How would I find angle B for this bridge? What formula would I use?
2 answers:
The bridge attached is drawn according to given dimensions, and it doesn't look right. Please double check the given dimensions.
Calculations:
Horizontal part of bottom chord below the 70 degree triangle
= 15.1*cos(70) = 5.16 (which is a major prt of the 6.3 units.
Height of vertical pieces DF and EH
= 15.1*sin(70) = 14.19
Note that structurally, DF and EH do not help in reducing stress on the bridge, since they are perpendicular to the bottom chord.
Therefore
angle B = atan(14.19/(6.3-5.16))=85.41 degrees
I believe the whole geometry does not look right, esthetically, and structurally, since the compression members are much longer than the tension members in the middle. (The vertical members carry no force.)
If you can review the input data, or post a new question, I will be glad to help.
Calculations:
Horizontal part of bottom chord below the 70 degree triangle
= 15.1*cos(70) = 5.16 (which is a major prt of the 6.3 units.
Height of vertical pieces DF and EH
= 15.1*sin(70) = 14.19
Note that structurally, DF and EH do not help in reducing stress on the bridge, since they are perpendicular to the bottom chord.
Therefore
angle B = atan(14.19/(6.3-5.16))=85.41 degrees
I believe the whole geometry does not look right, esthetically, and structurally, since the compression members are much longer than the tension members in the middle. (The vertical members carry no force.)
If you can review the input data, or post a new question, I will be glad to help.
The numbers on your diagram are inconsistent. You would need to start with the numbers you know for sure. Since I can't tell what those are, I can only give an outline of how you would solve the problem.
On your diagram, I have added the labels p, q, r, s to make it easier to talk about the different line segments.
The Pythagorean theorem tells you
pr² = pq² + qr²
rs² = qs² + qr²
The definitions of trigonometric ratios tell you
tan(P) = qr/pq . . . . . where angles P, R, S are the interior angles of ΔPRS
sin(P) = qr/pr
tan(S) = qr/qs
sin(S) = qr/rs
Once you decide what measurements you know for sure, then you can choose the formulas here for which you know 2 of the 3 variables and use the formula to find the remaining variable. Repeat the process until you have solved the whole figure.
It looks like your bridge is symmetrical about the center, so my angle S is your angle "b".
_____
For example, if we take the 15' and the 70° dimensions to be given (the ones you know for sure), then you can use the equations
tan(P) = qr/pq . . . . . to find pq
sin(P) = qr/pr . . . . . . to find pr (your "x")
tan(70°) = (15 ft)/pq
pq = (15 ft)/tan(70°) ≈ (15 ft)/2.74748 ≈ 5.45955 ft
sin(70°) = (15 ft)/pr
pr = (15 ft)/sin(70°) ≈ (15 ft)/0.939693 ≈ 15.9627 ft
From this you can determine the length qs to be
qs = ps - pq
qs = 6.3 ft - 5.45955 ft = 0.840446 ft
and that lets you determine angle S from
tan(S) = qr/qs
tan(S) = (15 ft)/(0.840446 ft) ≈ 17.8477
S = arctan(17.8477) = 86.79° . . . . . . . this is your angle "b" for the assumptions used here
The length of segment rs (your "a") can be determined either from the Pythagorean theorem formula or the trig formula.
rs = √(qs² + qr²) = √((0.840446 ft)² + (15 ft)²) ≈ √225.706 ft ≈ 15.0235 ft
_____
On the other hand, if you assume the bridge height is 15 feet and the support structure is symmetrical, as you seem to have shown*, then
pq = qs = (6.3 ft)/2 = 3.15 ft
tan(S) = 15/3.15 ≈ 4.76190
S = arctan(4.76190) = 78.14° . . . . . inconsistent with 70°
For this case, x = a
x = a = √(15² + 3.15²) ≈ 15.3272 . . . feet
___
* The single arc marking the base angles is a symbol that usually is used to mean the angles marked with that symbol are all congruent. This would mean all of the base angles of your 15' high bridge will be 78.14°, including your angle "b".
On your diagram, I have added the labels p, q, r, s to make it easier to talk about the different line segments.
The Pythagorean theorem tells you
pr² = pq² + qr²
rs² = qs² + qr²
The definitions of trigonometric ratios tell you
tan(P) = qr/pq . . . . . where angles P, R, S are the interior angles of ΔPRS
sin(P) = qr/pr
tan(S) = qr/qs
sin(S) = qr/rs
Once you decide what measurements you know for sure, then you can choose the formulas here for which you know 2 of the 3 variables and use the formula to find the remaining variable. Repeat the process until you have solved the whole figure.
It looks like your bridge is symmetrical about the center, so my angle S is your angle "b".
_____
For example, if we take the 15' and the 70° dimensions to be given (the ones you know for sure), then you can use the equations
tan(P) = qr/pq . . . . . to find pq
sin(P) = qr/pr . . . . . . to find pr (your "x")
tan(70°) = (15 ft)/pq
pq = (15 ft)/tan(70°) ≈ (15 ft)/2.74748 ≈ 5.45955 ft
sin(70°) = (15 ft)/pr
pr = (15 ft)/sin(70°) ≈ (15 ft)/0.939693 ≈ 15.9627 ft
From this you can determine the length qs to be
qs = ps - pq
qs = 6.3 ft - 5.45955 ft = 0.840446 ft
and that lets you determine angle S from
tan(S) = qr/qs
tan(S) = (15 ft)/(0.840446 ft) ≈ 17.8477
S = arctan(17.8477) = 86.79° . . . . . . . this is your angle "b" for the assumptions used here
The length of segment rs (your "a") can be determined either from the Pythagorean theorem formula or the trig formula.
rs = √(qs² + qr²) = √((0.840446 ft)² + (15 ft)²) ≈ √225.706 ft ≈ 15.0235 ft
_____
On the other hand, if you assume the bridge height is 15 feet and the support structure is symmetrical, as you seem to have shown*, then
pq = qs = (6.3 ft)/2 = 3.15 ft
tan(S) = 15/3.15 ≈ 4.76190
S = arctan(4.76190) = 78.14° . . . . . inconsistent with 70°
For this case, x = a
x = a = √(15² + 3.15²) ≈ 15.3272 . . . feet
___
* The single arc marking the base angles is a symbol that usually is used to mean the angles marked with that symbol are all congruent. This would mean all of the base angles of your 15' high bridge will be 78.14°, including your angle "b".
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