Chapter 8.1, Problem 74AYU

Estimating the Width of the Mississippi River A tourist at the top of the Gateway Arch (height, 630 feet) in St. Louis, Missouri, observes a boat moored on the Illinois side of the Mississippi River 2070 feet directly across from the Arch. She also observes a boat moored on the Missouri side directly across from the first boat (see diagram). Given that $B={\cot }^{-1}\frac{67}{55}$, estimate the width of the Mississippi River at the St. Louis riverfront.

Source: U.S. Army Corps of Engineers

To determine

The width of the Mississippi River at the ST. Louis riverfront.

Answer to Problem 74AYU

Mississippi River is approximately 1650 feet wide at the St Louis riverfront.

Explanation of Solution

Given:

A tourist at the top of the Gateway Arch (height, 630 feet) in St. Louis, Missouri, observes a boat moored on the Illinois side of the Mississippi River 2070 feet directly across from the Arch. She also observes a boat moored on the Missouri side directly across from the first boat (see diagram). Given that $B={\cot }^{-1}\frac{67}{55}$.

Formula used:

$\text{tanA }=\frac{\text{opp side}}{\text{base}}$

Calculation:

From the figure above,

$\tan \left(A+B\right)=\frac{2070}{630}$

$A+B={\tan }^{-1}\frac{2070}{630}$

$A={\tan }^{-1}\frac{2070}{630}-B$

$A={\tan }^{-1}\frac{2070}{630}-{\cot }^{-1}\frac{67}{55}$

$A\approx 33.69°$

Let $x$ be the distance between the Arch and the boat on the Missouri side.

$\tan A=\frac{x}{630}$

$x=630*\text{ tan}33.69$

$x=420$

Hence the Mississippi River is approximately $2070-420=1650$ feet wide at the St Louis riverfront.