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A woman’s glasses accidently fall off her face while she is looking out of a window in a tall building. The equation relating
$h$ , the height above the ground in feet, and
$t$ , the time in seconds her glasses have been falling, is
$h=64-16{t}^{2}.$
(a) How high was the woman’s face when her glasses fell off?
(b) How many seconds after the glasses fell did they hit the ground?
The length of a rectangle is 6 feet more than twice its width. The area is 8 square feet. Find the dimensions.
$\text{length}=8;\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{width}=1$
The length of a rectangle is 18 inches more than three times its width. The area is 81 square inches. Find the dimensions.
The length of a rectangle is two thirds its width. The area is 14 square meters. Find the dimensions.
$\text{width}=\sqrt{21}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{length}=\text{\hspace{0.17em}}\frac{2}{3}\sqrt{21}$
The length of a rectangle is four ninths its width. The area is 144 square feet. Find the dimensions.
The area of a triangle is 14 square inches. The base is 3 inches longer than the height. Find both the length of the base and height.
$b=7;\text{\hspace{0.17em}}\text{\hspace{0.17em}}h=4$
The area of a triangle is 34 square centimeters. The base is 1 cm longer than twice the height. Find both the length of the base and the height.
The product of two consecutive integers is 72. Find them.
$-9,-8\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}8,9$
The product of two consecutive negative integers is 42. Find them.
The product of two consecutive odd integers is 143. Find them. (
$-13,-11\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}11,13$
The product of two consecutive even integers is 168. Find them.
Three is added to an integer and that sum is doubled. When this result is multiplied by the original integer the product is 20. Find the integer.
$n=2,-5$
Four is added to three times an integer. When this sum and the original integer are multiplied, the product is $-1.$ Find the integer.
A box with no top and a square base is to be made by cutting out 2-inch squares from each corner and folding up the sides of a piece of cardboard.The volume of the box is to be 25 cubic inches. What size should the piece of cardboard be?
$4+\sqrt{12.5}\text{\hspace{0.17em}}\text{inches}$
A box with no top and a square base is to made by cutting out 8-inch squares from each corner and folding up the sides of a piece of cardboard. The volume of the box is to be 124 cubic inches. What size should the piece of cardboard be?
A study of the air quality in a particular city by an environmental group suggests that
$t$ years from now the level of carbon monoxide, in parts per million, will be
$A=0.1{t}^{2}+0.1t+\mathrm{2.2.}$
(a) What is the level, in parts per million, of carbon monoxide in the air now?
(b) How many years from now will the level of carbon monoxide be at 3 parts per million?
(a) carbon monoxide now
$2.2$ parts per million
(b)
$2.37\text{\hspace{0.17em}}\text{years}$
A similar study to that of problem 21 suggests
$A=0.3{t}^{2}+0.25t+\mathrm{3.0.}$
(a) What is the level, in parts per million, of carbon monoxide in the air now?
(b) How many years from now will the level of carbon monoxide be at 3.1 parts per million?
A contractor is to pour a concrete walkway around a wading pool that is 4 feet wide and 8 feet long. The area of the walkway and pool is to be 96 square feet. If the walkway is to be of uniform width, how wide should it be?
$x=2$
A very interesting application of quadratic equations is determining the length of a solar eclipse (the moon passing between the earth and sun). The length of a solar eclipse is found by solving the quadratic equation
${\left(a+bt\right)}^{2}+{\left(c+dt\right)}^{2}={\left(e+ft\right)}^{2}$
for
$t$ . The letters
$a,b,c,d,e,$ and
$f$ are constants that pertain to a particular eclipse. The equation is a quadratic equation in
$t$ and can be solved by the quadratic formula (and definitely a calculator). Two values of
$t$ will result. The length of the eclipse is just the difference of these
$t$ -values.
The following constants are from a solar eclipse that occurred on August 3, 431 B.C.
$\begin{array}{ccccccc}a& =& -619& & b& =& 1438\\ c& =& 912& & d& =& -833\\ e& =& 1890.5& & f& =& -2\end{array}$
Determine the length of this particular solar eclipse.
( [link] ) Find the sum: $\frac{2x+10}{{x}^{2}+x-2}+\frac{x+3}{{x}^{2}-3x+2}.$
$\frac{3x+14}{\left(x+2\right)\left(x-2\right)}$
(
[link] ) Solve the fractional equation
$\frac{4}{x+12}+\frac{3}{x+3}=\frac{4}{{x}^{2}+5x+6}.$
(
( [link] ) One pipe can fill a tank in 120 seconds and another pipe can fill the same tank in 90 seconds. How long will it take both pipes working together to fill the tank?
$51\frac{3}{7}$
( [link] ) Use the quadratic formula to solve $10{x}^{2}-3x-1=0.$
( [link] ) Use the quadratic formula to solve $4{x}^{2}-3x=0.$
$x=0,\frac{3}{4}$
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