Marianne has been collecting donations for her biscuit stall at the school summer fayre. There are some luxury gift tins of bisc
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The formula to find the slope of a line is m = 
where the x's and y's are your given coordinates and m is your slope. So, plug in your coordinates and solve.
m = <span>![\frac{y_2 - y_1}{x_2 - x_1} Plug in your coordinates m = [tex] \frac{-2 - 7}{8 - -1} Cancel out the double negative m = [tex] \frac{-2 - 7}{8 + 1} Simplify m = [tex] \frac{-9}{9} Divide m = -1 Now, plug that slope and one set of your given coordinates into point-slope form, [tex]y - y_1 = m(x - x_1)](https://tex.z-dn.net/?f=%20%5Cfrac%7By_2%20-%20y_1%7D%7Bx_2%20-%20x_1%7D%20%20%20Plug%20in%20your%20coordinates%20%3C%2Fspan%3Em%20%3D%20%3Cspan%3E%5Btex%5D%20%5Cfrac%7B-2%20-%207%7D%7B8%20-%20-1%7D%20%20%20Cancel%20out%20the%20double%20negative%20%3C%2Fspan%3Em%20%3D%20%3Cspan%3E%5Btex%5D%20%5Cfrac%7B-2%20-%207%7D%7B8%20%2B%201%7D%20%20%20Simplify%20%3C%2Fspan%3Em%20%3D%20%3Cspan%3E%5Btex%5D%20%5Cfrac%7B-9%7D%7B9%7D%20%20%20Divide%20m%20%3D%20-1%20%20%3C%2Fspan%3E%20Now%2C%20plug%20that%20slope%20and%20one%20set%20of%20your%20given%20coordinates%20into%20point-slope%20form%2C%20%5Btex%5Dy%20-%20y_1%20%3D%20m%28x%20-%20x_1%29)
. I'll use (-1, 7).
<span>
Plug in your points and slope
</span>y - 7 = -1(x - -1) Cancel out the double negative
y - 7 = -1(x + 1) Use the Distributive Property
y - 7 = -x - 1 Add 7 to both sides
y = -x + 6 </span>
m = <span>
<span>
</span>y - 7 = -1(x - -1) Cancel out the double negative
y - 7 = -1(x + 1) Use the Distributive Property
y - 7 = -x - 1 Add 7 to both sides
y = -x + 6 </span>
When the surrounding flaps are folded up, the base of the box will have dimensions by
, and the box will have a height of
. So the box has volume, as a function of
,
I don't know what technology is available to you, but we can determine an exact value for that maximizes the volume by using calculus.
Differentiating with respect to
gives
and setting this equal to 0 gives two critical points at
For the larger critical point we would get a negative volume, so we ignore that one. Then the largest volume would be about 168.5 cubic in.