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Chapter 7: Problem 30
Factor by grouping. $$6 a^{2}+a b-5 b^{2}$$
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Write an equation and solve. The hypotenuse of a right triangle is 2 in. longer than the longer leg. The shorter leg measures 2 in. less than the longer leg. Find the measure of the longer leg of the triangle
Organizers of fireworks shows use quadratic and linear equations to help them design their programs. Shells contain the chemicals which produce the bursts we see in the sky. At a fireworks show the shells are shot from mortars and when the chemicals inside the shells ignite they explode, producing the brilliant bursts we see in the night sky. Shell size determines how high a shell will travel before exploding and how big its burst will be when it does explode. Large shell sizes go higher and produce larger bursts than small shell sizes. Pyrotechnicians take these factors into account when designing the shows so that they can determine the size of the safe zone for spectators and so that shows can be synchronized with music. At a fireworks show, a 3 -in. shell is shot from a mortar at an angle of \(75^{\circ} .\) The height, \(y\) (in feet), of the shell \(t\) sec after being shot from the mortar is given by the quadratic equation $$y=-16 t^{2}+144 t$$ and the horizontal distance of the shell from the mortar, \(x\) (in feet), is given by the linear equation $$x=39 t$$ a) How high is the shell after 3 sec? b) What is the shell's horizontal distance from the mortar after 3 sec? c) The maximum height is reached when the shell explodes. How high is the shell when it bursts after 4.5 sec? d) What is the shell's horizontal distance from its launching point when it explodes? (Round to the nearest foot.) When a 10 -in. shell is shot from a mortar at an angle of \(75^{\circ},\) the height, \(y\) (in feet), of the shell \(t\) sec after being shot from the mortar is given by $$y=-16 t^{2}+264 t$$and the horizontal distance of the shell from the mortar, \(x\) (in feet), is given by$$x=71 t$$ e) How high is the shell after 3 sec? f) Find the shell's horizontal distance from the mortar after 3 sec. g) The shell explodes after 8.25 sec. What is its height when it bursts? h) What is the shell's horizontal distance from its launching point when it explodes? (Round to the nearest foot.) i) Compare your answers to a) and e). What is the difference in their heights after 3 sec? j) Compare your answers to c) and g.). What is the difference in the shells' heights when they burst? k) Assuming that the technicians timed the firings of the 3-in. shell and the 10 -in. shell so that they exploded at the same time, how far apart would their respective mortars need to be so that the 10 -in. shell would burst directly above the 3 -in. shell?
Factor completely. $$k^{4}-81$$
Write an equation and solve. A rectangular aquarium is 15 in. high, and its length is 8 in. more than its width. Find the length and width if the volume of the aquarium is 3600 in \(^{3}\).
Factor completely. $$d^{3}+1$$
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