Chapter 8, Problem 5CA

To determine

To explain the sequence formed by the curve radii arithmetic, geometric, or neither.

Answer to Problem 5CA

Arithmetic sequence.

Explanation of Solution

Given:

The track has 8 lanes that are each $1.22$ meters wide. The curve radius of lane 1 is $36.5$ meters.

Calculation:

Radius of the first curve $=36.5$

Radius of 2nd curve $=$ radius of the first curve $+$ gap between the curves $=36.5+1.22=37.72$

Radius of the 3rd curve $=36.5+2\times 1.22$

Radius of the 4th curve $=36.5+3\times 1.22$

and so on.

Therefore, the sequence is arithmetic.

To determine

To write a rule for the sequence formed by the curve radii.

Answer to Problem 5CA

  $36.5+\left(n-1\right)1.22$

Explanation of Solution

Given:

The track has 8 lanes that are each $1.22$ meters wide. The curve radius of lane 1 is $36.5$ meters.

Calculation:

Radius 2nd curve $-$ radius on first curve $=1.22$ , radius on (3rd-2nd) $=1.22$ ... and so on.

This implies the given situation corresponds to arithmetic sequence with first term $=36.5$ and common difference $=1.22$

Radius of the nth curve $=36.5+\left(n-1\right)1.22$

To determine

To explain world records must be set on tracks that have a curve radius of at most $50$ meters in the outside lane. Does the track shown meet the requirement.

Answer to Problem 5CA

Yes.

Explanation of Solution

Given:

You lake out a loan for $16,000 with an interest rate of 0.75% per month. At the end of each month, you make a payment of $300.

Calculation:

In this case $a=36.5$ , $n=8$ and $d=1.22$ i.e. the radius of 8th curve

  $\begin{array}{l}=a+\left(n-1\right)d\\ =36.5+7\times 1.22\\ =36.5+8.54\\ =45.04\end{array}$

This implies our curve is well within the range of atmost $50$ meters so track meets the requirement