Question

The area (a) of the region bounded by a parabola and a horizontal line is given by a = 2/3bh, where b and h are defined in the diagram. Find the area of the region determined by each pair of inequalities.

a) b > 0, h > 0
b) b < 0, h > 0
c) b > 0, h < 0
d) b < 0, h < 0

Answer 1

The area of the region bounded by a parabola and a horizontal line can only be calculated using positive values for base and height. Negative values do not make sense in physical geometry for the calculation of an area. Thus, a positive area is obtained only when both base and height are greater than zero.

So option (A) is the correct answer.

Step-by-step explanation:

The area (a) of the region bounded by a parabola and a horizontal line is given by the formula a = 2/3bh, where b is the length of the base of the parabolic region and h is the height from the base to the horizontal line. To find the area of the region determined by each pair of inequalities, simply apply the values of b and h according to the context of the inequality.

For b > 0, h > 0, The area is positive as both the base and height are positive.

For b < 0, h > 0: In this scenario, the concept of a negative base does not apply in terms of physical geometry for area computation, as both dimensions should be non-negative.

For b > 0, h < 0: Similarly, a negative height is not applicable for the calculation of area in physical geometry.

For b < 0, h < 0: Again, negative dimensions do not make sense in a physical context for calculating an area.

Only positive values for both b and h yield a meaningful area for the region bounded by a parabola and a horizontal line, as per the provided equation.