Item$2$

Time Remaining $54$ minutes $42$ seconds$00$:$54$:$42$

Item $2$

Time Remaining $54$ minutes $42$ seconds$00$:$54$:$42$

A manufacturer of programmable calculators is attempting to determine a reasonable free$-$service period for a model it will introduce shortly. The manager of product testing has indicated that the calculators have an expected life of $20$ months. Assume product life can be described by an exponential distribution.

T $/$ MTBF e$-$T $/$ MTBF T $/$ MTBF e$-$T $/$ MTBF T $/$ MTBF e$-$T $/$ MTBF

$0\mathrm{.}100\mathrm{.}90482\mathrm{.}600\mathrm{.}07435\mathrm{.}100\mathrm{.}0061$

$0\mathrm{.}200\mathrm{.}81872\mathrm{.}700\mathrm{.}06725\mathrm{.}200\mathrm{.}0055$

$0\mathrm{.}300\mathrm{.}74082\mathrm{.}800\mathrm{.}06085\mathrm{.}300\mathrm{.}0050$

$0\mathrm{.}400\mathrm{.}67032\mathrm{.}900\mathrm{.}05505\mathrm{.}400\mathrm{.}0045$

$0\mathrm{.}500\mathrm{.}60653\mathrm{.}000\mathrm{.}04985\mathrm{.}500\mathrm{.}0041$

$0\mathrm{.}600\mathrm{.}54883\mathrm{.}100\mathrm{.}04505\mathrm{.}600\mathrm{.}0037$

$0\mathrm{.}700\mathrm{.}49663\mathrm{.}200\mathrm{.}04085\mathrm{.}700\mathrm{.}0033$

$0\mathrm{.}800\mathrm{.}44933\mathrm{.}300\mathrm{.}03695\mathrm{.}800\mathrm{.}0030$

$0\mathrm{.}900\mathrm{.}40663\mathrm{.}400\mathrm{.}03345\mathrm{.}900\mathrm{.}0027$

$1\mathrm{.}000\mathrm{.}36793\mathrm{.}500\mathrm{.}03026\mathrm{.}000\mathrm{.}0025$

$1\mathrm{.}100\mathrm{.}33293\mathrm{.}600\mathrm{.}02736\mathrm{.}100\mathrm{.}0022$

$1\mathrm{.}200\mathrm{.}30123\mathrm{.}700\mathrm{.}02476\mathrm{.}200\mathrm{.}0020$

$1\mathrm{.}300\mathrm{.}27253\mathrm{.}800\mathrm{.}02246\mathrm{.}300\mathrm{.}0018$

$1\mathrm{.}400\mathrm{.}24663\mathrm{.}900\mathrm{.}02026\mathrm{.}400\mathrm{.}0017$

$1\mathrm{.}500\mathrm{.}22314\mathrm{.}000\mathrm{.}01836\mathrm{.}500\mathrm{.}0015$

$1\mathrm{.}600\mathrm{.}20194\mathrm{.}100\mathrm{.}01666\mathrm{.}600\mathrm{.}0014$

$1\mathrm{.}700\mathrm{.}18274\mathrm{.}200\mathrm{.}01506\mathrm{.}700\mathrm{.}0012$

$1\mathrm{.}800\mathrm{.}16534\mathrm{.}300\mathrm{.}01366\mathrm{.}800\mathrm{.}0011$

$1\mathrm{.}900\mathrm{.}14964\mathrm{.}400\mathrm{.}01236\mathrm{.}900\mathrm{.}0010$

$2\mathrm{.}000\mathrm{.}13534\mathrm{.}500\mathrm{.}01117\mathrm{.}000\mathrm{.}0009$

$2\mathrm{.}100\mathrm{.}12554\mathrm{.}600\mathrm{.}0101$

$2\mathrm{.}200\mathrm{.}11084\mathrm{.}700\mathrm{.}0091$

$2\mathrm{.}300\mathrm{.}10034\mathrm{.}800\mathrm{.}0082$

$2\mathrm{.}400\mathrm{.}09074\mathrm{.}900\mathrm{.}0074$

$2\mathrm{.}500\mathrm{.}08215\mathrm{.}000\mathrm{.}0067$

a$.$ If service contracts are offered for the expected life of the calculator, what percentage of those sold would be expected to fail during the service period? $($Round your answer to $4$ decimal places.$)$

b$.$ What service period would result in a failure rate of approximately $10$ percent? $($Round intermediate calculations to $2$ decimal places and final answer to the nearest whole number.$)$