Question

Speed of sound in the ocean. Measuring the speed of sound in the ocean is animportant part of marine research. One application is the study of climatechange. The speed of sound depends on the temperature, salinity, and depthbelow the surface. For a fixed temperature of 25 degrees Celsius and salinity of35 parts per thousand, the speed of sound is a function of the depth. At thesurface, the speed of sound is 1534 meters per second. For each increase indepth by 1 kilometer, the speed of sound increases by 17 meters per second.a. Explain why the function expressing the speed of sound in terms of depthis linear.b. Identify the slope and initial value of the linear function that gives thespeed of sound in terms of depth. Explain in practical terms what eachmeans.c. What increase in the speed of sound is caused by a 2-kilometer increase indepth?d. Use your answer to partc to determine the speed of sound when the depthis 2 kilometers.e. Use D for depth (in kilometers) and S for the speed of sound (in metersper second), and find a linear formula for S as a function of D.f. Use your formula from parte to calculate the speed of sound when thedepth is 2 kilometers.

Answer 1

a. The speed of sound in terms of depth is a linear function because the speed ouf the sound varies proportionality to the depth. In other words, every time the depth varies in one unit (1km) the speed of sound varies the same amount (17ms)

Be the independet variable X "depth" (measured in km) and the dependent variable Y "Speed of sound" (measured in ms), you can express the linear function as:

Y= 1534 + 17X

b. The initial value is the value of the speed of sound (Y) when the depth is zero (X), in this example you can consider the surface of the ocean as "X=0", so the initial value or y-intersect of this linear function is b= 1534 ms

The slope of the fuction represents how much does the dependet variable change every time the independent variable increases one unit, in this example we know that the speed of sound increases 17ms every time the depth increases 1km, so the slope of this function is:

m= 17 ms/km

c. and d. To know what will the speed of sound be at a 2km depth, you have to replace it in the formula we already determined. Remember that the depth is represented by the variable X, so in this case X=2

Y= 1534 + 17X

→ for X=2

Y= 1534 + 17*2= 1568ms

So, for a depth of 2km, the speed of sound will be 1568 ms.

e. We already solved this one in intem a. but instead of S and D we used Y and X. Using the given letters the formula is:

S= 1534 + 17D

f. the solution of this item is the same as in part c. You replace it in the formula and get that the speed of sound for a 2km depth is 1568ms