Explanation:
Finding the area of the canvas:
The canvas has sides measuring 3x - 3 and 4x + 1 inches. To find the area, we multiply the lengths of the sides. So, the area of the canvas is given by:
Area = (3x - 3) * (4x + 1)
= 12x^2 + x - 3
Therefore, the correct equation for the area of the canvas is A = 12x^2 + x - 3.
Finding the area of the circle created by the paint:
The rate at which the paint is spreading on the canvas is given by the function r(t) = 4t, where t represents time in minutes and r represents how far the paint has spread. The area of the circle created by the paint can be expressed as A[r(t)] = πr^2, where r(t) is the distance the paint has spread at time t.
Substituting the value of r(t) = 4t into the formula, we get:
A[r(t)] = π(4t)^2
= 16πt^2
Therefore, the area of the circle created by the paint is 16πt^2.
Determining if the circle will be at least 1300 in² in 5 minutes:
To check if the area of the circle will be at least 1300 in² in 5 minutes, we substitute t = 5 into the equation for the area:
A[r(5)] = 16π(5)^2
= 400π
Approximating the value of 400π, we get approximately, 1256.64.
Since 1256.64 is less than 1300, we can conclude that the area of the circle created by the paint after 5 minutes will not be at least 1300 in².