Question

the formula ( s(t)=1087.7sqrt{\frac{9t + 2617}{2457}} ) gives the speed of sound ( s(t) ), in feet per second, at a temperature of ( t ) degrees celsius. find the temperature of a blast furnace where sound travels 1750 ft/sec.
the temperature of a blast furnace where sound travels 1750 ft/sec is ( square^{circ}c ).
(do not round until the final answer. then round to the nearest tenth as needed.)

Explanation:

Step1: Substitute \( S(t) = 1750 \) into the formula

We have the equation \( 1750=1087.7\sqrt{\frac{9t + 2617}{2457}} \).
First, divide both sides by \( 1087.7 \): \( \frac{1750}{1087.7}=\sqrt{\frac{9t + 2617}{2457}} \).
So, \( \sqrt{\frac{9t + 2617}{2457}}\approx1.608 \).

Step2: Square both sides

Squaring both sides gives \( \frac{9t + 2617}{2457}=(1.608)^2\approx2.586 \).
Then multiply both sides by \( 2457 \): \( 9t+2617 = 2.586\times2457 \).
\( 9t+2617\approx6353.802 \).

Step3: Solve for \( t \)

Subtract \( 2617 \) from both sides: \( 9t=6353.802 - 2617=3736.802 \).
Divide both sides by \( 9 \): \( t=\frac{3736.802}{9}\approx415.2 \).

Answer:

\( 415.2^{\circ}C \)