Exercise 34 Page 883

A runner runs on a racetrack in the shape of a circular arc with a radius of 16.7 meters. We want to find the speed of the runner. To do so, we will use the Angle of Incline Formula. tanθ=v^2/gR We know that R=16.7 and g=9.8. We need to find the value of tanθ before finding the speed of the runner. We are given the value of the sine of θ. First consider the Quotient Identity for tangent. tanθ=sinθ/cosθTo find the value of the tangent we need to first find the cosine of θ. Recall one of the Pythagorean Identities considering both sinθ and cosθ. cos^2θ+sin^2θ=1 We know that sinθ= 14. Let's substitute this value into the identity and solve for cosθ.

cos^2θ+sin^2θ=1

Substitute

sinθ= 1/4

cos^2θ+( 1/4)^2=1

▼

Solve for cosθ

PowQuot

(a/b)^m=a^m/b^m

cos^2θ+1/16=1

SubEqn

LHS-1/16=RHS-1/16

cos^2θ=1-1/16

OneToFrac

Rewrite 1 as 16/16

cos^2θ=16/16-1/16

SubFrac

Subtract fractions

cos^2θ=15/16

SqrtEqn

sqrt(LHS)=sqrt(RHS)

sqrt(cos^2θ)=sqrt(15/16)

SqrtQuot

sqrt(a/b)=sqrt(a)/sqrt(b)

sqrt(cos^2θ)=sqrt(15)/4

SqrtToAbs

sqrt(a^2)=|a|

|cosθ|=sqrt(15)/4

Since the angle of incline is an acute angle, the value of its cosine needs to be positive. Therefore, cosθ= sqrt(15)4. Knowing the values for both sine and cosine we can use the previously mentioned ratio to find tanθ.

tanθ=sinθ/cosθ

SubstituteII

sinθ= 1/4, cosθ= sqrt(15)/4

tanθ=14/sqrt(15)4

▼

Evaluate right-hand side

DivFracByFracD

.a /b./.c /d.=a/b*d/c

tanθ=1/4 * 4/sqrt(15)

MultFrac

Multiply fractions

tanθ=4/4sqrt(15)

ReduceFrac

a/b=.a /4./.b /4.

tanθ=1/sqrt(15)

Now we have every value from the given formula except for the speed. Therefore, we can substitute g=9.8, R=16.7, and tanθ= 1sqrt(15) into the Angle of Incline Formula and solve for v.

tanθ=v^2/gR

SubstituteValues

Substitute values

1/sqrt(15)=v^2/9.8(16.7)

▼

Solve for v

Multiply

Multiply

1/sqrt(15)=v^2/163.66

MultEqn

LHS * 163.66=RHS* 163.66

1/sqrt(15)* 163.66=v^2

RearrangeEqn

Rearrange equation

v^2=1/sqrt(15) * 163.66

MoveRightFacToNumOne

1/b* a = a/b

v^2=163.66/sqrt(15)

SqrtEqn

sqrt(LHS)=sqrt(RHS)

sqrt(v^2)=sqrt(163.66/sqrt(15))

SqrtToAbs

sqrt(a^2)=|a|

|v|=sqrt(163.66/sqrt(15))

UseCalc

Use a calculator

|v|=6.500525...

RoundDec

Round to 1 decimal place(s)

|v|≈ 6.5

Note that since negative speed does not make sense, the value needs to be positive. Therefore, the speed of the runner is approximately 6.5.