The location P(t) of an object moving in the xy-plane at time t seconds is given by the equations P(t)=(x(t),y(t)), where x(t)=a +3t and y(t)=b +8t, a, b are constants and distances are measured in units of meters. The equations x(t), y(t) describe linear parametrized motion; see section 10.1 of the textbook for review. (h) The line along which the object is moving in the plane has the equation: y=___x+___.

(i) Let d(t) be the distance the object has traveled after t seconds. The formula for d(t) is:

(j) The instantaneous rate of change of d(t) at time t is:

(h) y =  -8/3·x + 16·a/3

(i) dt = √(x² + y²) = √( (a + 3·t)² + (b + 8·t)²)

(j)  

Step-by-step explanation:

(h) The location of the object = P(t) = (x(t), y(t))

Where;

x(t) = a + 3·t

y(t) = b + 8·t

Therefore we have;

At x = 0, t = -a/3. at y = 0, b = -8·t = -8·a/3

y = -8·a/3 + 8·t = -8/3(a - 3·t) = -8/3(x - 2·a) = -8/3·x + 16·a/3

y =  -8/3·x + 16·a/3

(i) dt = √(x² + y²) = √( (a + 3·t)² + (b + 8·t)²)

(j) The instantaneous rate of change of d(t), with a corresponding change in time, t, obtained by solving through an online platform, is given as follows;

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