To catch a ball, a professional baseball player leaps into the air with an initial velocity of 14 feet per second. Write a model for the height of the player above the ground after t seconds. After how many seconds does the player land on the ground? What would be the domain of this function?

The answer is below

Step-by-step explanation:

a) The baseball player leaps into the air with an initial velocity of  14 feet per second. This means that the player leaps from the ground, the initial height is therefore zero.

The height (y) is given by the formula:

y(t) = ut - (1/2)gt² + initial heigth

u = initial velocity = 14 ft/s, t = time taken, g = acceleration due to gravity = 32 ft/s². Substituting:

y(t) = 14t - (1/2) * 32 *t² + 0

y(t) = 14t - 16t²

b) when the player is on the ground, the height = 0. hence:

0 = 14t - 16t²

16t² - 14t = 0

t(16t - 14) = 0

t = 0 or 16t - 14 = 0

t = 0 or 16t = 14

t =0 or t = 0.875

Hence t = 0.875 seconds

c) The domain is the set of possible values for the time. Hence:

Domain = (0, 0.875] = 0 < t ≤ 0.875

answer: ratio of volume of cone (with 3 times the height and equal diameter) to volume of cylinder is 2 to 1. the value of π is not used.

step-by-step explanation:

radius is diameter/2 = 4.

area of circle is π r^2.

volume of cylinder is height × area of base = hπr^2 = 3(4^2) π = 48π cubic inches.

not knowing the formula for volume of cone, i calculate it using calculus (which is for calculating things).

volume of cone is integral 0 to h area(y) dy. area(y) is π r(y)^2. r(y) runs r to zero as y runs zero to h, r(0) = r, r(h) = 0.

r(y) = r - r/h y

area(y) = π (r - r/h y)^2 = π (r^2 - 2r^2y/h + r^2y^2/h^2) = π r^2(1-2y/h+y^2/h^2)

antiderivative area(y) = π r^2 (y-y^2/h+y^3/(3h^2))

integral 0 to h = π r^2(h-h^2/h+h^3/(3h^2) - (0)) = π r^2(h - h + h/3) = π r^2(h/3) = π/3 r^2 h

volume of this cone is π/3(4^2)18 cubic inches.

= 6 16 π = 96π cubic inches.

cone has twice the volume, 96/48.

cone height is 6 times the cylinder.

ratio of volume formula: π r^2 cancels.

cone over cylinder ratio is 6h/3 to h, 2 to 1.

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step-by-step explanation:

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