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) that they could resist the load was 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.) a. the holding time for a load of 33 lb is approximately □ sec.

Explanation:

Step1: Substitute weight into formula

Given $t = 31293w^{-1.5}$ and $w = 33$. Then $t=31293\times33^{-1.5}$.

Step2: Calculate $33^{-1.5}$

$33^{-1.5}=\frac{1}{33^{1.5}}=\frac{1}{\sqrt{33^{3}}}=\frac{1}{\sqrt{35937}}\approx\frac{1}{189.57}$.

Step3: Calculate holding - time

$t = 31293\times\frac{1}{189.57}\approx165$ seconds.

Step4: Analyze effect of tripling weight

Let the original weight be $w_1$ and the new weight be $w_2 = 3w_1$.
The original time is $t_1=31293w_1^{-1.5}$, and the new time is $t_2 = 31293w_2^{-1.5}=31293(3w_1)^{-1.5}$.
$t_2=31293\times3^{-1.5}\times w_1^{-1.5}$.
Since $3^{-1.5}=\frac{1}{3^{1.5}}=\frac{1}{\sqrt{27}}\approx\frac{1}{5.2}$.
The factor by which the holding - time changes is $\frac{t_2}{t_1}=3^{-1.5}\approx\frac{1}{5.2}\approx\frac{1}{5}$.

Answer:

a. 165
b. The holding time is changed by a factor of approximately $\frac{1}{5}$.