Homework Help

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October 14th 2007, 08:36 AM
vc15ao4

Homework Help

1) The time required for a pendulum to complete one round trip of motion is given by T=2pie(squareroof of (g/32). Solve the formula for g. Assume all variables represent positive quantities.

2)A local fitness club is gaining members at a constant rate since its grand opening. There were 75 members when the club opened and after 28 days, there were 271 members. Assume the relationship between the number of members, N, and the number of days since the grand opening, t, is linear. Express N in terms of t.

3)
Two cars leave the same intersection at 10 am. One travels north at a constant rate of 35 mph and the other travels east at a constant rate of 30 mph. At approximately what time will the two cars be 100 miles apart?

4)
The average age of a person when they first get married has been increasing at a constant rate (it is linear). In 1970, the average age was 20 and in 2000, it was 25 years of age. Let t represent the number of years since 1970. Express the average age in years, A, in terms of t.

5)
Sarah has decided to purchase a new vehicle. She has her eye on a new sports car or on a new SUV. The sports car costs $21,500 and has estimated operating costs of $1,800 per year. The SUV costs $19,800, but operating costs are estimated to be $2,300 per year. If x represents the number of years, choose the inequality that describes the number of years the SUV is less expensive than the sports car. Do not solve.

6)
A square garden is to be tilled and enclosed with a fence. The cost of preparing the soil will be $1 per square foot and the fence will cost $3 per foot. Find the dimensions of the garden that can be tilled and enclosed for $220.

7)
A rectangular picture has an area of 50 square inches. The picture is surrounded by a border of uniform width. The outer dimensions (picture with the border) are 8 inches by 10 inches. If the variable represents the width of the border, find the equation that would be used to find . Do not solve the equation. Simplify the equation. (Hint: Draw and label a picture.)

8)
A stone is projected upward with an initial speed of 112 ft./sec. The number of feet, , above the ground after seconds is given by sts=−16t2+112t. When will the stone be 160 feet above the ground?

9)
A manufacturer sells lamps for $6 each. At this price, he sells 3000 lamps. He wishes to raise the selling price and knows that only 1500 lamps will be sold if the selling price is $8 each. Given that the selling price, , and the number of lamps sold, pN, are linearly related, express N in terms of .
October 14th 2007, 01:25 PM
Jhevon

Quote:

Originally Posted by vc15ao4

1) The time required for a pendulum to complete one round trip of motion is given by T=2pie(squareroof of (g/32). Solve the formula for g. Assume all variables represent positive quantities.

$T = 2 \pi \sqrt {\frac g{32}}$ ........divide through by $2 \pi$

$\Rightarrow \frac T{2 \pi} = \sqrt {\frac g{32}}$ .........square both sides

$\Rightarrow \frac {T^2}{4 \pi^2} = \frac g{32}$ .........multiply both sides by 32

$\Rightarrow g = \frac {8T^2}{\pi^2}$
October 14th 2007, 01:30 PM
Jhevon

Quote:

Originally Posted by vc15ao4

2)A local fitness club is gaining members at a constant rate since its grand opening. There were 75 members when the club opened and after 28 days, there were 271 members. Assume the relationship between the number of members, N, and the number of days since the grand opening, t, is linear. Express N in terms of t.

the relationship is linear, thus we can express it in the form of the equation of a line, that is,

$N = mt + b$ ........where $m$ is the slope, $b$ is the y-intercept

when $t = 0$ , $N = 75$

thus, $75 = 0 + b \implies \boxed {b = 75}$

when $t = 28$ , $N = 271$

thus, $271 = 28m + 75$

now solve for $m$ and then plug it into $N = mt + 75$ to get the desired relationship
October 14th 2007, 01:42 PM
Jhevon

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Quote:

Originally Posted by vc15ao4

3)
Two cars leave the same intersection at 10 am. One travels north at a constant rate of 35 mph and the other travels east at a constant rate of 30 mph. At approximately what time will the two cars be 100 miles apart?

see the diagram below.

First let's define our variables:

Let $A$ be the car that travels north
Let $B$ be the car that travels east
Let $C$ be the point both cars start at
Let $d_A$ be the distance A travels
Let $d_B$ be the distance B travels
Let $t$ be the time in hours since 10 am.
Let $S$ be speed in general
Let $d$ be distance in general

Note that we want the distance AB in the diagram to be 100 miles. also note that we have a right triangle (we will use Pythagoras' theorem to solve our problem).

