2 = x^2 where x is the length of the bridge
The length of the bridge can be determined by using the Pythagorean theorem. The first side, which we will call A, has a length is 63 meters.(The river is 46 meters wide, plus the 10 meters inland on one side and the 7 meters inland on the other.) The second side, which we will call B, is 16 meters long, and that corresponds to the 16 meter height of the abutment. Note:Side A is perpendicular to side B because of the first minimum theorem. The third side, C, will be the length of the bridge. The numerical value for C is found by taking both sides A and B, squaring them, and adding the new values together. The result, 4225, is C squared. Finally, taking the square root of 4225 you get a result of 65 meters; which is the length of the bridge. To find the angle of elevation, the trigonometric functions of sine, cosine, and tangent must be used:
The sine of angle 1 = 16/65 = .264 The cosine of angle 1 = 63/65 = .969 The tangent of angle 1 = 16/63 = .254
* Since I have not done any of this in geometry class I was forced to look ahead in the book a few chapters.
Taking a calculator, and using any of the three values above, all that is needed to find the measure of the angle is to push the key that will give the reciprocal value of sine, cosine, or tangent. After doing this for each of the values, I found the measure of the angle of elevation to be 14.25 degrees.
( I am assuming that the bridge is flat and not arched in the middle like a normal bridge)
As shown by the picture a right triangle is created from the abuttment, the existing lenght, and the bridge which turns out to be the hypotenuse. The height of the abuttment is given as 16 meters and is a leg of the triangle created. the other leg of the triangle is found by adding the width of the river which is 46 meters, the distance the abuttment is away from the river which is 7 meters, and the distance the bridge will be from the edge of the river which is 10 meters. The distance calculated from the addition is 63 meters. To find the length of the bridge(which is the hypotenuse) I used the Pythagoreon Theorem.(a^2+b^2=c^2) Let a=16 meters and Let b=63 meters. I used the following calculations to find c: 16^2+63^2= c^2 c= (16^2+63^2)^.5 c= (4225)^.5 c= 65 meters The length of the bridge(or the hypotenuse of the triangle) is found to be 65 meters long. To find the angle of elevation (which is the angle whose sides in the triangle are the hypotenuse and the 63 meter leg) can be found by using a simple trigometric function. To find the angle of elevation for the bridge I took the inverse tangent(arc tangent) of the two legs of the triangle: a= 16 meters b= 63 meters angle of elevation= arc tan (16meters/63 meters) angle of elevation= 14.25 degrees ( the 14.25 degree angle of elevation is larger than the normal angle of elevation for a bridge would be in real life.) final answer: length of bridge= 65 meters angle of elevation= 14.25 degrees
To figure out the length of the bridge, Jill and Jimmy must use the Pythagorean theorem. This theorem is (a*a)+(b*b)=c*c. a is 16m, b is 63m (add 46m+10m+7m). After squaring these variables, the problem is 256+4096=c*c. c*c ends up being 4225, so c=65. Therefore the length of the bridge is 65 meters. But Jimmy and Jill should buy an extra meter or so of construction material in case of constructional error.
My name is Jennifer Strong and I am in a tenth grade honors geometry class at Shaler Area High School. Here is my answer for POW: (n^2 = n squared. r* = r degrees) The length of the bridge can be very easily determined using the Pythagorean Theorem. However, it must first be assumed that the abutment meets the river bank at a 90 degree angle. Second, all lines (the abutment, the bridge, and the bank to river to bank) must be assumed straight also. Since a^2 + b^2 = c^2, substituting the abutment for a (16 m) and the bank-river-bank for b (10+46+7=63 m), one gets 16^2 + 63^2 = 256 + 3939 = 4225 = c^2. c then equals the square root of 4225. Therefore, the length of the bridge, c , is 65 meters.
Knowing this, there are several ways to find the angle of elevation. These methods include: tan r* = a/b, cos r* = b/c, or sin r*=a/c Keeping the variables the same, tan r* = a/b seems the most applicable because we are given a and b in the original problem.
a/b = 16/63 = tan r* Then taking tan^-1 of each side, we are left with r* equals approximately 14.25*.
cos r*= b/c or sin r* = a/c would both be solved in a similar fashion.
Assume that the river is straight. The equation to find the length of the bridge is a squared times b squared = c squared. Which is sixty-three squared times sixteen squared = c squared. a is the length of the river (46m) plus the ten meters from the start of the bridge to the river and the seven meters to the abutment. b is the sixteen meters is the hight of the abutment. c is the length of the bridge which is the hypotenuse of the right triangle formed by the abutment and the ground length. When you work out the equation it gives you an answer of 65 meters.
