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Sagot :
Answer:
10 or 40
Step-by-step explanation:
We have the following equation
-4d²+200d-100= 1500
We want all the non zero on one side and so we subtract 1500
-4d²+200d-1600=0
Solve for d
4( -d²+50d-400)=0
factor (I can explain this part in depth if needed)
4(d-40)(d-10)=0
d=40
d=10
This means that the price either needs to be 40 or 10 dollars
The prices at which the theater would earn $1,500 in profit from the comedy show each weekend are $10 and $40 both.
How to evaluate a given mathematical expression with variables if values of the variables are known?
You can simply replace those variables with the value you know of them and then operate on those values to get a final value. This is the result of that expression at those values of the considered variables.
For this case, the profit function is specified as:
[tex]\small p\left(d\right)=-4d^2+200d-100[/tex]
where d denotes the price of each ticket of the theater.
We need such value of d, for which the profit evaluates to $1500
Since output of the expression is denoted by p(d), putting p(d) = 1500 gives us:
[tex]1500=p(d)\\1500=-4d^2+200d-100\\4d^2 - 200d + 1600 = 0\\d^2 -50d + 400 = 0\\d^2 - 40d - 10d + 400 = 0\\d(d-40) -10(d-40) = 0\\(d-10)(d-40) = 0\\\\[/tex]
This comes as 0 either because of (d-10) =0 or (d-40)=0 or both which means d =10 or d = 40 or both.
Checking both values of d and seeing if profit comes out at 1500, we get:
- Case 1: d = 10:
[tex]\small p\left(d\right)=-4d^2+200d-100\\p(10) = -4(10^2) + 200\times 10- 100\\p(10) = -400 + 2000 - 100 = 1500[/tex] (in dollars)
- Case 2: d=40:
[tex]\small p\left(d\right)=-4d^2+200d-100\\p(40) = -4(40)^2 + 200 \times 40 - 100\\p(40) = -6400 + 8000 - 100\\p(40) = 1500[/tex]
Thus, both prices of the ticket (as of $10 for each ticket and of $40 for each ticket) makes the theater earn $1500 as profit. (well since profits are same, keeping price at $10 would be good for general conditions)
Learn more about solving quadratic equations here:
https://brainly.com/question/26675692
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