Chapter 46, Problem 57A

To determine

(a)

To solve the given formula for $D$.

Answer to Problem 57A

  $D=\frac{L-\left(1.57d+2x\right)}{1.57}$.

Explanation of Solution

Given information:

Given that, the formula is used in machine trade calculations, $L=3.14\left(0.5D+0.5d\right)+2x$

  

Calculation:

We have been given the formula, $L=3.14\left(0.5D+0.5d\right)+2x$

We need to solve the formula for $D$.

The formula states that,

  $\Rightarrow L=3.14\left(0.5D+0.5d\right)+2x$

On expanding the parenthesis, we get

  $\Rightarrow L=1.57D+1.57d+2x$

On subtracting the sum of $1.57d+2x$ in both sides, we get

  $\Rightarrow L-\left(1.57d+2x\right)=1.57D$

Dividing by $1.57$ on both sides, we get

  $\Rightarrow D=\frac{L-\left(1.57d+2x\right)}{1.57}$

Hence, the formula that is $L=3.14\left(0.5D+0.5d\right)+2x$ can be solved for $D$ as $D=\frac{L-\left(1.57d+2x\right)}{1.57}$.

To determine

(b)

To perform the mathematical operation on given formula to get the value of $d$.

Answer to Problem 57A

The formula that is $L=3.14\left(0.5D+0.5d\right)+2x$ can be solved for $d$ as $d=\frac{L-\left(1.57D+2x\right)}{1.57}$.

Explanation of Solution

Given information:

Given that, the formula is used in machine trade calculations, $L=3.14\left(0.5D+0.5d\right)+2x$

  

Calculation:

We have been given the formula, $L=3.14\left(0.5D+0.5d\right)+2x$

We need to solve the formula for $d$.

The formula states that,

  $\Rightarrow L=3.14\left(0.5D+0.5d\right)+2x$

On expanding the parenthesis, we get

  $\Rightarrow L=1.57D+1.57d+2x$

On subtracting the sum of $1.57D+2x$ in both sides, we get

  $\Rightarrow L-\left(1.57D+2x\right)=1.57d$

Dividing by $1.57$ on both sides, we get

  $\Rightarrow d=\frac{L-\left(1.57D+2x\right)}{1.57}$

Hence, the formula that is $L=3.14\left(0.5D+0.5d\right)+2x$ can be solved for $d$ as $d=\frac{L-\left(1.57D+2x\right)}{1.57}$.

To determine

(c)

To solve the formula for $d$.

Answer to Problem 57A

  $x=\frac{L-\left(1.57D+1.57d\right)}{2}$.

Explanation of Solution

Given information:

Given that, the formula is used in machine trade calculations, $L=3.14\left(0.5D+0.5d\right)+2x$

  

Calculation:

We have been given the formula, $L=3.14\left(0.5D+0.5d\right)+2x$

We need to solve the formula for $x$.

The formula states that,

  $\Rightarrow L=3.14\left(0.5D+0.5d\right)+2x$

On expanding the parenthesis, we get

  $\Rightarrow L=1.57D+1.57d+2x$

On subtracting the sum of $1.57D+1.57d$ in both sides, we get

  $\Rightarrow L-\left(1.57D+1.57d\right)=2x$

Dividing by $2$ on both sides, we get

  $\Rightarrow x=\frac{L-\left(1.57D+1.57d\right)}{2}$

Hence, the formula that is $L=3.14\left(0.5D+0.5d\right)+2x$ can be solved for $x$ as $x=\frac{L-\left(1.57D+1.57d\right)}{2}$.