Suppose we have an ellipse parameterized by x = a cos(θ) and y = b sin(θ) with a and b positive. this is the ellipse described by the equation x 2/a2 + y 2/b2 = 1. it is known that the area of the ellipse is πab. the following argument shows that there is only one positive real number. what is wrong with the argument? suppose a and b are positive real numbers. we shall compute the area of the ellipse parameterized by x = a cos(θ) and y = b sin(θ) for 0 ≤ θ ≤ 2π by using polar coordinates. since r 2 = x 2 + y 2 , we have r 2 = a 2 cos2 (θ) + b 2 sin2 (θ) = b 2 + (a 2 − b 2 ) cos2 (θ). thus, πab = z 2π 0 z √ b 2+(a 2−b 2) cos2(θ) 0 r dr dθ = 1 2 z 2π 0 (b 2 + (a 2 − b 2 ) cos2 (θ)) dθ = 1 2 2π 0 (b 2 θ + (a 2 − b 2 )(θ/2 + sin(2θ)/4) = (a 2 + b 2 )π 2 so, 2ab = a 2 + b 2 , and hence (a − b) 2 = 0 which means a = b. since a and b were arbitrary positive real numbers, we conclude that there is only one positive real number

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/ˈplājəˌrizəm/

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noun

the practice of taking someone else's work or ideas and passing them off as one's own.

step-by-step explanation:

the answer is:

the lateral area is equal to

to find the lateral area of the square pyramid, first we need to find the perimeter of the base, and then, find the lateral area using the following formulas:

where,

s, is the base length

slant height, is the height of a the side of the pyramid

we are given:

now, calculating the perimeter of the base, we have:

then, calculating the lateral area:

hence, we have that the lateral area is equal to

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large crystals result from very slow cooling.

evaporation of water: this would produce an evaporite which is a sedimentary rock. like a salt pan in a desert. not possible to have volcanic rocks originate this way.

crystalization in molten earth: slow cooling over time. (hundreds to millions of years)

crystalization at earth's surface: this results in fast cooling. (a few hours to a few months or years)

crystalization from cooling off hot water: that would be a hydrothermal deposit. while associated with volcanics they are not volcanic rocks themselves and usually occur fairly rapidly.