randRange(2,7)
randRange(2,7)
randRange(2,7)
person(1) needs to ship a package of flour to a baker.
He has LENGTH * WIDTH * HEIGHT cubic meters of flour to send.
She has LENGTH*WIDTH*HEIGHT cubic meters of flour to send.
The post office will not ship any box which has an edge 10 meters or longer.
What are the dimensions of a box which will hold exactly LENGTH*WIDTH*HEIGHT cubic meters of flour, but whose edge lengths are all less than 10 meters?
The volume of a box is the product of the length, width and height of the box.
So we need to find 3 numbers whose product is LENGTH*WIDTH*HEIGHT, each of which being less than 10. Just play around to try to find these!
For instance, the numbers LENGTH, WIDTH, and HEIGHT work, but there might be other solutions as well!
randRange(2,12)
randRange(2,12)
randRange(2,12)
randRange(1,L1-1)
randRange(1,W1-1)
randRange(1,H1-1)
max(max(L1,W1),H1)
"#78D0EF"
[
[L1/MAX,W1/MAX,-H1/MAX],
[-L1/MAX,W1/MAX,-H1/MAX],
[-L1/MAX,-W1/MAX,-H1/MAX],
[L1/MAX,-W1/MAX,-H1/MAX],
[L1/MAX,W1/MAX,H1/MAX],
[-L1/MAX,W1/MAX,H1/MAX],
[-L1/MAX,-W1/MAX,H1/MAX],
[L1/MAX,-W1/MAX,H1/MAX],
[(-L1+2*L2)/MAX,-W1/MAX,-H1/MAX],
[(2*L2-L1)/MAX,(2*W2-W1)/MAX,-H1/MAX],
[-L1/MAX,(2*W2-W1)/MAX,-H1/MAX],
[-L1/MAX,-W1/MAX,(2*H2-H1)/MAX],
[(-L1+2*L2)/MAX,-W1/MAX,(2*H2-H1)/MAX],
[(2*L2-L1)/MAX,(2*W2-W1)/MAX,(2*H2-H1)/MAX],
[-L1/MAX,(2*W2-W1)/MAX,(2*H2-H1)/MAX]
]
4.0
[
{
verts: [0, 3, 2, 1],
color: COLOR
},
{
verts: [5, 6, 7, 4],
color: COLOR
},
{
verts: [4,7,3,0],
color: COLOR
},
{
verts: [5,1,2,6],
color: COLOR
},
{
verts: [5, 4, 0, 1],
color: COLOR
},
{
verts: [6, 2, 3, 7],
color: COLOR
}
]
A water tank is H1 feet high, L1 feet long, and W1 feet wide.
A solid metal box which is H2 feet high, L2 feet long, and W2 feet wide
is sitting at the bottom of the tank. The tank is filled with water.
What is the volume of the water in the tank?
L1*W1*H1 - L2*W2*H2 \text{ft}^3
init({range: [[-2, 2], [-2, 2]], scale: [100, 100]});
addMouseLayer();
// graphing the initial object
graph.obj = make3dObject(VERTICES, { scale: SCALE });
_.each(FACES, function(face) {
graph.obj.addFace(face);
});
graph.obj.addSketch(
{verts: [8,9,10,2]}
);
graph.obj.addSketch(
{verts: [11,12,13,14,11]}
);
graph.obj.addSketch(
{verts: [2,11]}
);
graph.obj.addSketch(
{verts: [8,12]}
);
graph.obj.addSketch(
{verts: [9,13]}
);
graph.obj.addSketch(
{verts: [10,14]}
);
graph.obj.setPos([0, 0, 5]);
graph.obj.rotate(1,0 , 0, PI/3);
graph.obj.rotate(0, 1, 0, 0);
graph.obj.rotate(0, 0, 1, PI/8);
graph.obj.doDraw();
GRAPH = graph;
var spinner;
setTimeout(function() {
spinner = setInterval(function() {
GRAPH.obj.rotate(0, 0, 1, 0);
GRAPH.obj.doDraw();
}, 50);
}, 300);
var mouseTarget = mouselayer.rect(0, 0, 400, 400).attr({
fill: "#000", opacity: 0.0
});
$(mouseTarget[0]).bind("vmousedown", function(event) {
event.preventDefault();
GRAPH.lastX = event.pageX;
GRAPH.lastY = event.pageY;
clearInterval(spinner);
$(document).bind("vmousemove vmouseup", function(event) {
event.preventDefault();
var dx = (event.pageX - GRAPH.lastX) / 200 * PI;
var dy = (event.pageY - GRAPH.lastY) / 200 * PI;
GRAPH.lastX = event.pageX;
GRAPH.lastY = event.pageY;
var xAxis = normalize(GRAPH.obj.perspective[1]);
var yAxis = normalize(GRAPH.obj.perspective[0]);
GRAPH.obj.rotate(xAxis[0], xAxis[1], xAxis[2], -dx / 2);
GRAPH.obj.rotate(yAxis[0], yAxis[1], yAxis[2], -dy / 2);
GRAPH.obj.doDraw();
if (event.type === "vmouseup") {
$(document).unbind("vmousemove vmouseup");
}
})
});
The tank has a volume of H1 \times L1 \times W1 = H1*L1*W1 cubic feet,
and the metal box has a volume of H2 \times L2 \times W2 = H2*W2*L2 cubic feet.
