Question

*Multiple step problem* I know, its a lot.. I will mark brainliest!

The function p(w)=230(1.1)^w represents the number of specialty items produced at the old factory w weeks after a change in management. The graph represents the number of specialty items produced at the new factory during the same time period.

(a) During Week 0, how many more specialty items were produced at the old factory than at the new factory? Explain
(b) Find and compare the growth rates in the weekly number of specialty items produced at each factory. Show your work
(c) When does the weekly number of specialty items produced at the new factory exceed the weekly number of specialty items produced at the old factory? Explain

Answer 1

A)During 0 week old factory have 40 more products than the new factory.
B)I am so sorry that I forgot how to find the. Rate of growth :)
C)at week 4 the new factory exceed the old factory because new factory produce 337 products and old factory only produce 336.74.


Hope it will help:)

Answer 2

(a) During week 0, 40 more items were produced at the old factory than at the new factory, computed using the exponential function and the graph.

(b) The growth rate of the new factory is 15%, while the growth rate of the old factory is 10%, making the comparison between them as 5%, computed using the exponential function and the graph.

(c) The weekly number of specialty items produced at the new factory exceeds the weekly number of specialty items produced at the old factory from the 5th week, computed using the exponential function and the graph.

What is the general form of an exponential function?

The general form of an exponential function is, $V = V_0(1 + r)^n$, where V is the final value, V₀ is the initial value, r is the growth rate, and n is the number of time periods.

How to solve the question?

We are given that the function $p(w) = 230(1.1)^w$ represents the number of specialty items produced at the old factory w weeks after a change in management, and the graph represents the number of specialty items produced at the new factory during the same time period.

(a) In the question, we are asked during week 0, how many more specialty items were produced at the old factory than at the new factory.

The number of specialty items produced at the old factory in week 0 can be calculated by substituting w = 0 in the function $p(w) = 230(1.1)^w$, representing the number of specialty items produced at the old factory w weeks after a change in management.

Thus, the number of specialty items produced at the old factory in week 0 = $230(1.1)^0 = 230*1 = 230$ {∵ a^0 = 1, for all values of a}.

The number of specialty items produced at the new factory can be calculated by using the graph, by checking the point on the graph where the value of the x-coordinate is 0, as the x-axis represents the number of weeks.

Thus, the number of specialty items produced at the old factory in week 1 = 190 {∵ (0, 190) is the point corresponding to week 0 in the graph}.

Thus, the difference is 230 - 190 = 40.

Thus, during week 0, 40 more items were produced at the old factory than at the new factory.

(b) In the question, we are asked to find and compare the growth rates in the weekly number of specialty items produced at each factory.

The general form of an exponential function is, $V = V_0(1 + r)^n$, where V is the final value, V₀ is the initial value, r is the growth rate, and n is the number of time periods.

To find the growth rates for the old factory, we compare the equation with the general form as follows:

$230(1.1)^w$ is equivalent to $V_0(1 + r)^n$,

or, 1.1 is equivalent to 1 + r,

or, r = 1.1 - 1 = 0.1 or 10%.

Thus, the growth rate for the old factory is 10%.

To find the growth rate for the new factory, we do as follows,

$505 = 190(1 + r)^7$ {Final value after 7 weeks is equal to the product of the initial value and (1 + r) raised to the power 7},

or, $r = {(505/190)^{(1/7)} - 1$,

or, r = 1.1499 - 1,

or, r = 0.1499 ≈ 0.15 = 15%.

Thus, the growth rate for the new factory is 15%.

Thus, the growth rate of the new factory is 15%, while the growth rate of the old factory is 10%, making the comparison between them 5%.

(c) In the question, we are asked to find the week in which the number of specialty items produced in the new factory exceeds the number of specialty items produced in the old factory.

The function representing the items produced at the old factory is, $p(w) = 230(1.1)^w$, while for the new factory can be shown as $q(w) = 190(1.15)^w$.

Thus, we need to find w, when $190(1.15)^w > 230(1.1)^w$,

or, ${(1.15)^w}/{(1.1)^w} > 230/190$,

or, $(1.04545)^w > 1.21053$,

or, $w log (1.04545) > log (1.21053)$,

or, w > 4.298035.

Thus, the weekly number of specialty items produced at the new factory exceeds the weekly number of specialty items produced at the old factory from the 5th week.

Learn more about exponential functions at

https://brainly.com/question/2501715

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