Suppose k grows at a rate of percent and l grows at a rate of
Percent. If
Then y grows at ________ percent.
A) 6
B) 5
C) 7
D) -6
E) 12

Question

Suppose k grows at a rate of  percent and l grows at a rate of 
Percent. If 
Then y grows at ________ percent.
A) 6 
B) 5 
C) 7 
D) -6 
E) 12

Quizplus · Accepted Answer

Since y is a function of both k and l, we need to use the multivariable chain rule of differentiation to solve this problem.

Let y = f(k,l)

Taking the partial derivative of y with respect to k, we get:

∂y/∂k = ∂f/∂k + (∂f/∂k)(dk/dt)

Using the given information, we can substitute the growth rates of k and l:

∂y/∂k = ∂f/∂k + 0.11k

Similarly, taking the partial derivative of y with respect to l, we get:

∂y/∂l = ∂f/∂l + 0.11l

Multiplying both sides by (l/y) and adding the two equations, we get:

(∂y/∂k)(k/y) + (∂y/∂l)(l/y) = ∂f/∂k (k/y) + ∂f/∂l (l/y) + 0.11(k+l)/y

Substituting the given values, we get:

(0.11eab38d_0771_2f2a_a5fb_cb07beb2e967_TB6622_11)(eab38d_0771_2f2b_a5fb_71b618675c9c_TB6622_11)/(8127193b2a8f_TB6622_11) + (0.11eab38d_0771_2f2b_a5fb_71b618675c9c_TB6622_11)(eab38d_0771_2f2a_a5fb_cb07beb2e967_TB6622_11)/(8127193b2a8f_TB6622_11) = 0 + 0 + 0.11

Simplifying the expression, we get:

0.11(2eab38d_0771_2f2a_a5fb_cb07beb2e967_TB6622_11)(2eab38d_0771_2f2b_a5fb_71b618675c9c_TB6622_11)/(8127193b2a8f_TB6622_11) = 0.11

Canceling out the common factor of 0.11 and solving for the growth rate of y, we get:

eab38d_0771_2f2c_a5fb_8127193b2a8f_TB6622_11 = 100(2eab38d_0771_2f2a_a5fb_cb07beb2e967_TB6622_11)(2eab38d_0771_2f2b_a5fb_71b618675c9c_TB6622_11)/(8127193b2a8f_TB6622_11)

Simplifying further, we get:

eab38d_0771_2f2c_a5fb_8127193b2a8f_TB6622_11 = 100(4eab38d_0771_2f2a_a5fb_cb07beb2e967_TB6622_11)(eab38d_0771_2f2b_a5fb_71b618675c9c_TB6622_11)/(8127193b2a8f_TB6622_11)

eab38d_0771_2f2c_a5fb_8127193b2a8f_TB6622_11 = 100(4k)(l)/(y)

Therefore, the growth rate of y is:

eab38d_0771_2f2c_a5fb_8127193b2a8f_TB6622_11 = 100(4k)(l)/(y) = 100(4)(11eab38d_0771_2f2a_a5fb_cb07beb2e967_TB6622_11)(11eab38d_0771_2f2b_a5fb_71b618675c9c_TB6622_11)/(8127193b2a8f_TB6622_11)

Simplifying, we get:

eab38d_0771_2f2c_a5fb_8127193b2a8f_TB6622_11 = 7

Therefore, the answer is (C) 7.

Suppose K Grows at a Rate of Percent and L