Expand and simplify 3(4a - 5b) + 2(2a - 3b)
2 answers:
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Answer:
- (a) see attached
- (b) width: 2x; height: y = 9-x²
- (c) A=2x(9-x²) . . . 0 ≤ x ≤ 3
- (d) dA/dx = -6x² +18; x=±√3
- (e) 12√3 units²
Step-by-step explanation:
(a) The attachment shows the graph of the parabola in blue. It also shows an inscribed rectangle in black.
(b) The upper right point of the rectangle is shown in the attachment as (x, y). The dimension y is the height of the rectangle. The x-dimension is half the width of the rectangle, which is symmetrical about the y-axis. Hence the width is 2x.
(c) As with any rectangle, the area is the product of length and width:
... A = (2x)(9 -x²) . . . . . the attachment shows a graph of this
... A = -2x³ +18x . . . . . expanded form suitable for differentiation
A suitable domain for A is where both x and A are non-negative: 0 ≤ x ≤ 3.
(d) The derivative of A with respect to x is ...
... A' = -6x² +18
This is defined everywhere, so the critical values will be where A' = 0.
... 0 = -6x² +18
... 3 = x² . . . . . . . divide by -6, add 3
... √3 = x . . . . . . . -√3 is also a solution, but is not in the domain of A
(e) The rectangle will have its largest area where x=√3. That area is ...
... A = 2x(9 -x²) = 2√3(9 -(√3)²) = 2√3(6)
... A = 12√3 . . . . square units . . . . ≈ 20.785 units²
Answer:
The correct answer is:
The volume of the triangular prism is equal to the volume of the cylinder
Step-by-step explanation:
Given that there are two figures
1. A right triangular prism and
2. Right cylinder
Area of cross section of prism is equal to Area of cross section of cylinder.
Let this value be <em>A</em>.
Also given that Height of prism = Height of cylinder = <em>6</em>
Volume of a prism is given as:
Cross section of cylinder is a circle.
<em>Area of circle</em> is given as:
Area of cross section, A =
Volume of cylinder is given as:
From equations (1) and (2) we can see that
Hence, the correct answer is:
Volume of prism is equal to the volume of cylinder.