Equations and Inequalities

[acarp]

Please help me with these few problems...

1)A city government built a $60 million sports arena. Some of the money was raised by selling bonds that pay simple interest at a rate of 11% annually. The remaining amount was obtained by borrowing money from an insurance company at a simple interest rate of 10%. How much was financed through the insurance company if the annual interest is $6.22 million? Round to the nearest million,

2) At 7 A.M. a snowplow, traveling at a constant speed, begins to clear a highway leading out of town. At 9 A.M. an automobile begins traveling the highway at a speed of 30 mi/hr and reaches the plow 30 minutes later. Find the speed of the snowplow.

3)Two children own two-way radios that have a maximum range of 7 miles. One leaves a certain point at 1:00 P.M., walking due north at a rate of 4 mi/hr. The other leaves the same point at 1:05 P.M., traveling due south at 6 mi/hr. When will they be unable to communicate with one another?

 

[acarp]

i have no clue on where to get started, i was sick during this lesson of my class and need help

 

[mmm4444bot]

I do not understand exercise (1). Does the $6.22 million annual interest come from the 10% investment, or is it the combined interest from both investments?

I generally do not complete homework exercises, but I'm making an exception in your case because I get the impression that you won't benefit from guidance (since you missed classes).

Exercises (2) and (3) involve a classic relationship between distance traveled (D), constant rate of travel (R), and elapsed time (T):

D = R * T

In exercise (2), I think that we first need to make the following assumption: the plow and the car both start from the same location.

We're told that the car drives at a constant rate of 30 mph for 30 minutes. Since we have both the rate (R) and the time (T), we can find the distance (D) travelled by both the car and the plow.

D = R * T

D = 30 * 0.5

D = 15

The car and plow each travelled 15 miles.

The exercise asks for the rate of the plow; as with most word problems, we need to pick a symbol to represent the unknown value for which we're asked.

It's a good habit to write it down.

x = the rate of the plow, in mph

Now, we can find this rate because we know the other two parts of the relationship for the plow: the distance and the time travelled.

The distance is 15 miles.

The time travelled is 2 hours more than the car's travel time because the plow got a two-hour "head start".

D = R * T

15 = x * 2.5

Solve for x, and you've got the plow's rate, in mph.

Exercise (3) can be tackled in different ways. Here is just one.

Since 4 + 6 = 10, the walking time is clearly less than an hour. In other words, the time it takes for them to become 7 miles apart will be a fraction of 1 hour. Knowing this in advance leads me to switch time units from hours to minutes.

x = the number of minutes required for the kids to be 7 miles apart

Since the time units are now minutes, we convert both given rates from mile/hour to mile/minute by dividing each rate by 60 minutes in 1 hour.

Northbound kid's rate: 4/60 = 1/15 mile/minute.

Southbound kid's rate: 6/60 = 1/10 mile/minute.

Let time start elapsing at 1:05 PM. In other words, x = 0 at 1:05 PM.

Now we can write an expression for the distance travelled by the northbound kid during their five-minute "head start".

D = R * T

D = (1/15) * 5

D = 1/3

During the five-minute "head start", the northbound kid travels 1/3 mile.

At this point, it's 1:05 PM, and time starts elapsing from x = 0.

Also, at this point, the kids are 1/3 mile apart, so they only have another 6 2/3 miles to go.

6 2/3 = 20/3 miles.

Since distance can be expressed as the product of rate and time, the remaining distance covered during the x minutes by the northbound kid is (1/15)x.

Likewise, the remaining distance covered during the x minutes by the southbound kid can be expressed as (1/10)x.

The sum of these two distances must be 20/3 miles.

(1/15)x + (1/10)x = 20/3

Solve for x, and you've got the total number of minutes elapsed from 1:05 PM for when they will be exactly 7 miles apart.

Report your answer in terms of an inequality for when their distance will be greater than 7 miles.

Please show some work, or explain what you're thinking, if you would like more help. We generally guide students versus completing their homework.