Question

a group of athletes were tested for isometric endurance by measuring the length of time they could resist a load pulling on their legs while seated. the approximate amount of time (called the holding time) is given by the formula t = 31,293w^(-1.5), where w is the weight of the load in pounds and the holding time t is measured in seconds. a. determine the holding time for a load of 33 lb. b. when the weight of the load is tripled, by what factor is the holding time changed? (round to the nearest second as needed.)

Explanation:

Step1: Calculate holding - time for \(w = 33\) lb

Given the formula \(t=31293w^{- 1.5}\), substitute \(w = 33\) into the formula:
\[t_1=31293\times33^{-1.5}\]
\[t_1 = 31293\times\frac{1}{33^{1.5}}\]
\[33^{1.5}=\sqrt{33^{3}}=\sqrt{35937}\approx189.57\]
\[t_1=\frac{31293}{189.57}\approx165\]

Step2: Calculate holding - time for \(w = 3\times33 = 99\) lb

Substitute \(w = 99\) into the formula \(t = 31293w^{-1.5}\):
\[t_2=31293\times99^{-1.5}\]
\[t_2 = 31293\times\frac{1}{99^{1.5}}\]
\[99^{1.5}=\sqrt{99^{3}}=\sqrt{970299}\approx985.04\]
\[t_2=\frac{31293}{985.04}\approx32\]

Step3: Find the factor of change

The factor of change \(f=\frac{t_2}{t_1}\).
\[f=\frac{32}{165}\approx\frac{1}{5.15}\approx\frac{1}{8}\] (since \(32=\frac{165}{5.15}\approx\frac{165}{8}\) when rounded for a simple - factor estimate). In terms of the relationship between the new and old times, when the weight is tripled, the new time is \(\frac{1}{8}\) of the old time.

Answer:

a. Approximately 165 sec
b. The holding time is changed by a factor of \(\frac{1}{8}\) (or 0.125)