Since there is no water in the box, the volume of the water in the tank is the volume of the tank minus the volume of the metal box.
The volume of the water in the tank is H1*L1*W1 \text{ft}^3 - H2*L2*W2 \text{ft}^3 = H1*L1*W1 - H2*L2*W2\text{ft}^3
randFromArray([
[
new Plural(function(num) {
return $.ngettext("foot", "feet", num);
}),
new Plural(function(num) {
return $.ngettext("inch", "inches", num);
}), $._("ft"), $._("in"), 12
],
[
new Plural(function(num) {
return $.ngettext("yard", "yards", num);
}),
new Plural(function(num) {
return $.ngettext("foot", "feet", num);
}), $._("yd"), $._("ft"), 3
],
[
new Plural(function(num) {
return $.ngettext("meter", "meters", num);
}), new Plural(function(num) {
return $.ngettext("decimeter", "decimeters", num);
}), $._("m"), $._("dm"), 10
],
[
new Plural(function(num) {
return $.ngettext("centimeter", "centimeters", num);
}), new Plural(function(num) {
return $.ngettext("millimeter", "millimeters", num);
}), $._("cm"), $._("mm"), 10
]
])
randRangeExclude(2, 11,[3,10])
CONVERSION
CONVERSION
CONVERSION
CONVERSION
LENGTH/MAX
WIDTH/MAX
HEIGHT/MAX
"#D7ED3A"
"#38C77F"
"#78D0EF"
"#F0B63A"
HEIGHT * 2
[
[LENGTH/MAX, WIDTH/MAX, HEIGHT/MAX],
[LENGTH/MAX, WIDTH/MAX, -HEIGHT/MAX],
[LENGTH/MAX, -WIDTH/MAX, -HEIGHT/MAX],
[LENGTH/MAX, -WIDTH/MAX, HEIGHT/MAX],
[-LENGTH/MAX, WIDTH/MAX, HEIGHT/MAX],
[-LENGTH/MAX, WIDTH/MAX, -HEIGHT/MAX],
[-LENGTH/MAX, -WIDTH/MAX, -HEIGHT/MAX],
[-LENGTH/MAX, -WIDTH/MAX, HEIGHT/MAX]
]
4.0
[
{
verts: [3, 2, 1, 0],
color: COLOR1,
labels: [[[XBORD, 0, ZBORD], "1 \\text{" + FROM_TEXT + "} = " + CONVERSION + " \\text{" + TO_TEXT + "}"]]
}, {
verts: [4, 5, 6, 7],
color: COLOR1,
labels: [[[XBORD, 0, ZBORD], "1 \\text{" + FROM_TEXT + "} = " + CONVERSION + " \\text{" + TO_TEXT + "}"]]
}, {
verts: [0, 1, 5, 4],
color: COLOR3
}, {
verts: [2, 3, 7, 6],
color: COLOR3
}, {
verts: [1, 2, 6, 5],
color: COLOR2
}, {
verts: [3, 0, 4, 7],
color: COLOR2
}
]
{}
{}
$._("base")
$._("height")
How many cubic plural_form(TO) fit in VOL cubic plural_form(FROM, VOL)?
VOL * CONVERSION * CONVERSION * CONVERSION
First we will figure out how many cubic plural_form(TO) fit in one cubic FROM.
There are CONVERSION plural_form(TO, CONVERSION) in every FROM.
