Question

The Gateway Arch in St. Louis was designed by Eero Saarinen and was constructed using the equation y 211.49-20.96 cosh 0.03291765x for the central curve of the arch, where x and y are in meters and xl s 91.20. (a) Graph the central curve. y y 200 200 150 150 100 100 50 50l o-100 -50 50 x 100 -100 -50 50 100 y 200 200 150 150- 100 100 50 50 o 200 200 150 150 100 100 50 50 -100 -50 50 100 -100 -50 50 100 (b) What is the height of the arch at its center? (Round your answer to two decimal places.) 190.53 m (c) At what points is the height 70 m? (Round your answers to two decimal places.) (x, y)-57.74,140 (smaller x-value) X (x, y) - (larger x-value) (d) What is the slope of the arch at the points in part (c)? (Round your answers to one decimal place.) (at the point with smaller x-value) (at the point with larger x-value)

Answer 1

PART(c):

The height function is given by

y(x211.49 20.96 cosh(0.03291765r)

put y = 70

70 211.49 20.96 cosh(0.03291765)

70 211.49 20 96e0.03291765 e 0.03291765 2

use the plugin

0.03291765 a=

1/a 70 211.49 20.96 2

70-211.49-20.96a1/a 2

1 |-141.49 -10.48 a - a

141.49 a 10.48

a 1 - 13.5009541985

a 1 13.5009541985a

After solving above quadratic eqn

a13.42647, a 0.07448

Now plug back the substitution

In 13.42647 0.03291765x 78.9007760763 13.42647 x 0.03291765

$e^{0.03291765x} = 0.07448 \Rightarrow x = -78.900670188$

Now find y values

Rounding off to 2decimals

y(-78.90) 70.00

Hence the points are

$(-78.90,\ 70.00) \Rightarrow smaller$

$(78.90,\ 70.00) \Rightarrow larger$

PART(d):

y(x211.49 20.96 cosh(0.03291765r)

diff wrt x

$y'(x) = 0-20.96(0.03291765)\sinh(0.03291765x)$

$y'(x) = -0.689953944\sinh(0.03291765x)$

Hence the slope is given by

$m = y'(x) = -0.689953944\sinh(0.03291765x)$

put the smaller value of x

$m = y'(-78.90) = -0.689953944\sinh(0.03291765 \times -78.90) = 4.6$

put the larger value of x

$m = y'(78.90) = -0.689953944\sinh(0.03291765 \times 78.90) = -4.6$

I hope this answer helps,
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