Answer:
2) $40 per shirt
3) 80/9 m = 8.9 m (nearest tenth)
Step-by-step explanation:
The first question has been answered here: https://brainly.com/question/27846862
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Question 2
Given information:
- Previous sales = 300 shirts at $20 each
- For every $1 increase in price, the number of sales diminishes by 5
Let [tex]x[/tex] = number of $1 increases
Let [tex]y[/tex] = total revenue (in dollars)
From the given information:
- Price change of shirt: [tex](20 + x)[/tex]
- Number of sales change: [tex](300 - 5x)[/tex]
Therefore, we can create a quadratic equation with the given information:
[tex]\implies y = (20+x)(300-5x)[/tex]
As we need to find the maximum amount of revenue, we need to find the vertex of [tex]y[/tex]. As the equation is already factored, the quickest way to do this is to the find the mid-point of the zeros (since a quadratic curve is symmetrical).
[tex]\begin{aligned}y & =0\\\implies (20+x)(300-5x) & =0\\\implies (20+x) & =0 \implies x=-20\\\implies (300-5x) & = 0 \implies x=60\end{aligned}[/tex]
[tex]\textsf{Midpoint}=\dfrac{-20+60}{2}=20[/tex]
Therefore, Sammy should have 20 $1 increases to maximize the revenue, so the new price will be:
[tex]\implies \$20 + 20 \times \$1 = \$40\: \sf per\:shirt[/tex]
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Question 3
Given information:
- Bridge is modeled as a parabola
- Width of bridge = 30 m
- Max height of bridge = 10 m high (in the middle)
As the bridge is modeled as a parabola, and we have been given the width and height, we can create a quadratic equation using the vertex form.
Vertex form: [tex]y=a(x-h)^2+k[/tex]
where:
- x is the horizontal distance of the bridge
- y is the height of the bridge
- (h, k) is the vertex
- a is some constant
Middle of the bridge = 30 m ÷ 2 = 15 m
Max height of the bridge = 10 m
Therefore, the vertex of the parabola is (15, 10)
[tex]\implies y=a(x-15)^2+10[/tex]
We know that when [tex]x = 0, y = 0[/tex]. Therefore, substitute these values into the equation and solve for a:
[tex]\implies 0=a(0-15)^2+10[/tex]
[tex]\implies 0=225a+10[/tex]
[tex]\implies 225a=-10[/tex]
[tex]\implies a=-\dfrac{10}{225}[/tex]
[tex]\implies a=-\dfrac{2}{45}[/tex]
Therefore, the equation of the parabola is:
[tex]\implies y=-\dfrac{2}{45}(x-15)^2+10 \quad \quad \textsf{for }0\leq x\leq 30[/tex]
The horizontal distance at 5 m right of the middle is:
[tex]\implies \dfrac{30}{2}+5=20\:\sf m[/tex]
Therefore, to find the height at this point, input [tex]x=20[/tex] into the equation and solve for y:
[tex]\implies -\dfrac{2}{45}(20-15)^2+10=\dfrac{80}{9}\:\sf m[/tex]
Therefore, the height of the bridge at 5 m to the right of the middle is:
[tex]\dfrac{80}{9}\:=8.9\: \sf m\:(nearest\:tenth)[/tex]
