Question

1. At what point of the parabola y= x^2-3x-5 is the tangent line parallel to 3x-y=2? Find the equation of the tangent line.
2. For the curve y=x^2+x at what point does the normal line at (0,0) intersects the tangent line at (1,2)?
3. A water bucket containing 10 gal of water develops a leak at time t=0 and the volume V of water in the bucket t seconds later is given by
V(t)=10(〖1-t/100)〗^2
until the bucket is empty.
a) At what time would the bucket be emptied?
b) At what rate is water leaking from the bucket after exactly 1 minute has passed?
c) When is the instantaneous rate of change of V equal to the average rate of change of V from t=0 to t=100?
4. A car is travelling at 100 fps when the driver suddenly applies the brakes. The position function of the skidding car is
s=100t-〖5t〗^2
how far and for how long does the car skid before it comes to a stop?

Answer 1

Answer on Question #62490, Physics / Mechanics | Relativity

1. At what point of the parabola $y = x^2 - 3x - 5$ is the tangent line parallel to $3x - y = 2$? Find the equation of the tangent line.

Solution:

$$\begin{array}{l} y = 3x - 3 \leftarrow \text{slope} = 3 \\ y = x^2 - 3x - 5 \\ \text{or } dy/dx = 2x - 3 = 3 \\ \text{or } 2x = 6 \\ \text{or } x = 3 \rightarrow y = (3)^2 - 3(3) - 5 = -5 \\ y = mx + b \\ -5 = (3)(3) + b \\ b = -14 \\ y = 3x - 14 \\ \end{array}$$

Answer: $y = 3x - 14$

2. For the curve $y = x^2 + x$ at what point does the normal line at $(0,0)$ intersect the tangent line at $(1,2)$?

Solution:

The first derivative of a curve gives the slope of the tangent to the curve at any point.

We are given the curve $y = x^2 + 2$

Now $y' = 2x$.

At the point $(0, 0)$, $y' = 2^*(0) = 0$

The equation of the tangent is $y - (0) = 0^*(x - 0)$

$$\Rightarrow y = 0$$

At the point $(1, 2)$, $y' = 2^* - 1 = 2$

The equation of the tangent is $y - 1 = 2^*(x - 2)$

$$\begin{array}{l} \Rightarrow y - 1 = 2x - 4 \\ \Rightarrow 2x - y - 3 = 0 \\ \end{array}$$

To determine the point of intersection of $y = 0$ and $2x - y - 3 = 0$

$$\begin{array}{l} 2x - y - 3 = 0 \\ y = 0 \\ \Rightarrow 2x - 3 = 0 \\ \Rightarrow x = 3/2 \\ \end{array}$$

Therefore the point of intersection of the required tangents is $(3/2, 0)$.

3. A water bucket containing 10 gal of water develops a leak at time $t=0$ and the volume $V$ of water in the bucket $t$ seconds later is given by $V(t)=10(1-t/100)^2$ until the bucket is empty.

a) At what time would the bucket be emptied?

b) At what rate is water leaking from the bucket after exactly 1 minute has passed?

c) When is the instantaneous rate of change of $V$ equal to the average rate of change of $V$ from $t=0$ to $t=100$?

Solution:

$$V(t) = 10[1 - (t/100)]^2$$

The rate of change of volume at any time $t$ is,

$$\begin{aligned} dV/dt &= 10.2[1 - (t/100)].(-1/100) \\ &= (-20/100)[1 - (t/100)] \\ &= (-1/5)[1 - (t/100)] \\ &= [(t/100) - 1]/5 \end{aligned}$$

b) At $t = 1$ minute = 60s

$$\begin{aligned} dV/dt &= [(60/100) - 1]/5 \\ &= [(0.6 - 1]/5 \\ &= -0.4/5 = -0.08 \text{ gallons/s} \end{aligned}$$

c) The average rate of change of volume from $t=0$ to $t=100$ is

$$V(ave) = [V(100) - V(0)]/(100 - 0)$$ $$\begin{aligned} V(100) &= 10[1 - (100/100)]^2 \\ &= 10[1 - 1]^2 \\ &= 10(0)^2 \\ &= 0 \text{ gallons} \end{aligned}$$ $$\begin{aligned} V(0) &= 10 \text{ gallons} \\ V(ave) &= (0 - 10)/(100 - 0) \\ &= -10/100 \\ &= -0.1 \text{ gallons} \end{aligned}$$

a) $dV/dt = [(t/100) - 1]/5$

So,

$$[(t/100) - 1]/5 = -0.1$$

(t/100) - 1 = -0.02

t/100 = 0.98

t = 98s

4. A car is travelling at 100 fps when the driver suddenly applies the brakes. The position function of the skidding car is s=100t- (5t)^2 how far and for how long does the car skid before it comes to a stop?

**Solution:**

x' = v(t) = 100 - 10t

0 = 100 - 10t

10t = 100

t = 10 s

And

x(10) = 100(10) - 5(10)²

= 1000 - 5(100)

= 500 ft

**Answer:** 10s and 500 ft

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