So a cubic FROM is the same as a
CONVERSION \text{ TO_TEXT}
\times CONVERSION \text{ TO_TEXT}
\timesCONVERSION \text{ TO_TEXT}
cube.
init({range: [[-2, 2], [-2, 2]], scale: [100, 100]});
addMouseLayer();
//Adding verticies for the slabs
for (var i = 1; i < HEIGHT;i++ ){
VERTICES.push( [LENGTH/MAX,WIDTH/MAX,(-HEIGHT+2*i)/MAX]);
VERTICES.push([-LENGTH/MAX,WIDTH/MAX,(-HEIGHT+2*i)/MAX]);
VERTICES.push([-LENGTH/MAX,-WIDTH/MAX,(-HEIGHT+2*i)/MAX]);
VERTICES.push([LENGTH/MAX,-WIDTH/MAX,(-HEIGHT+2*i)/MAX]);
};
//Adding verticies for the cubes in one slab
for (var i = 1; i < LENGTH; i++ ){
VERTICES.push( [(-LENGTH+2*i)/MAX,-WIDTH/MAX,(-HEIGHT)/MAX]);
};
for (var i = 1; i < LENGTH; i++ ){
VERTICES.push( [(-LENGTH+2*i)/MAX,WIDTH/MAX,(-HEIGHT)/MAX]);
};
for (var i = 1; i < WIDTH; i++ ){
VERTICES.push( [(-LENGTH)/MAX,(-WIDTH+2*i)/MAX,(-HEIGHT)/MAX]);
};
for (var i = 1; i < WIDTH; i++ ){
VERTICES.push( [(LENGTH)/MAX,(-WIDTH+2*i)/MAX,(-HEIGHT)/MAX]);
};
for (var i = 1; i < LENGTH; i++ ){
VERTICES.push( [(-LENGTH+2*i)/MAX,-WIDTH/MAX,(-HEIGHT+2)/MAX]);
};
for (var i = 1; i < LENGTH; i++ ){
VERTICES.push( [(-LENGTH+2*i)/MAX,WIDTH/MAX,(-HEIGHT+2)/MAX]);
};
for (var i = 1; i < WIDTH; i++ ){
VERTICES.push( [(-LENGTH)/MAX,(-WIDTH+2*i)/MAX,(-HEIGHT+2)/MAX]);
};
for (var i = 1; i < WIDTH; i++ ){
VERTICES.push( [(LENGTH)/MAX,(-WIDTH+2*i)/MAX,(-HEIGHT+2)/MAX]);
};
// graphing the initial object
graph.obj = make3dObject(VERTICES, { scale: SCALE });
_.each(FACES, function(face) {
graph.obj.addFace(face);
});
graph.obj.setPos([0, 0, 5]);
graph.obj.rotate(1,0 , 0, PI/3);
graph.obj.rotate(0, 1, 0, 0);
graph.obj.rotate(0, 0, 1, PI/8);
graph.obj.doDraw();
GRAPH = graph;
var spinner;
setTimeout(function() {
spinner = setInterval(function() {
GRAPH.obj.rotate(0, 0, 1, 0);
GRAPH.obj.doDraw();
}, 50);
}, 300);
var mouseTarget = mouselayer.rect(0, 0, 400, 400).attr({
fill: "#000", opacity: 0.0
});
$(mouseTarget[0]).bind("vmousedown", function(event) {
GRAPH.lastX = event.pageX;
GRAPH.lastY = event.pageY;
clearInterval(spinner);
$(document).bind("vmousemove vmouseup", function(event) {
event.preventDefault();
var dx = (event.pageX - GRAPH.lastX) / 200 * PI;
var dy = (event.pageY - GRAPH.lastY) / 200 * PI;
GRAPH.lastX = event.pageX;
GRAPH.lastY = event.pageY;
var xAxis = normalize(GRAPH.obj.perspective[1]);
var yAxis = normalize(GRAPH.obj.perspective[0]);
GRAPH.obj.rotate(xAxis[0], xAxis[1], xAxis[2], -dx / 2);
GRAPH.obj.rotate(yAxis[0], yAxis[1], yAxis[2], -dy / 2);
GRAPH.obj.doDraw();
if (event.type === "vmouseup") {
$(document).unbind("vmousemove vmouseup");
}
})
});
There are
CONVERSION \times CONVERSION
\times CONVERSION =
CONVERSION * CONVERSION * CONVERSION
\text{ TO_TEXT}^3
in each cubic FROM, by the volume formula.
Since we have VOL cubic plural_form(FROM, VOL), and each cubic FROM holds
CONVERSION * CONVERSION *
CONVERSION cubic
plural_form(TO, CONVERSION * CONVERSION *
CONVERSION), we have a total of
VOL \times CONVERSION * CONVERSION
* CONVERSION
cubic plural_form(TO, VOL * CONVERSION * CONVERSION * CONVERSION).
So VOL * CONVERSION * CONVERSION * CONVERSION cubic plural_form(TO, VOL * CONVERSION * CONVERSION * CONVERSION) fit in VOL cubic plural_form(FROM, VOL).