Question

Jose unlocks his cellphone by placing his right thumb on a square of 1 centimeter by 1 centimeter at the center of the screen. Upon recognizing his thumbprint, the square expands outward. The cellphone unlocks when the perimeter reaches 32 centimeters, taking a total of 2.5 seconds. What is the perimeter of the square after 1.5 seconds?

Answer 1

Answer: 20.8 cm

Explanation:

Notice that the corners of the squares form a line.

Set the x-axis to time and the y-axis to length of a side (not perimeter) to create coordinates.

At 0 seconds the side length is 1 --> (0, 1)

At 2.5 seconds the perimeter is 32 (side length is 32/4 = 8) --> (2.5, 8)

First, find the slope between the coordinates using
`m=(y_2-y_1)/(x_2-x_1)`

Let (x₁, y₁) = (0, 1) and (x₂, y₂) = (2.5, 8)

`m=(8-1)/(2.5-0)\quad =(7)/(2.5)\quad =(7)/((5)/(2))\quad =(14)/(5)\quad =2.8`

Next, use the Point-Slope formula to find the equation of the line:

`y-y_1=m(x-x_1)\\\\y-1=2.8(x-0)\\\\y=2.8x+1`

Lastly, find the side length (y) when x = 1.5

y = 2.8(1.5)x + 1

= 4.2 + 1

= 5.2

Perimeter of a square = 4 times the side length

P = 4(5.2)

= 20.8