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Answer:
For a rectangular solid with:
width = w
length = l
height = h
The volume is equal to:
V = w*l*h
in this case we know that:
width = w.
"length of 7 more than the width"
l = w + 7.
"a height that is equivalent to 15 decreased by 3 times the width."
h = 15 - 3*w
Then the volume will be:
V = w*l*h = w*(w + 7)*(15 - 3w) = (w^2 + 7*w)*(15 - 3*w)
V = ( -3*w^3 + 15*w^2 + 105*w - 21*w^2)
V = (-3*w^3 - 6*w^2 + 105*w)
Now, the maximum volume will be for the value of w such that:
V'(w) = 0.
and:
V''(W) < 0
Where:
dV/dw = V'(w).
dV'/dw = V''(w)
Then first we need to differentiate the equation for the volume.
V'(w) = dV/dw = ( 3*(-3*w^2) + 2*(-6*w) + 105)
V'(w) = -9*w^2 - 12*w + 105.
Then we need to find the solution for:
-9*w^2 - 12*w + 105 = 0.
We can use the Bhaskara formula, and we will get:
Then the two solutions are:
w₁ = (+12 - 62.6)/(-18) = 2.81
w₂ = (+12 + 62.6)/(-18) = -15.5
But we can not have a negative width, so we can just discard the second solution.
Now let's check the second condition for the maximum, we must have:
V''(2.81) < 0.
V'' = dV'/dw = 2*(-9*w) - 12 = -18*w - 12
V''(2.81) = -18*2.81 - 12 = -62.58 < 0 .
Then the volume is maximized when w = 2.81, and the maximum volume will be:
V(2.81) = (-3*(2.81)^3 - 6*(2.81)^2 + 105*2.81) = 180.1
The length of line segment BC is 10.5 inches.
Given that
The equation tan(55 degrees) equals 15/b, which can be used to find the length of Line segment AC.
Triangle ABC is shown.
Angle BCA is a right angle and angle BAC is 55 degrees.
The length of AC is b and the length of hypotenuse BC is 15 centimeters.
We have to determine
What is the length of Line segment AC?
According to the question
Triangle ABC is shown.
Angle BCA is a right angle and angle BAC is 55 degrees.
The length of AC is b and the length of hypotenuse BC is 15 centimeters.
Here, BC is not the hypotenuse of the right triangle ABC.
The length line segment AC is given by,
In triangle ABC
Where = 55 degrees, Perpendicular = 15 and base is b.
Substitute all the value in the equation
Hence, the length of line segment BC is 10.5 inches.
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