Now, $S = \frac dt$

$\Rightarrow d = St$

Thus, the distance car A travels in time $t$ is given by:

$d_A = 35t$

and the distance car B travels in time $t$ is given by:

$d_B = 30t$

thus, by Pythagoras' theorem, we want $t$ such that, $(35t)^2 + (30t)^2 = 100$

now solve for $t$ . that will give you the number of hours since 10 am the cars were traveling for. then you can tell what time they will be 100 miles apart
October 14th 2007, 02:03 PM
Jhevon

Quote:

Originally Posted by vc15ao4

4)
The average age of a person when they first get married has been increasing at a constant rate (it is linear). In 1970, the average age was 20 and in 2000, it was 25 years of age. Let t represent the number of years since 1970. Express the average age in years, A, in terms of t.

do the same thing i did with number 2
October 14th 2007, 02:08 PM
Jhevon

Quote:

Originally Posted by vc15ao4

5)
Sarah has decided to purchase a new vehicle. She has her eye on a new sports car or on a new SUV. The sports car costs $21,500 and has estimated operating costs of $1,800 per year. The SUV costs $19,800, but operating costs are estimated to be $2,300 per year. If x represents the number of years, choose the inequality that describes the number of years the SUV is less expensive than the sports car. Do not solve.

the total cost (cost plus operating cost) of the sports car for $x$ years is $21500 + 1800x$

the total cost of the SUV for $x$ years is $19800 + 2300x$

thus we want $x$ such that $19800 + 2300x < 21500 + 1800x$
October 14th 2007, 02:14 PM
Jhevon

Quote:

Originally Posted by vc15ao4

6)
A square garden is to be tilled and enclosed with a fence. The cost of preparing the soil will be $1 per square foot and the fence will cost $3 per foot. Find the dimensions of the garden that can be tilled and enclosed for $220.

Let $x$ be the side length of the garden in feet.

then the fence will cost $3(4x) = 12x$ ...since there are four sides of length $x$

the soil preparation cost will be $x^2$ ...since it cost a dollar per unit area and $x^2$ is the area of a square with side length $x$

thus we want:

$12x + x^2 = 220$

this is a quadratic equation, solve it for $x$ to get the dimensions of the garden.

this quadratic can be factored by foiling, but if you can't see the factors, just use the quadratic formula (you know what that is, right?)
October 14th 2007, 02:25 PM
Jhevon

1 Attachment(s)

Quote:

Originally Posted by vc15ao4

7)
A rectangular picture has an area of 50 square inches. The picture is surrounded by a border of uniform width. The outer dimensions (picture with the border) are 8 inches by 10 inches. If the variable represents the width of the border, find the equation that would be used to find . Do not solve the equation. Simplify the equation. (Hint: Draw and label a picture.)

you have things missing in this question. i suppose you want to find the equation that gives us the width of the border.

see the diagram below

Let $x$ be the width of the border.
Then the length of the picture is $10 - 2x$
and the width of the picture is $8 - 2x$

since $\mbox{Area } = \mbox { Length } \times \mbox { Width }$

we have that $50 = (10 - 2x)(8 - 2x)$

and you can simplify that to get a quadratic. you don't have to solve it
October 14th 2007, 02:47 PM
Jhevon

Quote:

Originally Posted by vc15ao4

8)
A stone is projected upward with an initial speed of 112 ft./sec. The number of feet, , above the ground after seconds is given by sts=−16t2+112t. When will the stone be 160 feet above the ground?

just solve the quadratic $-16t^2 + 112t = 160$

that is, solve $t^2 - 7t + 10 = 0$ for $t$

(the equation can be factored by foiling, you will get two times. one will be when the stone is on its way up, the other when it is on its way down)
October 14th 2007, 03:00 PM
Jhevon

Quote:

Originally Posted by vc15ao4

9)
A manufacturer sells lamps for $6 each. At this price, he sells 3000 lamps. He wishes to raise the selling price and knows that only 1500 lamps will be sold if the selling price is $8 each. Given that the selling price, , and the number of lamps sold, pN, are linearly related, express N in terms of .

again, since it's linear, we have $N = m_pN + b$ where $m$ is the slope and $b$ is the y-intercept

we know that when $_pN = 6$ , $N = 3000$

so, $3000 = 6m + b$

also, when $_pN = 8$ , $N = 1500$

so, $1500 = 8m + b$

thus we have the system of linear equations:

$6m + b = 3000$ ..............(1)
$8m + b = 1500$ ..............(2)

solve this system for m and b and plug their values into $N = m_pN + b$ to find the desired relationship

ALTERNATE SOLUTION:

recall that, for a line, the slope m, of a line connecting the points $(x_1,y_1)$ and $(x_2,y_2)$ , is given by $m = \frac {y_2 - y_1}{x_2 - x_1}$

here, we can treat the $N$ 's as the $y$ 's and the $_pN$ 's as the $x$ 's

so here, $m = \frac {1500 - 3000}{8 - 6}$

then we can use any one point in te point slope form to get the equation of the line. call $(x_1, y_1)$ say $(8,1500)$

then $N - 1500 = m(_pN - 8)$

Now solve for $N$