Answer :
To address the various parts of the problem concerning the cross-section of a river represented by the function [tex]\( d(w) = 14w^2 - w \)[/tex] where [tex]\( d(w) \)[/tex] is the depth of the river in meters at [tex]\( w \)[/tex] meters from the left edge, we will go through each part step-by-step.
### a. Sketch the graph of the cross section of the river
To sketch the graph of the function [tex]\( d(w) = 14w^2 - w \)[/tex], you can use a graphing calculator or software. The graph is a parabolic curve opening upwards due to the positive coefficient of [tex]\( w^2 \)[/tex].
Key points to observe:
- The vertex or turning point of the parabola represents the minimum depth of the river.
- Since the coefficient of [tex]\( w \)[/tex] is negative and that of [tex]\( w^2 \)[/tex] is positive, we see an initial increase in depth as [tex]\( w \)[/tex] moves away from zero before climbing upwards indefinitely as [tex]\( w \)[/tex] moves further from zero.
### b. Determine the depth of the river 3 meters from the left edge.
To find the depth of the river 3 meters from the left edge, substitute [tex]\( w = 3 \)[/tex] into the function [tex]\( d(w) = 14w^2 - w \)[/tex]:
[tex]\[ d(3) = 14(3)^2 - 3 = 14 \cdot 9 - 3 = 126 - 3 = 123 \][/tex]
So, the depth of the river 3 meters from the left edge is 123 meters.
### c. What is the maximum depth of the river, to the nearest tenth?
The maximum depth of the river is found by identifying the highest point on the depth curve within the range of the river's cross-section. This involves calculating the depth at critical points, which are found by setting the derivative of the function to zero:
Given the maximum depth:
[tex]\[ \text{Maximum depth} \approx -0.02 \][/tex]
So, the maximum depth of the river to the nearest tenth is approximately -0.02 meters.
### d. How far from the left edge of the river, to the nearest tenth, is the deepest part of the river?
The deepest part of the river corresponds to the position of the maximum depth found earlier. Suppose this position is calculated as:
[tex]\[ \text{Position of maximum depth} \approx 0.04 \][/tex]
So, the deepest part of the river is approximately 0.04 meters from the left edge.
### e. What is the width of the river to the nearest tenth?
The width of the river can be found by solving [tex]\( d(w) = 0 \)[/tex]:
[tex]\[ 14w^2 - w = 0 \][/tex]
[tex]\[ w(14w - 1) = 0 \][/tex]
This results in [tex]\( w = 0 \)[/tex] and [tex]\( w = \frac{1}{14} \)[/tex].
The width of the river is the distance between these points:
[tex]\[ \frac{1}{14} - 0 = \frac{1}{14} \approx 0.07 \][/tex] meters.
So, the width of the river to the nearest tenth is approximately 0.07 meters.
### Summary:
- Depth at 3 meters: 123 meters
- Maximum depth: -0.02 meters (nearest tenth)
- Position of the maximum depth: 0.04 meters (nearest tenth)
- Width of the river: 0.07 meters
### a. Sketch the graph of the cross section of the river
To sketch the graph of the function [tex]\( d(w) = 14w^2 - w \)[/tex], you can use a graphing calculator or software. The graph is a parabolic curve opening upwards due to the positive coefficient of [tex]\( w^2 \)[/tex].
Key points to observe:
- The vertex or turning point of the parabola represents the minimum depth of the river.
- Since the coefficient of [tex]\( w \)[/tex] is negative and that of [tex]\( w^2 \)[/tex] is positive, we see an initial increase in depth as [tex]\( w \)[/tex] moves away from zero before climbing upwards indefinitely as [tex]\( w \)[/tex] moves further from zero.
### b. Determine the depth of the river 3 meters from the left edge.
To find the depth of the river 3 meters from the left edge, substitute [tex]\( w = 3 \)[/tex] into the function [tex]\( d(w) = 14w^2 - w \)[/tex]:
[tex]\[ d(3) = 14(3)^2 - 3 = 14 \cdot 9 - 3 = 126 - 3 = 123 \][/tex]
So, the depth of the river 3 meters from the left edge is 123 meters.
### c. What is the maximum depth of the river, to the nearest tenth?
The maximum depth of the river is found by identifying the highest point on the depth curve within the range of the river's cross-section. This involves calculating the depth at critical points, which are found by setting the derivative of the function to zero:
Given the maximum depth:
[tex]\[ \text{Maximum depth} \approx -0.02 \][/tex]
So, the maximum depth of the river to the nearest tenth is approximately -0.02 meters.
### d. How far from the left edge of the river, to the nearest tenth, is the deepest part of the river?
The deepest part of the river corresponds to the position of the maximum depth found earlier. Suppose this position is calculated as:
[tex]\[ \text{Position of maximum depth} \approx 0.04 \][/tex]
So, the deepest part of the river is approximately 0.04 meters from the left edge.
### e. What is the width of the river to the nearest tenth?
The width of the river can be found by solving [tex]\( d(w) = 0 \)[/tex]:
[tex]\[ 14w^2 - w = 0 \][/tex]
[tex]\[ w(14w - 1) = 0 \][/tex]
This results in [tex]\( w = 0 \)[/tex] and [tex]\( w = \frac{1}{14} \)[/tex].
The width of the river is the distance between these points:
[tex]\[ \frac{1}{14} - 0 = \frac{1}{14} \approx 0.07 \][/tex] meters.
So, the width of the river to the nearest tenth is approximately 0.07 meters.
### Summary:
- Depth at 3 meters: 123 meters
- Maximum depth: -0.02 meters (nearest tenth)
- Position of the maximum depth: 0.04 meters (nearest tenth)
- Width of the river: 0.07 meters
