Chapter 6.1, Problem 19E

(a)

To determine

The distance travelled by the car using the rectangles.

Answer to Problem 19E

The total area is 0.898 mile.

Explanation of Solution

Given information:

The velocity of a vintage sports car accelerating from 0 to 142 mi/h in 36 sec (10 thousandths of an hour).

Formula used:

LRAM approximation method is used.

Calculation:

For left-hand point rectangle approximation method (LRAM), area is divided into rectangles. Total time is divided into equal intervals and at left-hand point of interval calculates the velocity which is equal to height of rectangle then calculates the area or small rectangles then adds all the area. Total area gives the distance travelled. There is no need to measure height because we have recorded data.

For LRAM,

  $\begin{array}{l}\text{total area}=0.001\left(0\right)+0.001\left(40\right)+0.001\left(162\right)+0.001\left(82\right)\\ +0.001\left(96\right)+0.001\left(108\right)+0.001\left(116\right)\\ +0.001\left(125\right)+0.00\left(132\right)+0.001\left(137\right)+0.001\left(142\right)\end{array}$

All values which are added above, these are the areas of small rectangles.

  $\begin{array}{l}\text{total area}=0+0.040+0.062+0.082+0.096+0.108\\ +0.116+0.125+0.132+0.137\\ \text{total area}=0.898\text{mile}\end{array}$

Conclusion:

The total area is 0.898 mile.

(b)

To determine

The half-way point value.

Answer to Problem 19E

The speed is 116mph

Explanation of Solution

Given information:

The velocity of a vintage sports car accelerating from 0 to 142 mi/h in 36 sec (10 thousandths of an hour).

Formula used:

LRAM approximation is used.

Calculation:

Halfway point is half of distance travelled.

  $\begin{array}{l}\text{halfway point}=\frac{0.969}{2}\\ \text{halfway point}=0.4845\text{mile}\end{array}$

Now finding that up to where this distance covers.

LRAM sums up to 0.006 hour,

Sum = 0.388 mile

RRAM sums up to 0.006 hour,

Sum = 0.504 mile

Take average of two sums,

  $\begin{array}{l}\text{average}=\frac{0.388+0.504}{2}\\ \text{average}=0.4460\text{mile}\end{array}$

LRAM sums up to 0.007 hour,

Sum = 0.504 mile

RRAM sums up to 0.006 hour,

Sum = 0.629 mile

Take average of two sums,

  $\begin{array}{l}\text{average}=\frac{0.388+0.504}{2}\\ \text{average}=0.5665\text{ mile}\end{array}$

So, from above calculation the halfway point is lie between 0.006 hour and 0.0007 hour, since it is closer to the distance covered in 0.006 hour, so we take 0.0006 hour.

  $\begin{array}{l}\text{time taken}=0.006\text{hour}\\ \text{=0}\text{.006}\times \text{3600}\text{sec}\\ \text{time taken}=21.6\sec \end{array}$

Since from table given in question, speed of car at 0.006 hours is 116 mi/h

  $\text{speed}=116\text{mph}$

Conclusion:

The speed is 116mph