On some tubes of caulking, the diameter of the circular

nozzle opening can be adjusted to produce lines of
varying thickness. The table shows the length l of
caulking obtained from a tube when the nozzle opening
has diameter d and cross-sectional area A.
48. Determine whether l varies inversely with d. If so,
write an equation relating l and d.
49. Determine whether l varies inversely with A. If so,
write an equation relating l and A.
50. Find the length of caulking you get from a tube
whose nozzle opening has a diameter of �
34
� inch.
d (in.) A (in.2) l (in.)

18
� �2
π
5�6 1440

14
� �6
π
4 �
360

38�
�2
9
5
π
6 �
160

12
� �1
π
6 �
90
no
yes; l = } 4
8
5
A
π} 40 in.

dndjd

To determine whether l (length of caulking) varies inversely with d (diameter of nozzle opening), we need to check if the product of l and d is constant for all given values.

To find the equation relating l and d, we need to find the value of the constant by dividing the product of l and d in one set of data by the corresponding values in another set of data. Let's take the first set of data: d = 1/8 inch, A = (π/4)(5/6)^2 = 0.654 in^2, l = 1440 in.

(l1 * d1) = (l2 * d2)
(1440 * 1/8) = (l2 * 34/32)
l2 = (1440 * 1/8) * (32/34)
l2 = 1440/17

So, the equation relating l and d is l = (1440/17)d.

Moving on to the second question, to determine whether l varies inversely with A (cross-sectional area), we need to check if their product is constant for all given values.

To find the equation relating l and A, we follow the same process as before. Let's take the first set of data: d = 1/8 inch, A = 0.654 in^2, l = 1440 in.

(l1 * A1) = (l2 * A2)
(1440 * 0.654) = (l2 * 0.654)
l2 = (1440 * 0.654) / 0.654
l2 = 1440

So, the equation relating l and A is l = 1440.

Now, for the third question, we need to find the length of caulking obtained from a tube with a nozzle opening diameter of 3/4 inch.

Using the equation from question 48, l = (1440/17)d, we substitute d = 3/4 inch:
l = (1440/17) * 3/4
l = (1440 * 3) / (17 * 4)
l = 4320 / 68
l = 63.53 inches

Therefore, the length of caulking obtained from a tube with a nozzle opening diameter of 3/4 inch is approximately 63.53 inches.