A water trough is 8m long and its cross-section is an isosceles triangle which is 60cm wide at the top, and the height is 60cm. The trough is not full.
Give an expression for V, the volume of water in the trough in cm^3, when the depth of the water is d cm.
Thank you in advance!!
draw a diagram. You can see that when the depth is d, the width of the cross-section is also d. So, the area of the cross-section is 1/2 d^2.
v = 1/2 d^2 * 800 cm^3
To find the expression for the volume of water in the trough, we need to calculate the area of the cross-section and then multiply it by the length of the trough.
The cross-section of the trough is an isosceles triangle with a width of 60 cm at the top and a height of 60 cm. However, the depth of the water, d, will determine the width at the bottom of the triangle.
To find the width at the bottom, we can use similar triangles. The height of the water in the trough is the same as the height of the triangle, which is 60 cm. The height of the dry part of the trough is 60 cm - d cm.
Using the similar triangles property, we can set up the following proportion:
(height of the dry part of the trough) / (width of the dry part of the trough) = (height of the triangle) / (width of the triangle)
(60 cm - d cm) / (width of the dry part of the trough) = 60 cm / 60 cm
Simplifying the right side gives:
(60 cm - d cm) / (width of the dry part of the trough) = 1
Cross-multiplying gives:
60 cm - d cm = (width of the dry part of the trough)
Now, we can express the width of the dry part of the trough in terms of the depth of the water, d:
width of the dry part of the trough = 60 cm - d cm
Using this expression, we can calculate the area of the cross-section of the trough.
Area of the cross-section = (1/2) * (width at the top + width of the dry part) * height
= (1/2) * (60 cm + (60 cm - d cm)) * 60 cm
Therefore, the expression for V, the volume of water in the trough, in cm^3, when the depth of the water is d cm is:
V = (1/2) * (60 cm + (60 cm - d cm)) * 60 cm * 8 m
To convert this to cm^3, we multiply by 100 (since 1 m = 100 cm):
V = (1/2) * (60 cm + (60 cm - d cm)) * 60 cm * 8 m * 100 cm
Simplifying this expression will give the final expression for V in terms of d.