A running track of 440 ft is to be laid out enclosing a football...

A running track of

440 \mathrm{ft}

is to be laid out enclosing a football field, the shape of which is a rectangle with a semi-circle at each end. If the area of the rectangular portion is to be maximum, then the lengths of its side are

Options

A $70 \mathrm{ft}$ and $110 \mathrm{ft}$
B $80 \mathrm{ft}$ and $120 \mathrm{ft}$
C $35 \mathrm{ft}$ and $110 \mathrm{ft}$
D $35 \mathrm{ft}$ and $120 \mathrm{ft}$

Correct answer

C. $35 \mathrm{ft}$ and $110 \mathrm{ft}$

Step-by-step solution

Perimeter

=440 \mathrm{ft}\Rightarrow \quad 2 x+\pi r+\pi r=4402 x+2 \pi r=440

\begin{aligned} A & =\text{ Area\text{ of } the rectangular portion }= x 2 r \\ A & =x \frac{(440-2 x)}{\pi}=\frac{1}{\pi}\left(440 x-2 x²\right) \end{aligned}

Let \frac{d A}{d x}=\frac{1}{\pi}(440-4 x)=0

\begin{aligned}& \Rightarrow x=110 \text{ for which } \frac{d^2 A}{d x^2}< 0 \\ & \Rightarrow A \text{ is maximum when } x=110 \\ & \Rightarrow 2 r=\frac{440-2 x}{\pi}=\frac{440-220}{22 / 7}=70 \\ \text{ and } \therefore \quad r=35 \mathrm{ft} \text{ and } x=110 \mathrm{ft} \end{aligned}

Practice Application of Derivatives on Quantrex Academy →

More from Application of Derivatives

The maximum area of rectangle inscribed in a circle of diameter R is BITSAT 2024 Consider the function f(x)=(|x-1|)/(x^2) , then f(x) is BITSAT 2024 The maximum volume (in cu. units) of the cylinder which can be inscribed in a sphere of… BITSAT 2024 If the angle made by the tangent at the point (x_0, y_0) on the curve x=12(t+ t t)… BITSAT 2024 The altitude of a cone is 20 cm and its semivertical angle is 30^ ^ ∘ . If the… BITSAT 2024 The point of inflexion for the curve y=(x-a)^n , where n is odd integer and n ≥ 3 is BITSAT 2024 The population p(t) at time t of a certain mouse species satisfies the differential… BITSAT 2024 The function f(x)= (1+x)-(2 x)/(2+x) is increasing on BITSAT 2023 If f(x)= x, g(x)= 2 x, h(x)= 3 x and I(x)= x , then which of the following option is… BITSAT 2023 The interval in which the function f(x)=(4 x^2+1)/(x) is decreasing is : BITSAT 2023 The curve given by x+y=e^ x y has a tangent parallel to the Y-axis at the point BITSAT 2023 The function f(x)= ^ -1 ( x+ x) is an increasing function in BITSAT 2023 A spherical balloon is filled with 4500 π cubic meters of helium gas. If a leak in the… BITSAT 2022 The points at which the tangent passes through the origin for the curve y=4 x^ 3 -2 x^ 5… BITSAT 2021 If the radius of a sphere is measured as 9 cm with an error of 0.03 cm , then find the… BITSAT 2021 f(x)= ( e^ 2 x -1 e^ 2 x +1 )is BITSAT 2021 Full Application of Derivatives list Application of Derivatives JEE Maths PYQs JEE Mathematics PYQs All BITSAT PYQs