Math 181 Quiz #3 Solutions

1. (5 points) Mold grows in a circular colony on a Petri dish containing an agar gel with
the appropriate nutriment. The area A of the mold colony (in mm²) is a quadratic
function of the number t of hours since the start of the experiment , so that it has the
form

A = at² + bt + c.

The measurement given in the experiments is as fol low

(i) What is the formula for the area of the colony in terms of time in hours.
(ii) Predict the area of the colony when t = 60 hours.
(iii) According to the formula you get, what is the time when the area is 80mm2?
Solution :
(i) Plug the data back to the equation, one has linear system

so that

a = 0.00831,
b = 0.0699,
c = 1.35.

Then the quadratic equation is

A (t) = 0.00831t² + 0.0699t + 1.35.

(ii) When t = 60, according to the equation, one has

A(60) = 0.00831 × 60² + 0.0699 × 60 + 1.35 = 35.5mm².

(iii) When A = 80, then the quadratic equation is

80 = 0.00831t² + 0.0699t + 1.35 => 0.00831t² + 0.0699t − 78.65 = 0.

Therefore, by the quadratic formula, one has

Disregarding the negative one , one has t = 93.2 hours.

2. (6 points) A man stands on the edge of a deep pit in the ground. He stoops and
throws a stone upward, so that it first rises and then drops into the pit. The height
H of the stone (in feet) above the ground at time t seconds after the rock is dropped
is given by the data
The height is given by a quadratic equation of the form

H(t) = at² + bt + c.

(i) What is the formula for the height of the rock in terms of time t.
(ii) Tell the height of the rock after 0.25 seconds.
(iii) Tell the time when the stone passes the ground level again.
(iv) The stone hits the bottom of the pit after 6.13 seconds. How deep is the pit?
Solution:
(i) Plug the data back to the equation, one has linear system

so that

a = −16,
b = 18,
c = 0.

Then the quadratic equation is

H(t) = −16t² + 18t.

(ii) H(0.25) = −16 × (0.25)² + 18 × 0.25 = 3.5 feet.
(iii) Passing ground again means

0 = −16t² + 18t => t = 0, or t = 9/8 = 1.125 sec.

Disregard t = 0, which is the initial starting time, the answer is t = 1.125.
(iv) H(6.13) = −16 × (6.13)² + 18 × 6.13 = −490.9, so that the depth of the pit is
490.9 feet.

3. (9 points) Mold growth experiment is done with the data given below. Based on the
given data, find linear regression . (Recall: for linear regression, you suppose to have
equation . The matrix expression of the regression should be

where .)

Solution: According the formula, you should have

Therefore, the system reads

so



Then the regression line reads

y = 0.754x − 5.12.

Figure 1