Chapter 5.4, Problem 45E

To determine

To find: The cost of building of a pier.

Answer to Problem 45E

The total cost of construction of pier of length 50-foot is $23570.23\text{dollars}$ .

Explanation of Solution

Given data: A 50-foot pier is to be built for which the construction cost increases as $C\left(x\right)=100\sqrt{x}$ dollar per foot, where x is the distance from the shore.

Method/Formula used:

A pier can be constructed at the cost $c\left(t\right)\text{dollar}\text{/ft}$ .Then the total cost C of construction of pier of length L is given by

  $C\left(L\right)={\displaystyle \underset{0}{\overset{L}{\int }}c\left(x\right)dx}$ … (1)

  $Formula\text{:}{\displaystyle \int x^{n}dx=}\frac{x^{n+1}}{n+1}$

Calculations:

A 50-foot pier is to be built for which the construction cost increases as $C\left(x\right)=100\sqrt{x}$ dollar per foot, where x is the distance from the shore. A 50-foot pier is to be constructed.

Substitute 50 for L in result (1), the total cost of constructing 50- feet pier is

  $\begin{array}{c}C\left(50\right)={\displaystyle \underset{0}{\overset{50}{\int }}\left(100\sqrt{x}\right)dx}\\ ={\left[100\left(\frac{x^{3/2}}{3/2}\right)\right]}_0^{50}\\ =\frac{200}{3}{\left[x^{3/2}\right]}_0^{50}\\ =\frac{200}{3}\left[{\left(50\right)}^{3/2}-{\left(0\right)}^{3/2}\right]\end{array}$

Solve further,

  $\begin{array}{c}C\left(50\right)=\frac{200}{3}\left[{\left(50\right)}^{3/2}-{\left(0\right)}^{3/2}\right]\\ =\frac{200}{3}{\left(50\right)}^{3/2}\\ =23570.23\text{dollars}\end{array}$

Therefore, the total cost of construction of pier of length 50-foot is $23570.23\text{dollars}$ .