Answer
$2$ inches.
Work Step by Step
The given values are.
Length of the painting: $16$ inches.
Width of the painting: $10$ inches.
Let the uniform width be $t$.
Thus, the length of the painting and frame is $16+2t$
and the width of the painting and frame is $10+2t$.
Combined area of the painting and frame is $(16+2t)(10+2t)$.
We are given that the comboined area of the painting and frame is $280$, so we have.
$\Rightarrow (16+2t)(10+2t)=280$.
Apply the distributive property.
$\Rightarrow 16\cdot 10+16\cdot 2t+2t\cdot 10+2t\cdot 2t=280$.
Simplify.
$\Rightarrow 160+32t+20t+4t^2=280$
Subtract $280$ from both sides.
$\Rightarrow 160+32t+20t+4t^2-280=280-280$
Add like terms.
$\Rightarrow 4t^2+52t-120=0$
Divide both sides by $4$.
$\Rightarrow \frac{1}{4}(4t^2+52t-120)=\frac{1}{4}(0)$
Apply distributive property and simplify.
$\Rightarrow t^2+13t-30=0$
Rewrite the middle term $13t$ as $15t-2t$.
$\Rightarrow t^2+15t-2t-30=0$
Group terms.
$\Rightarrow (t^2+15t)+(-2t-30)=0$
Factor each group.
$\Rightarrow t(t+15)-2(t+15)=0$
Factor out $(t+15)$.
$\Rightarrow (t+15)(t-2)=0$
Set each factor equal to zero.
$t+15=0$ or $t-2=0$
Isolate $t$.
$t=-15$ or $t=2$.
Take the positive value because $t$ represents a dimension and it has to be positive.
The uniform width of the frame is $2$ inches.
