Chapter 46, Problem 51A

To determine

(a)

To perform the mathematical operation as indicated.

Answer to Problem 51A

The formula $\left(1\right)$ that is $A=ab$ can be solved for $a$ as $a=\frac{A}{b}$.

Explanation of Solution

Given information:

Given that, the dimensions shown in the figure can be found using these two formulas.

  $\left(1.\right)A=ab\left(2.\right)d=\sqrt{a^2+b^2}$

  

Calculation:

We have been given the formula, $\left(1.\right)A=aband\left(2.\right)d=\sqrt{a^2+b^2}$ to determine the dimensions shown in the figure

We need to solve the formula $\left(1\right)$ for $a$.

Formula $\left(1\right)$ states that,

  $\Rightarrow A=ab$

Dividing by $b$ on both sides, we get

  $\Rightarrow a=\frac{A}{b}$

Hence, the formula $\left(1\right)$ that is $A=ab$ can be solved for $a$ as $a=\frac{A}{b}$.

To determine

(b)

To solve formula $\left(1\right)$ for $b$.

Answer to Problem 51A

  $b=\frac{A}{a}$

Explanation of Solution

Given information:

Given that, the dimensions shown in the figure can be found using these two formulas.

  $\left(1.\right)A=ab\left(2.\right)d=\sqrt{a^2+b^2}$

  

Calculation:

We have been given the formula, $\left(1.\right)A=aband\left(2.\right)d=\sqrt{a^2+b^2}$ to determine the dimensions shown in the figure.

We need to solve the formula $\left(1\right)$ for $b$.

Formula $\left(1\right)$ states that,

  $\Rightarrow A=ab$

Dividing by $a$ on both sides, we get

  $\Rightarrow b=\frac{A}{a}$

Hence, the formula $\left(1\right)$ that is $A=ab$ can be solved for $b$ as $b=\frac{A}{a}$.

To determine

(c)

To perform the mathematical operation on formula $\left(2\right)$ for $a$ as indicated.

Answer to Problem 51A

  $a=\sqrt{d^2-b^2}$.

Explanation of Solution

Given information:

Given that, the dimensions shown in the figure can be found using these two formulas.

  $\left(1.\right)A=ab\left(2.\right)d=\sqrt{a^2+b^2}$

  

Calculation:

We have been given the formula, $\left(1.\right)A=aband\left(2.\right)d=\sqrt{a^2+b^2}$ to determine the dimensions shown in the figure.

We need to solve the formula $\left(2\right)$ for $a$.

Formula $\left(2\right)$ states that,

  $\Rightarrow d=\sqrt{a^2+b^2}$

Taking square of both sides, we get

  $\Rightarrow d^2=a^2+b^2$

Subtracting $b^2$ on both sides, we get

  $\Rightarrow d^2-b^2=a^2$

Taking the square root on both sides, we get

  $\Rightarrow a=\sqrt{d^2-b^2}$

Hence, the formula $\left(2\right)$ that is $d=\sqrt{a^2+b^2}$ can be solved for $a$ as $a=\sqrt{d^2-b^2}$.

To determine

(d)

To solve formula $\left(2\right)$ for $b$.

Answer to Problem 51A

  $b=\sqrt{d^2-a^2}$.

Explanation of Solution

Given information:

Given that, the dimensions shown in the figure can be found using these two formulas.

  $\left(1.\right)A=ab\left(2.\right)d=\sqrt{a^2+b^2}$

  

Calculation:

We have been given the formula, $\left(1.\right)A=aband\left(2.\right)d=\sqrt{a^2+b^2}$ to determine the dimensions shown in the figure.

We need to solve the formula $\left(2\right)$ for $b$.

Formula $\left(2\right)$ states that,

  $\Rightarrow d=\sqrt{a^2+b^2}$

Taking square of both sides, we get

  $\Rightarrow d^2=a^2+b^2$

Subtracting $a^2$ on both sides, we get

  $\Rightarrow d^2-a^2=b^2$

Taking the square root on both sides, we get

  $\Rightarrow b=\sqrt{d^2-a^2}$

Hence, the formula $\left(2\right)$ that is $d=\sqrt{a^2+b^2}$ can be solved for $b$ as $b=\sqrt{d^2-a^2}$.