3969 + 256 = c squared. when you take the square-root of the left side you get an answer of 65 meters. To find the angle of elevation of the bridge you have to use a tangent. To come about with this answer I took a/b and pressed the tan-1 key on my calculator and got an answer of 14.25 degrees
The length of the bridge is sixty-five meters, assuming the river is straight, the abuttment is at a right angle to the shore, and the bridge starts at ground level(on the shore opposite the abuttment). Then, the length across the river, up the abuttment, and the bridge connecting them is a right triangle. This allows the Pythagorean Theorem to be used to find the length of the bridge. Substituting sixteen for a(the height of the abuttment), and sixty-three for b(the distance across the river, inshore ten meters, and inshore to the abuttment) into the equation of: a squared+ b squared= c squared. This gives me the equation: 256+3969= c squared. Therefore, c squared is 4225, and its square root is 65 meters, the length of the bridge.
To find the angle of elevation of the bridge, I was forced to use a tangent. I found the tangent by dividing 16(side a) by 63(side b). This gave me a value of .2539. Then by pressing the tan-1 key on my calculator, it gave me the angle of elevation which was 14.25 degrees.
Thomas Niebel Grade 10 Shaler High
******************************* From Dan Schrom <email@example.com>
First, to find the length of the bridge, you must use the Pythagorean Theorem. The PT says that A^2 + B^2 = C^2. In other words, the sum of the squares of the two legs in a right triangle equal the the square of the hypotnuse. Although you must say that the abuttment forms a 90 degree angle. This holds true because
63^2 + 16^2 = 65^2. After you find the length of the hypotnuse(length of bridge), you can then find the angle of elevation by using sin. sin = a = a/c = 16/65 = 14.25 degrees. ! 65 ! ! ________/_____________________________/____!
10 + 43 + 7
Therefore, the length of the bridge equals 65meters and the angle of elevation is 14.25 degrees.
Gino Perrotte I am a tenth grade honors geometry student. This is the first time that I have used the NEXT computer system.
To find out how long the bridge will be, I used the pythagorean theorem. To use this theorem you must assume that the abuttment is perpendicular to the ground. The legs of the right triangle will be the river and the abuttment. The length of the abuttment is given to be 16 meters. The length of the river is given to be 46 meters. Then you have to add 10 meters and 7 meters to the river length because that is how far in they want the brigde to start. Then I applied the Pythagorean theorem, which is c^2= a^2 + b^2. Doing the math: C^2= 16^2 + 63^2; c^2= 4225; take the square root of both sides and you get c= 65. The hypotenuse of the triangle is 65 meters long, and the hypotenuse is the bridge. Therefore the bridge is 65 meters long. To find the angle of elevation, the equation a/c sin^-1 is used. In the triangle the abuttment is side a, the river is side b, and the bridge is side c. Substituting the numbers into the equation you get: (16/65) sin^-1= 14.25 degrees The angle of elevation is 14.25 degrees. | | 16m |
My name is Alyssa McGrath and this is my solution to the problem of the week. I attend Shaler Area High School in Pittsburgh, PA. I am in 10th grade and I'm in Honors Geometry.
For the 1st part of the problem I found the length of the bridge. To find it I added 10m+46m+7m. The sum is 63m (the base of the triangle). The hight is 16m as stated in the problem. I assumed the angle here would be 90 degrees, so I used the Pythagorean Theorem to find the length of the hypotenuse (final side). The theorem gives you the equation: a^2+b^2=c^2
When the values were filled in it became: 16^2+63^2=c^2 4225=c^2 65=c This means the hypotenuse is 65 meters. To find the angle of elevation I looked in my book to find the formulas for: Sin = a/c =0.246 Cos = b/c =0.969 Tan = a/b =0.254 I then looked on the table of Trigonometric Ratios. When the three numbers matched up the chart said the angle of elevation is 14 degrees. c !
_________ _______! ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ b I couldn't draw the hypotenuse but it forms the angle of elevation.
Looking at the bridge from down river, you can see it forms a right triangle. You can find the length of the bridge by finding the lengths of the two legs then using the Pythagorean Theorem. You can use this because one of the angles in the triangle is a right angle. The two legs are the overall distances at the base, which is 7 + 46 + 10, and the abuttment, which is 16 meters high. Therefore, 16^2 + 63^2 = the square of the bridge length. Next, 256 + 3969 = x^2, 4225 = x^2, and x = 65. So the bridge is 65 meters long. Tim Schnurr, Fairfield HS, grade 10.
Solution by Sandy Broten, Grade 10, Edgerton High School, WI
The bridge forms a right triangle. We know two dimentions of the triangle (height is 16 meters and span is 63 meters) so knowing the Pythagoream Theorem you can find the length of the third side. 2 2 2 16 + 63 = x 256 + 3969 = 4225 the square root of 4225 = 65, so the bridge will be 65 m long.
Solution by Jeremy Goede, Grade 10, Edgerton High School, WI
The river is 46 m wide, add 17 m for where you start the bridge and where the abuttment is which equals 63 m. This is a right triangle, so sou square 63 and 16 and add them to find the square of the length of the bridge. 63^2 + 16^2 = 4225 and the square root of 4225 is 65. The bridge is 65 m long.
To find the angle of elevation,
Let x be the angle of elevation.
tan x = opposite leg / adjacent leg = 16/63 = .253968254