#### Answer

$a=9$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To find the value of $
a
$ such that the given equation, $
at^2+24t+16=0
,$ will have $1$ rational solution, equate the discriminant to $0$ and solve for the variable.
$\bf{\text{Solution Details:}}$
In the equation above, $a=
a
,$ $b=
24
,$ and $c=
16
.$ Using the Discriminant Formula which is given by $b^2-4ac,$ the discriminant is
\begin{array}{l}\require{cancel}
(24)^2-4(a)(16)
\\\\=
576-64a
.\end{array}
Equating the discriminant to $0$ so that the given equation will have $1$ rational solution, then
\begin{array}{l}\require{cancel}
576-64a=0
.\end{array}
Using the properties of equality to isolate the variable results to
\begin{array}{l}\require{cancel}
-64a=-576
\\\\
a=\dfrac{-576}{-64}
\\\\
a=9
.\end{array}
Hence, the given equation has $1$ rational solution when $
a=9
.$