Question 751632: Hello, I tried to solve this question using the answer to part a (25+3x) However, in solving i got two real roots and neither could be dismissed as both satisfied the equation. The question is:
Q 48) An apprentice carpenter has designed a wooden coffee table. He expects to sell 25 of these tables for $95 each, at an upcoming artisan market. By conducting a survey, he determines that for each $2 reduction in the price of the table he will likely gain 3 sales.
a) An expression that describes the price of each table, based on the number of reductions, is 95-2x, where x represents the number of $2 reductions in price. Write a similar expression to describe the number of tables the carpenter can expect to sell, based on the number of reductions.
b) Write an equation to describe the carpenters income as a product of the price of each table and the expected number of tables sold.
c) The carpenter hopes to earn $3600 to pay for his time, materials, and sales booth, as well as make a small profit. Determine the number of $2 reductions he can apply to the price of the tables to make $3600.
d) How many reductions will produce maximum income? Indicate the number of tables sold and the price of each one when maximum income occurs
Thank you
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! since this is a quadratic equation, it will have roots that will satisfy the equation.
those roots can be real or imaginary.
since the roots for this equation are real, the graph of the equation will cross the x-axis at those roots.
the formula for your equation is y = (95-2x) * (25+3x)
x is the number of reductions in price.
95 is the starting price and 25 is the starting number of units sold.
each reduction in price is represented by 95 - 2x.
each increase in number of sales is represented by 25 + 3x.
you don't want to dismiss these roots.
they belong there.
the graph of your equation looks like this:
since x can't be less than 0, you ignore anything on the graph that is to the left of the y-axis.
when x = 0, your revenue is 95 * 25 = 2375.
your expression of (25 + 3x) to model the increase in sales is correct.
the formula for revenue is y = (95-2x)*(25+3x)
with one reduction in price, the revenue becomes 93 * 28 = 2604
with 2 reductions in price, the revenue becomes 91 * 31 = 2821
to solve for how many reductions in price are required to make revenue of 3600, you would need to solve the equation 3600 = (95 - 2x) * (25 + 3x)
you can also look at the graph and get an idea of where that value would be.
it appears that the value of x will be between 6 and 7 when the value of y is 3600.
the graph can only approximate the answer.
to find the exact answer, you need algebra.
the equation is already in the form to solve for the roots.
set the equation to 0 and you get:
95 - 2x) * (25 + 3x) = 0
this leads to:
95 - 2x = 0 of 25 + 3x = 0
you gets roots at x = 95/2 and x = -25/3.
in decimal form, these roots are at x = 47.5 and x = -8.33333.
look at the graph to confirm the graph crosses the axis at these values of x.
to solve for when y = 3600 (revenue = 3600), you need to set the equation equal to 3600 and solve.
you get (95-2x) * (25 + 3x) = 3600
to solve this, you need to make it equal to 0 by subtracting 3600 from both sides of the equation and then solving for those roots.
you get (95 - 2x) * (25 + 3x) - 3600 = 0
you will need to convert this into quadratic form and then solve either by factoring or by using the quadratic formula.
you do this by multiplying the factors out.
the quadratic equation becomes y = 2375 + 235x - 6x^2 - 3600 = 0
after combining like terms, this equation becomes:
-6x^2 + 235x - 1225 = 0 which is then multiplied by -1 on both sides of the equation to get:
6x^2 - 235x + 1225 = 0
this is done because having a positive coefficient on the x^2 term is easier to factor.
i didn't bother looking for factors.
i just used the quadratic formula which will find the roots of any quadratic equation, even if those roots are imaginary.
in this equation, the roots are real.
solving this equation using the quadratic formula gets you:
x = 32.97513223
or
x = 6.191534433
at each of these values, the value of y will be 3600 in the original equation.
this was confirmed to be true so the solution is good.
maximum income will be at the maximum point of the quadratic equation.
this point can be found using the formula of x = -b / 2a.
the quadratic formula we are solving for is the original quadratic equation, not the adjusted equation we used to find the value of x when y = 3600.
that original equation is:
(95 - 2x) * (25 + 3x) = 0
the quadratic form of that equation is ax^2 + bx + c = 0
we need to multiply the factor out again in order to get the quadratic form, which is:
-6x^2 + 235x + 2375 = 0
multiply both sides of that equation by -1 to get:
6x^2 - 235x - 2375 = 0
a is the coefficient of the x^2 term.
b is the coefficient of the x term.
c is the constant.
this makes:
a = 6
b = -235
c = -2375
the formula for the max / min point of a quadratic equation is:
x = -b/2a
replace a and b with their respective values and you get:
x = 235/12 = 19.5833333
when x = 19.583333333..., the quadratic equation should be at it's maximum value.
you can confirm that by looking at the graph to see if it's reasonable.
i looked at the graph and it appears to be reasonable.
i also used a graphing calculator to solve for the maximum point and the graphing calculator confirms it as well.
the graphing calculator tells me that the maximum value of the graph is qhen x = 19.583333...... and the value of y at that point is equal to 4676.0417
without the graphing calculator, you would need to use the original quadratic equation and replace x with 19.5833333.... and solve for y.
i believe that answers all the questions.
since you can't sell a part of a table, you would want to find the nearest integer that contains the maximum value on the graph.
try 19 and try 20.
since the graph is symmetric, i would assume 20.
19 gets you 4674
20 gets you 4675
both are close but 20 gets the nod.
let me know if you have any questions regarding this solution.
the equation of (95-2x)*(25+3x) is correct.
you will get 2 roots that will not go away because they belong there.
you solve for the roots by setting the original equation equal to 0.
you solve for find 3400 by setting y = 3400 and then subtracting it from both sides of the equation so the equation is once again in standard form and you then solve for that equation to get its roots which will be the value of x that will supply you with a value of 3400 when you plug that value into the original equation.
the trick for solving the quadratic equation when y = 3600 is to subtract 3600 from both sides of the equation to place it in standard form and then solve for the roots of the modified equation using the quadratic formula.
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