Question

a square image of an object on a video game has an initial side length of 25 pixels. to make the object appear closer, the side length of the square is increased by 10 pixels per second. this length change is represented by the function ( s(t) = 25 + 10t ), where ( t ) is time in seconds.
if the area of the square is ( a(s) = s^2 ), find the formula for the area as a function of time, and then determine ( a(s(3)) )
( a(t) ) equals ( square t^2 + square t + 625 ).
the area of the square image after 3 seconds is ( square ) square pixels.

Explanation:

Step1: Substitute s(t) into A(s)

$A(t) = (25 + 10t)^2$

Step2: Expand the squared binomial

Use $(a+b)^2 = a^2 + 2ab + b^2$:
$A(t) = 25^2 + 2\times25\times10t + (10t)^2 = 625 + 500t + 100t^2$

Step3: Rearrange to match given form

$A(t) = 100t^2 + 500t + 625$

Step4: Calculate s(3) first

$s(3) = 25 + 10\times3 = 25 + 30 = 55$

Step5: Compute A(s(3))

$A(s(3)) = 55^2 = 3025$

Answer:

$A(t)$ equals $\boldsymbol{100}t^2 + \boldsymbol{500}t + 625$.
The area of the square image after 3 seconds is $\boldsymbol{3025}$ square pixels.