Exercise 15 Page 587

To determine if Eva will reach a height of

10

feet, we first need to substitute

10

for

h

into the given equation.

h = - 16 t^{2} + 22.3 t + 2 ⇓ 10 = - 16 t^{2} + 22.3 t + 2

Since we are not interested in the solutions themselves, we will use the discriminant of the quadratic equation to determine the number of solutions if exist. In the Quadratic Formula,

b^{2} - 4 a c

is the discriminant.

a x^{2} + b x + c = 0 ⇕ x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

Let's first rewrite the given equation in standard form.

10 = - 16 t^{2} + 22.3 t + 2

SubEqn

$LHS - 10 = RHS - 10$

0 = - 16 t^{2} + 22.3 t - 8

RearrangeEqn

Rearrange equation

- 16 t^{2} + 22.3 t - 8 = 0

Having rewritten the equation, we can now identify the values of

a,

b,

and

c .

10 = - 16 t^{2} + 22.3 t + 2 ⇕ 0 = - 16 t^{2} + 22.3 t + (- 8)

Finally, let's evaluate the discriminant.

b^{2} - 4 a c

SubstituteValues

Substitute values

22.3^{2} - 4 (- 16) (- 8)

▼

Simplify

CalcPow

Calculate power

497.29 - 4 (- 16) (- 8)

MultNegNegTwoPar

$(- a) (- b) = a \cdot b$

497.29 - 4 (128)

MultNegPos

$(- a) b = - a b$

497.29 - 512

SubTerm

Subtract term

- 14.71

Since the discriminant is

- 14.71,

the quadratic equation has no real solutions. Therefore, Eva will not reach a height of

10

feet.

Extra

Further information

If the discriminant is greater than zero, the equation will have two real solutions. If it is equal to zero, the equation will have one real solution. Finally, if the discriminant is less than zero, the equation will have no real solutions.