Question

The number of bees spotted in Amelie's garden can also be modeled by the function √ B(x) = 50 k + 2x where x is the daily high temperature, in degrees Farenheit, and k is a positive constant. When the number of bees spotted is 100, the daily high temperature is increasing at a rate of 2◦F per day. According to this model, how quickly is the number of bees changing with respect to time when 100 bees are spotted?

Answer 1

According to this model, the rate of increase of bees spotted with respect to time when 100 bees are spotted = 80 bees per day.

Explanation:

√B(x) = 50 k + 2x

where x is the daily high temperature, in degrees Farenheit.

k is a positive constant.

When the number of bees spotted = 100,

The daily high temperature is increasing at a rate of 2°F per day; (dx/dt) = 2 °F/day

How quickly is the number of bees changing with respect to time when 100 bees are spotted?

√B(x) = 50k + 2x

Taking a time derivative of both sides,

(d/dt) [√B] = (d/dt) [50 k + 2x]

(1/2) (B⁻⁰•⁵) (dB/dt) = (d/dt)(50k) + (d/dt)(2x)

½(B⁻⁰•⁵) (dB/dt) = 50 (dk/dt) + 2 (dx/dt)

B = 100 bees

(dB/dt) = ?

(dk/dt) = 0 (since k is a positive constant)

(dx/dt) = 2°F/day

½(100⁻⁰•⁵) (dB/dt) = 50 (0) + 2 (2)

½(10⁻¹) (dB/dt) = 4

(1/2)(1/10) (dB/dt) = 4

(dB/dt) = 4×2×10 = 80

(dB/dt) = 80 bees per day

Hope this Helps!!!