Exercise 49 Page 263

An equation is given to find the height h in feet of a projectile t seconds after it is shot upwards from ground level. h=96t-16t^2 We will find the value(s) of t when the height of a projectile is 96 feet. Let's first substitute h= 96 into the equation. 96=96t-16t^2Notice that it is a quadratic equation. Let's rewrite the equation so that all of the terms are on the left-hand side and then simplify it as much as possible.

96=96t-16t^2

AddEqn

LHS+16t^2=RHS+16t^2

16t^2+96=96t

SubEqn

LHS-96t=RHS-96t

16t^2-96t+96=0

DivEqn

.LHS /16.=.RHS /16.

t^2-6t+6=0

Let's recall the Quadratic Formula used to find solution(s) to quadratic equations. ax^2+ bx+ c=0 [0.5em] ⇕ [0.5em] x=- b± sqrt(b^2-4 a c)/2 a To solve our equation we first need to identify the values of a, b, and c. t^2-6t+6=0 [0.5em] ⇕ [0.5em] 1t^2+( - 6)t+ 6=0 We see that a= 1, b= - 6, and c= 6. Let's substitute these values into the Quadratic Formula.

t=- b±sqrt(b^2-4ac)/2a

SubstituteValues

Substitute values

t=-( - 6)±sqrt(( - 6)^2-4( 1)( 6))/2( 1)

▼

Simplify

NegNeg

- (- a)=a

t=6±sqrt((- 6)^2-4(1)(6))/2(1)

IdPropMult

Identity Property of Multiplication

t=6±sqrt((- 6)^2-4(6))/2

CalcPow

Calculate power

t=6±sqrt(36-4(6))/2

Multiply

Multiply

t=6±sqrt(36-24)/2

SubTerms

Subtract terms

t=6±sqrt(12)/2

SplitIntoFactors

Split into factors

t=6±sqrt(4* 3)/2

SimpRoot

Simplify root

t=6 ± 2sqrt(3)/2

FactorOut

Factor out 2

t=2(3 ± sqrt(3))/2

ReduceFrac

a/b=.a /2./.b /2.

t=3 ± sqrt(3)

The solutions for this equation are t=3 ± sqrt(3). Let's calculate them separately using a calculator and round to the nearest tenth.

lt_1=3+sqrt(3) t_2=3-sqrt(3)

CalcRoot

Calculate root

lt_1=3+1.732050... t_2=3-1.732050...

AddSubTerms

Add and subtract terms

lt_1=4.732050... t_2=1.267949...

RoundDec

Round to 1 decimal place(s)

lt_1=4.7 t_2=1.3

Therefore, the height of the projectile will be 96 feet on the way up at about 1.3 seconds and on the way down at about 4.7 seconds.