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Question

formative homework help!!

The quantity of a substance can be modeled by the function k(t) that satisfies the differential equation dk/ dt =-1/90 (k-450). One point on this function is k(2) = 900. Based on this model, use a linear approximation to the graph of k(t) at t = 2 to estimate the quantity of the substance at t = 2.1.

Answer 1

Hi Sarah,

This question is asking you to find the equation of a tangent line at a given point, then use that tangent line to approximate a nearby point on the curve. When we determine the equation of a line, we use the Point-Slope Form:

y - y1 = m(x - x1)

Our unknowns are the Point (x1,y1) and the Slope (m).

Fortunately, they gave you a point to use: k(2) = 900. This implies that x1 = 2 and y1 = 900.

y - 900 = m(x - 2)

They also gave you a means of determining the slope. Since the derivative predicts the slope of the tangent line at any point on the curve, we can use dk/dt to figure out what m is when at (2,900)

m = -1/90 (900 - 450) = -1/90 * 450 = -5

So now we have our tangent line modeled completely:

y - 900 = -5(x - 2)

Or I guess it would be more appropriate to state:

k - 900 = -5(t - 2)

Same thing. Anyway. They want you to use this line to estimate k when t = 2.1. So,

k - 900 = -5(2.1 - 2)

k - 900 = -5(0.1)

k - 900 = -0.5

k = 899.5

Hope this helps!