# A rectangle has dimensions 0.7x and 5 - 3x. What value of x gives the maximum area and what is the maximum area?

Mar 29, 2016

at $x = 0.8 \dot{3}$
$1.458 \dot{3} u n i {t}^{2}$

#### Explanation:

Given dimensions of the rectangle are: $0.7 x \mathmr{and} \left(5 - 3 x\right)$

Area of the rectangle$A = l \times b$, insering given values we obtain
Area of the rectangle$A = 0.7 x \times \left(5 - 3 x\right)$,
$A = 3.5 x - 2.1 {x}^{2}$,
To maximize the area the first derivative is to be set equal to $0$
or $3.5 - 2.1 \times 2 x = 0$, solving for $x$ we obtain
$3.5 - 4.2 x = 0$
or $x = \frac{3.5}{4.2} = \frac{5}{6}$
$\implies x = 0.8 \dot{3}$
To confirm that it is a maximum we need to evaluate second derivative at the obtained value of $x$
Second derivative $= - 4.2$. Since it is a negative quantity, therefore it is a maxima.

Maximum Area $= 0.7 \times 0.8 \dot{3} \times \left(5 - 3 \times 0.8 \dot{3}\right)$
$= 0.58 \dot{3} \times 2.5$
$= 1.458 \dot{3} u n i {t}^{2}$

Mar 29, 2016

Maximum area = $1.4361 {\text{ units}}^{2}$ to 4 decimal places

$x = \frac{0.35}{4.2} = 0.083 \overline{3}$ to 4 decimal places

($\text{ "x=2809/3000 " }$ as an exact value )

#### Explanation:

Let the area be $a$

From the question the area is:

$\text{ } a = 0.7 x \left(5 - 3 x\right)$

$\text{ } \implies a = 0.35 x - 2.1 {x}^{2}$

Rate of change in $\left(\text{change in "a)/("change in } x\right)$

Using calculus: shortcut method

$\frac{\delta a}{\delta x} = 0.35 - 4.2 x$

Rate of change is zero at maximum area

$\implies \frac{\delta a}{\delta x} = 0 = 0.35 - 4.2 x$

$\implies 4.2 x = 0.35$

$\implies x = \frac{0.35}{4.2} = 0.083 \overline{3}$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~
to find the fractional solution:

$10 x = 8.3 \overline{3}$

so $10 x - x = 8.427$

$x = \frac{8.427}{9} = \frac{8427}{9000} \text{ }$ Mmmmmm!
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Thus the exact area is

$0.7 \times \frac{8427}{9000} \times \left(5 - 3 \times \frac{8427}{9000}\right)$

The approximate area is

$3.2772 - 1.8411 = 1.4361$ to 4 decimal places