Module 4 Lesson 6 Formative Assessment Q
1. Find the point on the curve $$f(x)=-x^2$$ that is nearest to the point $$A(3,0)$$.
Write an equation and take a derivative.
(2,-4)
*(1,-1)
(0,0)
(3,-9)
2. A window has a lower section in the shape of a rectangle and an upper portion in the shape of a
semicircle surmounted on the upper side of this rectangle. The perimeter of the window is 15 units. Find
the ratio of the radius of the circle to the height of the rectangle so that the area of the window is a
maximum.
Write an equation and take a derivative.
$$\frac{\pi}{2}$$
$$\frac{\pi}{1}$$
$$\frac{1}{2}$$
*$$\frac{1}{1}$$
3. Triangle ABC is inscribed in a semicircle with diameter BC and radius OC measures 10 cm. Find the
value of angle B that makes the maximum area of triangle ABC.
Write an equation and take a derivative.
$$\frac{\pi}{3}$$
*$$\frac{\pi}{4}$$
$$\frac{\pi}{6}$$
$$\frac{\pi}{8}$$
4. A long rectangular sheet of metal, one foot wide, is to be made unto a gutter by turning up two sides
at right angles to the sheet. How many inches should be turned up to give the gutter its maximum
capacity?
Write an equation and take a derivative.
1.5 in
2 in
2.25 in
*3 in
5. A rectangle is inscribed in the ellipse $$\frac{x^2}{20}+\frac{y^2}{12}=1$$ . What are the dimensions
of the rectangle with maximum area?
Write an equation and take a derivative.
6.153 x 5.201
*6.325 x 4.899
6.225 x 4.899
6.253 x 4.899
Module 4 Lesson 6 Summative Assessment Q
1. Find a point on the curve $$x^2-9y=0$$ that is closest to the point $$P(5,-2)$$ .
(-3,1)
*(3,1)
(2,5)
(-1,4)
2. A window has the shape of a rectangle surmounted by a semicircle. The total perimeter of the
window is 14 feet and the window has a maximum area. Which of the following gives the approximate
ratio of the radius of the semicircle to the height of the rectangle?
1:1.333
1:1.9603
*1:1
2.213:1
3. An isosceles triangle ABC with apex $$\angle{B}=2\theta$$ is inscribed in a circle of radius 10 cm. Find
the angle $$\theta$$ for which the triangle ABC will have maximum area.
$$\frac{\pi}{12}$$
*$$\frac{\pi}{6}$$
$$\frac{\pi}{4}$$
$$\frac{\pi}{3}$$
4. An open box can be made from a square piece of material by cutting equal squares from each corner
and turning up the sides. Find the dimensions of the box of maximum volume if the material has
dimension 6 cm by 6 cm.
*4 cm x 4 cm x 1 cm
3 cm x 3 cm x 3 cm
4.5 cm x 4.5 cm x 0.75 cm
5 cm x 5 cm x 0.5 cm
5. A rectangle is inscribed in the ellipse $$\frac{x^2}{20}+\frac{y^2}{12}=1$$ . What is the maximum
perimeter of the rectangle?
*22.627
22.467
21.590
20.627
Activity
Choose two problems from those below. Justify your solutions in
paragraph form. Be sure to cite any properties or rules used.
1. What is the maximum area of a rectangle that can be inscribed in
the curve $$\frac{x^2}{9}+\frac{y^2}{4}=1$$ ?
2. Find the ratio of length to width of a rectangle with the greatest
area that can be inscribed in a circle of radius r.
3. Given an equilateral triangle with all sides equal to 12 units, find
the area of the largest rectangle inscribed in the triangle.
4. Find the volume of the right circular cone with maximum volume
that can be inscribed in a sphere of radius 3 units.
5. A right circular cylinder is inscribed in a cone with height 12
inches and a radius of the base of 4 inches. Find the maximum
volume of the cylinder.
You are to read and respond to at least two other student's posts
discussing their justification or providing an alternate solution.
Click Add a new discussion topic to submit your post.