Question

A builder wants to build the bridge whose cross section is shown in the diagram. Two companies offer simple bids on building the bridge according to the distance b and the number of​ I-beams. If the cross section could be redesigned for cost reasons in terms of​ b, what is the distance b for which the companies would have the same bid on the​ project?
EMPIRE DESIGN
​BUILD-EM-UP CO.
Bid form
Bid form
​ Proposal:
​ Proposal:
​$300,000b
​$320,000b
​ $60,000 per​ I-beam
​ $40,000 per​ I-beam

Answer 1

The length of the bridge is the distance from the beginning to the end.

The distance b between each beam is 9ft.

Let:

$I \to$ I-Beam

$d_I \to$ distance between I beam and the bridge

$b \to$ distance between each I beam

Given that:

$I = \frac 34 ft$

$d_I = 3 ft$

$Length = 55ft\ 6in$ --- length of the  bridge

From the diagram (see attachment), there are: 6 I-beams.

So, the length of the 6 I-beams is:

$L_1 = 6 \times I$

$L_1 = 6 \times \frac 34$

$L_1 = \frac {18}4$

$L_1 = 4.5ft$

There are 2 I-beams beside the bridge

So, the distance between the 2 I-beams and the bridge is:

$d_1 =2 \times d_I$

$d_1 =2 \times 3ft$

$d_1 =6ft$

There are 5 spaces between the I-beams

So, the length of the total spaces is:

$L_2 = 5 \times b$

$L_2 = 5b$

The total length is:

$Length = L_1 + d_1 + L_2$

So, we have:

$4.5ft + 6ft + 5b = 55ft\ 6in$

Collect like terms

$5b = 55ft\ 6in - 4.5ft - 6ft$

$5b = 44.5ft\ 6in$

Convert inches to feet

$5b = 44.5ft\ + \frac{6}{12}ft$

$5b = 44.5ft\ + 0.5ft$

$5b = 45ft$

Divide both sides by 5

$b = 9ft$

Hence, the distance (b) between each beam is 9ft.

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