The trough is in the shape of a triangular prism. It is 5 ft long and its vertical cross sections are isosceles triangles with base 2ft and height 3ft. Water is being siphoned out of the trough at the rate of 2 cubic feet per minute. At any... time t, let h be the depth and V be the volumeof water in the trough.
if a restaurant has a dinning room 92 ft by 27 ft what is the area of the dinning room What is the perimeter of the dinning room?
To find an equation that relates the depth of water (h) to the time (t) and the volume of water (V) in the trough, we can start by understanding the geometric properties of the trough.
The trough is in the shape of a triangular prism, which means it has a triangular cross-section that is repeated along its length. The base of each triangular cross-section has a length of 2 ft and a height of 3 ft. The length of the trough is given as 5 ft.
Now, let's consider the volume of the trough at any given time t. The volume of the trough can be thought of as the sum of the volumes of the individual triangular cross-sections along its length.
The volume (V) of a single triangular cross-section can be calculated using the formula for the volume of a prism:
V = (base * height * length) / 2,
where the base is the length of the base of the cross-section, the height is the height of the cross-section, and the length is the length of the trough.
In this case, the base is 2 ft, the height is 3 ft, and the length is 5 ft. Therefore, the volume (V) of a single triangular cross-section is:
V = (2 * 3 * 5) / 2 = 15 ft^3.
Since the trough consists of several identical triangular cross-sections, the total volume at any time t is given by:
V(t) = 15t,
where t represents the time in minutes.
Next, let's determine the depth of water (h) in the trough at any time t. The depth can be determined by dividing the volume of water (V) by the area of the triangular cross-section.
The area of a triangle can be calculated using the formula:
Area = (base * height) / 2.
In this case, the base is 2 ft and the height can be thought of as the unknown depth h. Therefore, the area of the triangular cross-section is:
Area = (2 * h) / 2 = h ft^2.
To find the depth (h) at any time t, we can rearrange the equation for the volume (V):
V(t) = Area * length = h * 5.
Since we already know that V(t) = 15t, we can substitute this into the equation:
15t = h * 5.
Finally, we can solve for the depth (h) as a function of time (t):
h(t) = (15t) / 5 = 3t.
So, the equation that relates the depth of water (h) to the time (t) is h(t) = 3t.