An environmentalist studies the sound level of a waterfall. The sound level, S, in decibel… — SAT

An environmentalist studies the sound level of a waterfall. The sound level, $S$ , in decibels ( $\text{dB}$ ), at a distance of $d$ meters from the base of the waterfall is modeled by the equation $S = 92 - 0.15d$ , where $0 \leq d \leq 200$ . What is the best interpretation of the $S$ -intercept of the graph of this equation in the $dS$ -plane?

A
The sound level of the waterfall decreases by $0.15\text{ decibels}$ for each meter of distance from its base.
B
The sound level of the waterfall is $91.85\text{ decibels}$ at its base.
The sound level of the waterfall is $92\text{ decibels}$ at its base.Answer
D
The distance from the base where the sound level is $0\text{ decibels}$ is $92\text{ meters}$ .

Answer

The sound level of the waterfall is

92\text{ decibels}

at its base.

The

S

-intercept of the graph in the

dS

-plane occurs where the horizontal variable,

d

, is equal to

0

. Substituting

d = 0

into the equation yields

S = 92 - 0.15(0) = 92

. Since

d

represents the distance from the base of the waterfall in meters and

S

represents the sound level in decibels, this means that at a distance of

0

meters (the base of the waterfall), the sound level is

92\text{ decibels}

Step-by-Step Solution

Identify the definition of the

S

-intercept in the context of the

dS

-plane.

The

S

-intercept is the point on the graph where the independent variable,

d

(distance), is equal to

0

By definition, the vertical intercept of a graph occurs where the horizontal coordinate is zero.

Substitute

d = 0

into the linear equation.

S = 92 - 0.15(0) = 92

Evaluating the equation at

d = 0

determines the value of the sound level at zero meters from the base.

Interpret the mathematical result in the real-world context.

At a distance of

0

meters (which is the base of the waterfall), the sound level is

92\text{ decibels}

This connects the coordinate point

(0, 92)

to the physical properties of the scenario.

Key Concept

Interpreting the vertical intercept of a linear relationship in context

Estimated Time:1m 30s

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