1) Effect of rainfall on predators and prey: The population of lions, L, in Afri

Question

1) Effect of rainfall on predators and prey: The population of lions, L, in Africa greatly depends on the population of their prey, P, which consists mainly of large mammals with a preference for zebras, impalas, wildebeest, buffalo, and warthogs. The extension of vegetation, V , affects the population of prey, which feed on the vegetation The vegetation, and the vegetation if effected by the amount of rainfall, r. These dependencies can be as L = f(P), P = g(V ) and V = h(r). Suppose data was collected on an African wildlife preserve and the functions f, g, h were approximated as follows:

L = f(P) = (1/2)P^2 , P = g(V ) = 2V, V = h(r) = r^1/2 .

A change in the abundance of rainfall will affect the amount of vegetation. This change in vegetation will affect the population of prey which in turn will affect the population of lions on the reserve. Therefore, the amount of rainfall affects the lion population. Compute the rate of change in the population of lions with respect to the rainfall, dL/dr.

2) Tumor Growth: Tumors can differ significantly in shape. For example, an adenocarcinoma (a cancerous tumor originating in glandular tissue) tends to be more irregularly shaped while an adenoma (a benign tumor originating in glandular tissue) tends to be more regularly shaped (for example close to spherical). Assume were studying a tumor that is approximately spherical. A tumor grows such that its radius expands at a constant rate k. Determine the rate of growth of the volume of tumor when the radius is 5 millimeters.

(dV/dt) = 4r^2(dr/dt)

3) Carbon monoxide level: An environmental study of a certain community suggests that the average daily level of carbon monoxide in the air may be modeled by the formula

C(p) = sqrroot(0.5p^2 + 17)

parts per million when the population is p thousand. It is estimated that t years from now, the population of the community will be

p(t) = 3.1 + 0.1t^2

thousand. At what rate will the carbon monoxide level be changing with respect to time 3 years from now?

4) Drug Sensitivity: It is extremely important for doctors to understand the characteristics of the drugs they prescribe to patients. The strength of the drug is given by R(M) where M measures the dosage, i.e. the amount of medicine absorbed in the blood, and the sensitivity of the patients body to the drug is the derivative of R with respect to M. For a certain drug, the drug strength is described by

R(M) = 2M sqrroot(10 + 0.5M)

where M is given in milligrams. Find R0 (50), the sensitivity to a dose of 50 mg.

Explanation / Answer

(1) given

L = f(P) = (1/2)P^2 , P = g(V ) = 2V, V = h(r) = r^1/2

dL/dr = d/dr ((1/2)P^2 ) ==> d/dr ((1/2)(2V)^2 ) ==>d/dr ((1/2)( r^1/2)^2 )

==> d/dr ((1/2) r ==> 1/2

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( 3 ) given   C(p) = sqrroot(0.5p^2 + 17) and p(t) = 3.1 + 0.1t^2

C'(p) =d/dt( sqrroot(0.5p^2 + 17) )

===> d/dt( sqrroot(0.5(3.1 + 0.1t^2)^2 + 17) )

==> 1/( 2 * sqrroot(0.5(3.1 + 0.1t^2)^2 + 17) ) ( d /dt 0.5(3.1 + 0.1t^2)^2 + 17) )

==> 1/( 2 * sqrroot(0.5(3.1 + 0.1t^2)^2 + 17) ) ( 0.2 t ( 0.1 t^2 + 3.1 ))

===> C'(p) = ( 0.1t ( 0.1 t^2 + 3.1 )) / (sqrroot(0.5(3.1 + 0.1t^2)^2 + 17) )

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( 4 ) R(M) = 2M sqrroot(10 + 0.5M)

R'(M) = d/dM (2M sqrroot(10 + 0.5M))

   ==>sqrroot(10 + 0.5M)d/dM(2M) + (2M) d/dM(sqrroot(10 + 0.5M))

   ==> 2 * sqrroot(10 + 0.5M) + (2M) (1/( 2 *(sqrroot(10 + 0.5M) ) ) d/dM((10 + 0.5M))

=====> 2 * sqrroot(10 + 0.5M) + (2M) (1/( 2 *(sqrroot(10 + 0.5M) ) ) (0.5)

==> 2 * ( sqrroot(10 + 0.5M) + 0.25 M / (sqrroot(10 + 0.5M) ) )

==> R'(50) ==>2 * ( sqrroot(10 + 0.5(50)) + 0.25 (50) / (sqrroot(10 + 0.5(50) )

R'(50)====> 16.05793mg

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