Question

a rock sits on the edge of bridge that passes over a river. when a car drives by, the rock falls off the bridge into the river. the graph below shows the height of the rock above the river as it falls from the bridge. graph: x-axis time (seconds) from 0 to 1.25, y-axis height above river (feet) from 0 to 16. curve starts at (0,14), ends at (1.0,0). what is the domain of the function that describes the height of the rock above the river? a. 0 ≤ x ≤ 14 b. 0 ≤ y ≤ 14 c. 0 ≤ x ≤ 0.94 d. 0 ≤ y ≤ 0.94 (note: likely typo in option c/d, intended 1.0?)

Explanation:

Step1: Recall Domain Definition

Domain is the set of all input (x - values) for a function. Here, x represents time (seconds).

Step2: Analyze the Graph's x - axis

From the graph, the x - values start at 0 (when the rock starts falling) and end at 1.0 (when it hits the river). Wait, but looking at the options, option C is \(0\leq x\leq0.94\)? Wait, maybe a typo, but let's re - evaluate. Wait, the x - axis is time, and the graph's x - range: the leftmost x is 0, and the rightmost x (where y = 0) is around 1.0, but the options have C as \(0\leq x\leq0.94\) (maybe a rounding). Also, domain is about x (input), so we can eliminate options with y (range is about y). So options B and D are about y, so eliminate. Option A is about x from 0 to 14, but 14 is the initial y - value (height). So the correct domain should be the x - values, so looking at the options, the only one with x as the variable for domain is A? Wait, no, wait the graph: the x - axis goes from 0 to 1.0, but the options: Wait, maybe I misread. Wait the options:

A: \(0\leq x\leq14\) – no, 14 is y - value.

Wait, maybe the original problem has a typo, but let's think again. Domain is x (time) values. The rock starts at x = 0 (time 0) and stops at x when y = 0 (when it hits the river). From the graph, the x - intercept is at x = 1.0, but the options:

Wait, the options:

A: \(0\leq x\leq14\) – x is time, 14 is height, so no.

B: \(0\leq y\leq14\) – y is range (height), so that's range, not domain.

C: \(0\leq x\leq0.94\) – maybe a rounding of 1.0?

D: \(0\leq y\leq0.94\) – y is range, so no.

So the domain is the set of x - values (time) from 0 to when it hits the river (x = 1.0, approximately 0.94 maybe due to graph scaling). And domain is about x, so the correct option is A? Wait no, A has x from 0 to 14, which is wrong. Wait, maybe I made a mistake. Wait the y - axis is height (14 at x = 0), x - axis is time. So domain is x ∈ [0, t], where t is the time when it hits the river. The graph shows that at x = 0, y = 14, and at x = 1.0, y = 0. So the domain is \(0\leq x\leq1.0\) (or approximately 0.94). So among the options, the only one with x as the variable for domain is A (even though 14 is wrong) or C. Wait, maybe the options have a typo, but let's check the variables:

Domain is the set of independent variable (x, time) values. So we need an option with x. So options A and C have x. Option A: \(0\leq x\leq14\) – 14 is the initial height (y - value), so that's incorrect. Option C: \(0\leq x\leq0.94\) – 0.94 is approximately the time when it hits the river (x - value). So the correct answer should be C? Wait, no, the x - axis in the graph goes to 1.0, so 0.94 is close. And domain is x, so the correct option is C? Wait, but let's re - check the options:

A: \(0\leq x\leq14\) – x is time, 14 is height, wrong.

B: \(0\leq y\leq14\) – y is height, that's range, wrong.

C: \(0\leq x\leq0.94\) – x is time, from 0 to when it hits the river, correct (domain).

D: \(0\leq y\leq0.94\) – y is height, wrong.

So the correct answer is C.

Answer:

C. \(0\leq x\leq0.94\)