The daily profit P (in thousands of dollars) from the sale of televisions is a function of the number x of televisions sold (in hundreds). The formula for this function is: P = -5x² + 13x - 4 What are the break-even sales amounts (the sales amounts that result in no profit or loss)?
Accepted Solution
A:
Answer:The break-even sales amounts is 36 or 224.Step-by-step explanation:Consider the provided function. [tex]P= -5x^2 + 13x - 4[/tex]Where x is the number of televisions sold (in hundreds) and P is the profit.We need to calculate the break-even sales amounts.the break-even sales amounts is the sales amounts that result in no profit or loss.That means substitute P=0 and solve for x.[tex]-5x^2 + 13x - 4=0[/tex][tex]5x^2-13x+4=0[/tex][tex]\mathrm{For\:a\:quadratic\:equation\:of\:the\:form\:}ax^2+bx+c=0\mathrm{\:the\:solutions\:are\:}\\x_{1,\:2}=\frac{-b\pm \sqrt{b^2-4ac}}{2a}[/tex]Substitute a=4, b=-13 and c=4 in above formula.[tex]x_{1,\:2}=\frac{-\left(-13\right)\pm \sqrt{\left(-13\right)^2-4\cdot \:5\cdot \:4}}{2\cdot \:5}[/tex][tex]x_{1,\:2}=\frac{13\pm\sqrt{169-80}}{10}[/tex][tex]x_{1,\:2}=\frac{13\pm\sqrt{89}}{10}[/tex][tex]x=\frac{13+\sqrt{89}}{10},\:x=\frac{13-\sqrt{89}}{10}\\x\approx2.243, \ or\ x\approx 0.357[/tex]Therefore, the break-even sales amounts is 36 or 224.