ALGEBRA WORD PROBLEM!!! PLEASE HELP!!?

May 2021 6 166 Report

ALGEBRA WORD PROBLEM!!! PLEASE HELP!!?

The depth of a pond is 180 cm and is being reduced by 1 cm per week. The depth of a second pond is 160 cm and is being reduced by 1/2 cm per week. If the depths of both ponds continue to be reduced at these constant rates, in about how many weeks will the ponds have the same depth?

A. 10

B. 20

C. 40

D. 80

E. 140

who ever solves can you please show steps

Science & Mathematics / Mathematics

Verified answer

What we are solving for is # of weeks. Isolating the variables, we have:

H_1 = 180 cm <---- initial height

r_1 = 1 cm/week <---- rate of change

H_2 = 160 cm <---- initial height

r_2 = 0.5 cm/week <---- rate of change

We're finding number of weeks after which the height of both ponds are the same.

Height of pond 1: H_1 - r_1*#ofWeeks

Height of pond 2: H_2 - r_2*#ofWeeks

A rate of change multiplied by an amount of time is equal to an amount of change. So rate*time=total change. After 1 week, pond 1 has height 180 cm - (1 cm/week)*1week = 180 cm - 1cm = 179 cm. After 2 weeks, pond 1 has height 180 cm - (1cm/week)*2weeks = 180 cm - 2 cm = 178 cm. And so on.

Because we are finding when the heights are equal, we set the terms for each pond's height equal to each other and solve for #ofWeeks:

H_1 - r_1*#ofWeeks = H_2 - r_2*#ofWeeks

H_1 - r_1*#ofWeeks - H_2 = -r_2*#ofWeeks

H_1 - H_2 = r_1*#ofWeeks - r_2*#ofWeeks

H_1 - H_2 = (r_1 - r_2)*#ofWeeks

(H_1 - H_2) / (r_1 - r_2) = #ofWeeks

(180 cm - 160 cm) / (1 cm/week - 0.5 cm/week) = #ofWeeks

40 weeks

*****ANSWER*****

# of weeks = 40

Answer C

Ian H

It might help you understand if I explain the answer as just numbers.

180 - 40 = 160 - 40/2

That's all there is to it.

But they want to encourage you to form an equation. You start by giving the unknown number of weeks a name. We call it n.

If the depth of the pond is being reduced from 180 cm by 1 cm per week what will its depth be in n weeks ? It depth will be 180 - n.

If the depth of the second pond is being reduced from 160 cm by 1/2 cm per week what will that depth be in n weeks ? That depth will be 160 - n/2.

The word same means equal so that tells you that you can write an equation.

They are the same when 180 - n = 160 - n/2

You solve that by arranging the numbers together on one side and the letters on the other.

180 - 160 = n - n/2

n/2 = 20

n = 40

In 40 weeks the depth will be the same because

180 - 40 = 160 - 20

This is much more explanation than even the teacher usually provides, but I hope it will help you to learn the method of changing words to equations which you will probably need many times yet.

Regards - Ian